A rectangle has four sides and angles at 90 degree.
What is a rectangle?A plane figure with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square. A rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.
Given: shape of a rectangle
A rectangle is closed flat shape, having four sides, and each angle equal to 90 degrees. The opposite sides of the rectangle are equal and parallel.
Hence, A rectangle is a closed 2-D shape, having 4 sides, 4 corners, and 4 right angles (90°).
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another term for positive correlation is ______. group of answer choices direct correlation indirect correlation nondirectional correlation unidirectional correlation
Another term for positive correlation is "direct correlation." In a direct correlation, as one variable increases, the other variable also tends to increase.
This implies a positive linear relationship between the variables. For example, if we observe that as the number of hours spent studying increases, the test scores also increase, we can say that there is a direct correlation between study hours and test scores.
It indicates that there is a consistent and predictable relationship between the variables, with both moving in the same direction. The terms "indirect correlation," "nondirectional correlation," and "unidirectional correlation" do not accurately describe a positive correlation.
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66. y= 3, 7, 5, 11, 14, sst= _________ question 49 options: a) 8. b) 80. c) 13.2. d) 12.4.
The sum of squares total y= 3, 7, 5, 11, 14, sst = 80. The answer to the question is b) 80.
To calculate the sum of squares total (SST), we need to find the total variability of the data from the mean.
First, we need to find the mean of the data:
mean = (3 + 7 + 5 + 11 + 14) / 5 = 8
Next, we calculate the sum of the squared differences between each data point and the mean:
(3 - 8)^2 + (7 - 8)^2 + (5 - 8)^2 + (11 - 8)^2 + (14 - 8)^2 = 2 + 1 + 9 + 9 + 36 = 57
Therefore, the sum of squares total (SST) is 57.
So the answer is not one of the options given in the question.
mean = (sum of all numbers) / (number of numbers)
So, in this case:
mean = (3 + 7 + 5 + 11 + 14) / 5 = 8
Next, we need to calculate the sum of squares total using the formula:
sst = Σ(y - mean)
where Σ represents the sum of all values in the set.
Substituting in the values from the set, we get:
sst = (3 - 8)2 + (7 - 8)2 + (5 - 8)2 + (11 - 8)2 + (14 - 8)2
sst = [tex](-5)^2 + (-1)^2 + (-3)^2 + 3^2 + 6^2[/tex]
sst = 25 + 1 + 9 + 9 + 36
sst = 80
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Vera makes a shipping container from cardboard the container is shaped like a triangular prism each base is a triangle with a height of 3 inches in a base of 8 inches she uses a total of 956 in.² to make the container what is the containers length (HURRYY PLEASE)
The calculated length of the container is 70.11 inches
How to calculate the length of the containerFrom the question, we have the following parameters that can be used in our computation:
Shape = triangular prism
Height = 3 inches
Base = 8 inches
Surface area = 956 square inches
The slant lengths of the triangular sides are calculated using
a² = (8/2)² - 3²
a = √7
The surface area of a triangular prism is calculated as
SA = bh + (a + a + c) * l
So, we have
3 * 8 + (8 + √7 + √7) * l = 956
So, we have
(8 + √7 + √7) * l = 932
Divide
l = 70.11
Hence, the length of the container is 70.11 inches
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Answer:
Step-by-step explanation:
To find the length of the container, we need to determine the area of the two triangular bases.
The formula for the area of a triangle is: Area = (base * height) / 2.
Let's calculate the area of one triangular base:
Base = 8 inches
Height = 3 inches
Area of one triangular base = (8 * 3) / 2 = 12 square inches.
Since there are two triangular bases, the total area of the bases is 2 * 12 = 24 square inches.
We are given that the total area of the container is 956 square inches.
Total area of the container = 2 * Area of one triangular base + Lateral surface area
Lateral surface area = Total area of the container - 2 * Area of one triangular base
Lateral surface area = 956 - 24 = 932 square inches.
The lateral surface area of a triangular prism is given by the formula: Lateral surface area = perimeter of the base * height.
The perimeter of a triangular base is the sum of the lengths of its sides. Since it is an isosceles triangle with a base of 8 inches, the two equal sides will have a length of 8 inches as well.
Perimeter of the base = 8 + 8 + 8 = 24 inches.
Now, we can find the length of the container by rearranging the formula for the lateral surface area:
Lateral surface area = perimeter of the base * height
932 = 24 * length
length = 932 / 24
length ≈ 38.83 inches (rounded to two decimal places)
Therefore, the length of the container is approximately 38.83 inches.
Find the domain of the vector-valued function. (Enter your answer using interval notation.) r(t) = √(16 – t^2i) + t^2j − 6tk
The domain of the vector-valued function is:
[-4, 4]
The domain of the vector-valued function r(t), we need to determine the values of t that make the function well-defined.
The first component of the vector function is given by:
√(16 – t²i)
The square root is only defined for non-negative values.
Thus, we must have:
16 – t²i ≥ 0
Solving for t, we get:
-4 ≤ t ≤ 4
Next, there are no restrictions on the second component of the vector function, so it is defined for all values of t.
Finally, the third component of the vector function is defined for all values of t.
We must identify the values of t that give the vector-valued function r(t) a well-defined domain.
Keep in mind that the vector function's initial component is supplied by: (16 - t2i).
Only positive numbers can be used to define the square root.
Therefore, we require:
16 – t²i ≥ 0
When we solve for t, we obtain: -4 t 4.
The second component of the vector function is unrestricted and is defined for all values of t.
The vector function's third component is thus specified for all values of t.
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For a vector-valued function r(t) to have a domain, all its component functions must be defined.
In this case, the first component function is √(16 – t^2), which is defined only for values of t such that 16 - t^2 is nonnegative, since the square root of a negative number is undefined in the real numbers. Therefore, we must have:
16 - t^2 ≥ 0
Solving for t, we get:
-4 ≤ t ≤ 4
This gives the domain of the first component function as the closed interval [-4, 4].
The second and third component functions, t^2 and -6t, are defined for all real numbers.
Therefore, the domain of the vector-valued function r(t) is the same as the domain of its first component function, which is:
[-4, 4]
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Use a parameterization to find the flux doubleintegral_S F middot n do of F = 5xy i - 2z k outward (normal away from the z-axis) through the cone z = 6 squareroot x^2 +y^2 0 lessthanorequalto z lessthanorequalto 6. The flux is (Type an exact answer, using pi as needed.)
The flux of the vector field F through the cone is zero.
To find the flux of the vector field F = 5xy i - 2z k outward through the cone z = 6 square root x^2 +y^2 with 0 ≤ z ≤ 6, we need to first parameterize the cone. Let x = r cos θ and y = r sin θ, where r ≥ 0 and 0 ≤ θ ≤ 2π, then we have z = 6r for the cone.
Now we can compute the unit normal vector n as n = (zr/6) cos θ i + (zr/6) sin θ j + (z/6) k, and then calculate the dot product F · n as F · n = 5xy (zr/6) - 2z (z/6) = (5/6)zr^2 cos θ sin θ - z^2/3.
The double integral of F · n over the cone is then given by:
doubleintegral_S F · n dS = doubleintegral_R (5/6)zr^2 cos θ sin θ - z^2/3 r dr dθ
where R is the region in the xy-plane that corresponds to the base of the cone.
Integrating with respect to r first, from 0 to 6, we get:
doubleintegral_S F · n dS = integral_0^(2π) integral_0^6 (5/18)z^3 cos θ sin θ - (1/9)z^3 r dr dθ
Evaluating the integral with respect to r and then θ, we obtain:
doubleintegral_S F · n dS = 0
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9. find a particular solution for y 00 4y 0 3y = 1 1 e t using transfer functions, impulse response, and convolutions. (other methods are not accepted)
the point P_0(2,1,2) lies on the tangent plane, we can use it to find the equation of the normal line:
x - 2 = 2
We start by finding the characteristic equation:
r^2 + 4r + 3 = 0
Solving for r, we get:
r = -1 or r = -3
So the complementary solution is:
y_c(t) = c_1 e^{-t} + c_2 e^{-3t}
Next, we need to find the transfer function H(s):
s^2 Y(s) - s y(0) - y'(0) + 4s Y(s) - 4y(0) + 3Y(s) = 1/s + 1/(s-1)
Applying the initial conditions y(0) = 0 and y'(0) = 1, we get:
(s^2 + 4s + 3) Y(s) = 1/s + 1/(s-1) + 4
Y(s) = [1/(s+1) + 1/(s+3) + 4/(s^2 + 4s + 3)] / (s^2 + 4s + 3)
We can factor the denominator of the second term in the numerator:
Y(s) = [1/(s+1) + 1/(s+3) + 4/((s+1)(s+3))] / [(s+1)(s+3)]
Using partial fraction decomposition, we get:
Y(s) = [2/(s+1) - 1/(s+3) + 1/((s+1)(s+3))] / (s+1) + [-1/(s+1) + 2/(s+3) - 1/((s+1)(s+3))] / (s+3)
Taking the inverse Laplace transform, we get:
y(t) = 2e^{-t} - e^{-3t} + (1/2)(1 - e^{-t}) - (1/2)(1 - e^{-3t})
So the general solution is:
y(t) = y_c(t) + y_p(t) = c_1 e^{-t} + c_2 e^{-3t} + 2e^{-t} - e^{-3t} + (1/2)(1 - e^{-t}) - (1/2)(1 - e^{-3t})
To find a particular solution, we need to solve for the unknown coefficients. Using the initial conditions y(0) = 1 and y'(0) = 0, we get:
c_1 + c_2 + 3/2 = 1
-c_1 - 3c_2 - 1/2 = 0
Solving this system of equations, we get:
c_1 = -2/5
c_2 = 9/10
So the particular solution is:
y_p(t) = (-2/5) e^{-t} + (9/10) e^{-3t} + (1/2)(1 - e^{-t}) - (1/2)(1 - e^{-3t})
Finally, the tangent plane at P_0(2,1,2) is given by the equation:
2x + 4y + 3z = 24
which corresponds to option (B) in the given choices.
To find the normal line, we first need to find the normal vector to the tangent plane, which is simply:
n = <2, 4, 3>
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A single bus fare costs $2. 35. A monthly pass costs $45. 75. Alia estimates that she will ride the bus 25 times this month. Matthew estimates that he will ride the bus 18 times. Should they both buy monthly passes?
Answer: They both buy monthly passes.
Step-by-step explanation: Let's first calculate how much Alia and Matthew would pay if they both bought individual bus fares for the number of times they plan to ride the bus:
Alia: 25 rides x $2.35 per ride = $58.75
Matthew: 18 rides x $2.35 per ride = $42.30
Now let's see how much they would pay if they both bought monthly passes:
Alia: $45.75
Matthew: $45.75
Since the cost of buying individual bus fares is more than the cost of buying monthly passes, it would be more economical for both Alia and Matthew to buy monthly passes.
Therefore, yes, they both should buy monthly passes.
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Let F(x, y) = (9x + 5y)i + (2x – 7y?)j. Let D be the rectangle {(x, y)|0 < x < 2,0 Sy < 1} and let C be the boundary of D, oriented counterclockwise. (a) (4 points) Use Green's Theorem to compute the circulation f F. dr. Your solution should involve a double integral. (b) (2 points) Is F equal to V f for some function f? Use your work from part (a) to justify your answer. (c) (4 points) Use Green's Theorem to compute the flux f(F. n)ds where n denotes the outward-pointing unit normal vector. Your solution should involve a double integral. (d) (2 points) Is F equal to V x G for some vector field G? Use your work from part (C) to justify your answer.
The circulation of F around C is -16. No, F is not equal to V f for some function f. The flux of F across C is -8. No, F is not equal to V x G for any vector field G.
(a) The circulation of F around C is -16.
Using Green's Theorem, we can write the circulation of F as the line integral around the boundary of D:
∮CF · dr = ∬D (∂Q/∂x - ∂P/∂y) dA
where P = 9x + 5y, Q = 2x - 7y, and dr = dx i + dy j.
Taking the partial derivatives, we get:
∂Q/∂x - ∂P/∂y = 2 - 9 = -7.
Thus, the circulation of F around C is:
∮CF · dr = ∬D -7 dA = -7(area of D) = -16.
(b) No, F is not equal to V f for some function f.
If F were equal to the gradient of some scalar function f, then the circulation of F around any closed path would be zero. However, we just calculated that the circulation of F around C is -16, which means F cannot be expressed as the gradient of any scalar function.
(c) The flux of F across C is -8.
Using Green's Theorem, we can write the flux of F across C as the line integral around the boundary of D:
∮CF · ds = ∬D (∂P/∂x + ∂Q/∂y) dA
Taking the partial derivatives, we get:
∂P/∂x + ∂Q/∂y = 9 - 7 = 2.
Thus, the flux of F across C is:
∮CF · ds = ∬D 2 dA = 2(area of D) = -8.
(d) No, F is not equal to V x G for any vector field G.
If F were equal to the curl of some vector field G, then the flux of F across any closed surface would be zero. However, we just calculated that the flux of F across C is -8, which means F cannot be expressed as the curl of any vector field.
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helppp
Amy is shopping for a new couch. She
finds one that she likes for $800, but
her budget is $640. How much of a
discount does she need in order to be
able to afford the couch?
Answer:
She would need a 20% discount.
Step-by-step explanation:
800x = 640 Divide both sides by 800
x = .8
640 is 80% of 800
100% - 80% = 20%
Check
800(.2) = 160 This is the discount needed.
800 - 160 = 640
Answer:
20%
Step-by-step explanation:
I'm sure there's some actual calculation to find this answer, but we'll figure it out with trial and error:
First, 50% off of $800 is 0.5 * 800 = 400, and 800 - 400 = $400 price.
We see that we need a smaller discount as a minimum to afford, so let's try:
30% off: 0.3 * 800 = 240, and 800 - 240 = $560 as new price.
20% off: 0.2 * 800 = 160, and 800 - 160 = $640 as new price, which is the exact number of Amy's budget (and a lucky guess)!
So, if there is a 20% discount, the new price will be $640, which is the exact same as Amy's budget.
If I helped, please consider making this answer brainliest ;)
**EDIT**
The answer above this is what you should absolutely make brainliest. They used the calculation I mentioned, but I was too lazy to search up
The cartesian product of two sets is a set of pairs combining all elements from the first set with each of the elements in the second set. T/F
True. The cartesian product of two sets is a set of pairs combining all elements from the first set with each of the elements in the second set.
The cartesian product of two sets A and B, denoted by A × B, is the set of all possible ordered pairs where the first element comes from set A and the second element comes from set B. In other words, each element in set A is combined with every element in set B to form a pair.
For example, let A = {1, 2} and B = {3, 4}. The cartesian product A × B would be {(1, 3), (1, 4), (2, 3), (2, 4)}, which includes all possible combinations of elements from A and B.
The cartesian product is a fundamental concept in set theory and plays a crucial role in various areas of mathematics, including algebra, combinatorics, and geometry. It allows for the systematic exploration of all possible combinations between sets and is often used in defining relations, functions, and mappings between different mathematical structures.
Therefore, it is true that the cartesian product of two sets is a set of pairs combining all elements from the first set with each of the elements in the second set.
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The polynomial -2 x^2 + 500x represents the budget surplus of the town of Alphaville for the year 2010. Alphaville’s surplus in 2011 can be modeled by -1. 5 x^2 + 400x. If x represents the yearly tax revenue in thousands, by how much did Alphaville’s budget surplus increase from 2010 to 2011? If Alphaville took in $750,000 in tax revenue in 2011, what was the budget surplus that year?
Alphaville's budget surplus increased by $25,000 from 2010 to 2011. In 2011, with a tax revenue of $750,000, the budget surplus was $75,000.
To find the increase in Alphaville's budget surplus from 2010 to 2011, we need to calculate the difference between the two surplus functions: (-1.5x^2 + 400x) - (-2x^2 + 500x). Simplifying the expression, we get -1.5x^2 + 400x + 2x^2 - 500x = 0.5x^2 - 100x.
Next, we substitute the tax revenue of $750,000 into the equation to find the budget surplus for 2011. Plugging in x = 750, we get 0.5(750)^2 - 100(750) = 281,250 - 75,000 = $206,250.
Therefore, Alphaville's budget surplus increased by $25,000 ($206,250 - $181,250) from 2010 to 2011. In 2011, with a tax revenue of $750,000, the budget surplus was $206,250.
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Use a power series to approximate the definite integral to six decimal places. ∫1/20arctan(x/2)dx
The definite integral is approximately 0.121548.
We can use the power series expansion of arctan(x) to approximate the given integral.
Recall that the power series expansion of arctan(x) is:
arctan(x) = x - (1/3)x³ + (1/5)x⁵ - (1/7)x⁷ + ...
We can substitute x/2 into the power series to get:
arctan(x/2) = (x/2) - (1/3)(x/2)³ + (1/5)(x/2)⁵ - (1/7)(x/2)⁷ + ...
Now we can integrate term by term to get:
∫[0,1/2] arctan(x/2)dx
= [(1/2)x² - (1/18)x⁴ + (1/50)x⁶ - (1/98)x⁸ + ...] evaluated from 0 to 1/2
= (1/2)(1/2)² - (1/18)(1/2)⁴ + (1/50)(1/2)⁶ - (1/98)(1/2)⁸ + ...
= 0.122078...
To approximate the integral to six decimal places, we need to sum up enough terms in the power series to ensure that the absolute value of the next term is less than or equal to 0.000001.
We can use a calculator or a computer program to find that the ninth term of the power series is -0.000002378. Therefore, the sum of the first eight terms gives an approximation of the integral to six decimal places:
0.122078 - 0.000523 - 0.000007 + 0.000000 + ...
≈ 0.121548
Therefore, the definite integral is approximately 0.121548 to six decimal places.
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The annual numbers of industrial accidents in a motor plant for the past 9 years are 300, 250, 110, 435, 693, 250, 375, 420 & 460
Find the 3rd
The third highest number of industrial accidents in the motor plant over the past 9 years is 375.
In summary, the third highest number of industrial accidents in the motor plant over the past 9 years is 375.
To find the third highest number of industrial accidents, we need to sort the given numbers in descending order and identify the third value.
The given numbers are: 300, 250, 110, 435, 693, 250, 375, 420, and 460.
Arranging these numbers in descending order: 693, 460, 435, 420, 375, 300, 250, 250, 110.
The third highest number is 435, but we are looking for the third number in the original order. Since 435 is the second highest in the original order, we continue down the list.
The next highest number is 420, which is the third highest in the original order. However, we are still looking for the fourth highest number.
The third highest number in the original order is 375. This is the number we are looking for.
Therefore, the third highest number of industrial accidents in the motor plant over the past 9 years is 375.
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let d:c[infinity](r)→c[infinity](r)d:c[infinity](r)→c[infinity](r) and d2:c[infinity](r)→c[infinity](r)d2:c[infinity](r)→c[infinity](r) be the linear transformations defined by the first derivative
The linear transformations d and d2 are defined by taking the first derivative of a function in the space of smooth functions c[infinity](r). In other words, given a function f in c[infinity](r), d(f) is the function that represents the rate of change of f at each point in r, while d2(f) represents the rate of change of d(f).
To understand this concept better, consider an example of a function f(x) = x² in the interval r = [0, 1]. The derivative of f is f'(x) = 2x, which represents the slope of the tangent line to the curve of f at each point x in the interval. Thus, d(f)(x) = 2x. Similarly, the second derivative of f is f''(x) = 2, which represents the curvature of the curve of f at each point x in the interval. Thus, d2(f)(x) = 2.
These linear transformations are important in the study of differential equations and calculus. They allow us to represent the behavior of functions in terms of their rates of change, and to derive new functions from existing ones based on these rates of change. Additionally, these transformations have applications in physics, engineering, and other areas of science where the study of rates of change is essential.
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What is the value of this expression?
4 5/8+ 5/6 - 1 3/4
Enter your answer as a mixed number in simplest form by filling in the boxes.
A recipe uses 40g of chocolate chips and 150g of flour
What is the ratio of chocolate chips to flour in its simplest form
Answer: 4:15
Step-by-step explanation:
divide both by 10
Answer:4:15
Step-by-step explanation:Im just built different tbh bro
from a sample of 300, with h0=>.75, alpha= .05 and sample proportion = 0.68, you _________ hypothesis.
Answer: Therefore, the answer is that you reject the null hypothesis.
Step-by-step explanation:
To determine whether we can reject or fail to reject the null hypothesis, we need to perform a hypothesis test.
Null hypothesis (H0): The true population proportion is 0.75.
Alternative hypothesis (Ha): The true population proportion is not 0.75.
We can use a one-sample z-test to test this hypothesis. The test statistic is calculated as:
z = (p - P) / sqrt(P(1-P) / n)
where p is the sample proportion, P is the hypothesized population proportion (0.75), and n is the sample size.
Plugging in the values from the problem, we get:
z = (0.68 - 0.75) / sqrt(0.75 * 0.25 / 300)
z = -2.67
Using a standard normal distribution table, we find that the probability of getting a z-score of -2.67 or less is 0.0038. Since this probability is less than the level of significance (alpha) of 0.05, we can reject the null hypothesis.
Therefore, the answer is that you reject the null hypothesis.
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A truck carrying 8.37 pounds of sand travels to a construction yard and loses 1.9 pounds of sand along the way. How much sand does the truck have when it arrives at the yard
Answer:
6.47 lbs.
Step-by-step explanation:
Honestly I can't tell if there is something missing from this problem (like a variable, or some kind of rate as in the truck loses 1.9 lbs of sand per mile or whatever) or if it's just straight subtraction.
8.37-1.9 = 6.47 lbs.
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.h(x) = integral^ex_1 3 ln(t) dt h'(x) =
The derivative of the function h(x) is h'(x) = 3 x ln(x) - 3 x.
The function h(x) is defined as h(x) = ∫1^x 3 ln(t) dt. To find its derivative, we can use the Part 1 of the Fundamental Theorem of Calculus, which states that if f(x) is continuous on [a,b] and F(x) is an antiderivative of f(x), then the derivative of the integral ∫a^x f(t) dt is simply f(x).
In our case, we have f(t) = 3 ln(t), which is continuous on [1, e]. We can find an antiderivative of f(t) by integrating it with respect to t:
∫ 3 ln(t) dt = 3 t ln(t) - 3 t + C
where C is the constant of integration.
Using this antiderivative, we can apply the Fundamental Theorem of Calculus to find the derivative of h(x):
h'(x) = d/dx [∫1^x 3 ln(t) dt]
h'(x) = 3 x ln(x) - 3 x
Therefore, the derivative of the function h(x) is h'(x) = 3 x ln(x) - 3 x.
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General motors stock fell from $39.57 per share in 2013 to 28.72 per share during
2016. If you bought and sold 8 shares at these prices what was your loss as a percent of
the purchase price?
Given that General Motors' stock fell from $39.57 per share in 2013 to $28.72 per share in 2016.
If a person bought and sold 8 shares at these prices, the loss as a percent of the purchase price is as follows:
First, calculate the total cost of purchasing 8 shares in 2013.
It is given that the price of each share was $39.57 per share in 2013.
Hence the total cost of purchasing 8 shares in 2013 will be
= 8 × $39.57
= $316.56.
Now, calculate the revenue received by selling 8 shares in 2016.
It is given that the price of each share was $28.72 per share in 2016.
Hence the total revenue received by selling 8 shares in 2016 will be
= 8 × $28.72
= $229.76.
The loss will be the difference between the purchase cost and selling price i.e loss = Purchase cost - Selling price
= $316.56 - $229.76
= $86.8
Therefore, the loss incurred on the purchase and selling of 8 shares is $86.8.
Now, calculate the loss percentage.
The formula for loss percentage is given by the formula:
Loss percentage = (Loss/Cost price) × 100.
Loss = $86.8 and Cost price = $316.56
∴ Loss percentage = (86.8/316.56) × 100
= 27.4%.
Therefore, the loss percentage is 27.4%.
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Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
y =
0 3 sin4 t dt
integral.gif
ex
y?' =
The derivative of the function y = ∫0^(3sin(4t)) ex dt with respect to t is y'(t) = (3/4) (ex cos(4t)).
To use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function y = ∫0^(3sin(4t)) ex dt, we need to first understand what the theorem states.
Part 1 of the Fundamental Theorem of Calculus states that if a function f(x) is continuous on the closed interval [a, b], and if F(x) is any antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a), or ∫[a,b] f(x) dx = F(b) - F(a).
In other words, the theorem provides a way to calculate the definite integral of a function by evaluating the difference between two antiderivatives of the function.
Now, let's apply this theorem to the function y = ∫0^(3sin(4t)) ex dt. To do this, we need to first find an antiderivative of the integrand ex.
The antiderivative of ex is simply ex itself, so we have:
∫ ex dt = ex + C, where C is the constant of integration.
Now, we can use this antiderivative to find an antiderivative of the integrand in our original function y. Let u = 4t, so that du/dt = 4 and dt = du/4. Then, we have:
y = ∫0^(3sin(4t)) ex dt = ∫0^(3sin(u)) ex (du/4) = (1/4) ∫0^(3sin(u)) ex du
Let F(u) = ∫ ex du = ex + C, where C is a constant of integration. Then, we have:
y = (1/4) F(3sin(u)) - (1/4) F(0) = (1/4) (ex)|_0^(3sin(u)) = (1/4) (ex - 1)
Using the chain rule again, we have:
d/dt (3sin(u)) = 3cos(u) (du/dt) = 3cos(4t)
Substituting this expression back into the previous equation, we get:
y'(t) = (1/4) (ex) (3cos(4t)) = (3/4) (ex cos(4t))
Therefore, the derivative of the function y = ∫0^(3sin(4t)) ex dt with respect to t is y'(t) = (3/4) (ex cos(4t)).
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Suppose that a is the set {1,2,3,4,5,6} and r is a relation on a defined by r={(a,b)|adividesb} . what is the cardinality of r ?
The cardinality of the set a and relation r such that r = {(a, b) | a divides b} is equal to 14.
Set is defined as,
{1,2,3,4,5,6}
The relation r defined on set a as 'r = {(a, b) | a divides b}. means that for each pair (a, b) in r, the element a divides the element b.
To find the cardinality of r,
Count the number of ordered pairs (a, b) that satisfy the condition of a dividing b.
Let us go through each element in set a and determine the values of b for which a divides b.
For a = 1, any element b ∈ a will satisfy the condition .
Since 1 divides any number. So, there are 6 pairs with 1 as the first element,
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6).
For a = 2, the elements b that satisfy 2 divides b are 2, 4, and 6. So, there are 3 pairs with 2 as the first element,
(2, 2), (2, 4), (2, 6).
For a = 3, the elements b that satisfy 3 divides b are 3 and 6. So, there are 2 pairs with 3 as the first element,
(3, 3), (3, 6).
For a = 4, the elements b that satisfy 4 divides b are 4. So, there is 1 pair with 4 as the first element,
(4, 4).
For a = 5, the elements b that satisfy 5 divides b are 5. So, there is 1 pair with 5 as the first element,
(5, 5).
For a = 6, the element b that satisfies 6 divides b is 6. So, there is 1 pair with 6 as the first element,
(6, 6).
Adding up the counts for each value of a, we get,
6 + 3 + 2 + 1 + 1 + 1 = 14
Therefore, the cardinality of the relation r is 14.
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Find the power series for (x)=24x^3/(1−x^4)^2 in the form ∑=1[infinity].form.Hint: First, find the power series for (x)=6/1−x^4. Then differentiate.(Express numbers in exact form. Use symbolic notation and fractions where needed.)
Okay, here are the steps to find the power series for f(x) = 24x^3 / (1 - x^4)^2:
1) First, find the power series for g(x) = 6 / (1 - x^4). This is a geometric series:
g(x) = 6 * (1 - x^4)^-1 = 6 * (1 + x^4 + x^8 + x^12 + ...)
2) This power series has terms:
6 + 6x^4 + 6x^8 + 6x^12 + ...
3) Now, differentiate this series term-by-term:
g'(x) = 24x^3 + 32x^7 + 48x^11 + ...
4) Finally, square this differentiated series:
(g'(x))^2 = (24x^3 + 32x^7 + 48x^11 + ...) ^2
5) Combine like terms and simplify:
(g'(x))^2 = 24^2 x^6 + 2(24)(32) x^11 + 2(24)(48) x^{15} + ...
So the power series for f(x) = 24x^3 / (1 - x^4)^2 is:
f(x) = 24^2 x^6 + 48x^11 + 96x^{15} + ...
In exact form with fractions:
f(x) = 24^2 x^6 + (48/11) x^11 + (96/15) x^{15} + ...
Does this make sense? Let me know if any part of the explanation needs more clarification.
The power series for(x)=24x³/(1−x⁴)² is ∑=[∞]6(n+1)(4n)x⁴ⁿ+².
To find the power series for (x)=24x³/(1−x⁴)^2 in the form ∑=1[∞],
We first need to find the power series for (x)=6/1−x⁴.
Using the formula for a geometric series,
a, ar, ar^2, ar^3, ...
where a is the first term, r is the common ratio, and the nth term is given by ar^(n-1).
we have:
(x)=6/1−x⁴ = 6(1 + x⁴ + x⁸ + x¹² + ...)
Now, we differentiate both sides of the equation:⁸⁷¹²
(x)'= 24x³/(1−x^4)² = 6(4x³ + 8x⁷ + 12x¹¹ + ...)
Thus, the power series for (x)=24x³/(1−x⁴)² is:
∑=1[∞] 6(n+1)(4n)x⁴ⁿ+²
where n starts from 0.
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1. Find the derivative of the function.
g(x) = sec−1(9ex)
Find g'(x)=?
2. Evaluate the integral. (Use C for the constant of integration.)
ex(8 + ex)5 dxEvaluate the integral. (Use C for the constant of integration.) | e*(8 + e*)5 dx
1. The derivative of the function is g'(x) = 9eˣ/(81e²ˣ - 1). 2. The integral is (8 + eˣ)⁶/6 + C, where C is the constant of integration.
1. Let y = sec⁽⁻¹⁾(9ex)
Then, taking the secant on both sides,
sec y = 9ex
Differentiating both sides w.r.t x:
sec y tan y (dy/dx) = 9eˣ
(dy/dx) = (9eˣ)/(sec y tan y)
Now, from the right triangle with hypotenuse sec y, we have:
[tex]tan y = \sqrt{sec^2 y - 1} = \sqrt{(81e^{2x} - 1)/(81e^{2x})}[/tex]
sec y = 9eˣ
Substituting these in the expression for dy/dx, we get:
[tex]g'(x) = (9e^x)/\sqrt{(81e^{2x} - 1)/(81e^{2x})} * 1/\sqrt{(81e^{2x} - 1)/(81e^{2x})}[/tex]
g'(x) = 9eˣ/(81e²ˣ - 1)
2. We can solve this integral using substitution.
Let u = 8 + eˣ, du/dx = eˣ
Substituting these in the given integral, we get:
Integral of eˣ * (8 + eˣ)⁵ dx = Integral of u⁵ du = u⁶/6 + C
= (8 + eˣ)⁶/6 + C, where C is the constant of integration.
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1) The older John gets, the taller he is.
independent and dependent variables
I'm not quite sure what the question is, but I think that John's age would be the independent variable, and John's height would be the dependent variable.
5. The table shows the student population of Richmond High School this year.
Grade 11 (J)
Grade 12 (S)
Total
Girls (G) Boys (B) Total
150
210 360
200 140 340
350 350 700
What is
P(G|J)?
The probability of a student being a girl given that they are in grade 11 is approximately 0.4167 or 41.67%.
The table provided represents the student population of Richmond High School for this year. Let's break down the information in the table:
Grade 11 (J): This row represents the student population in grade 11.
Grade 12 (S): This row represents the student population in grade 12.
Total: This row represents the total number of students in each category.
Girls (G) Boys (B) Total: This row represents the gender distribution within each grade and the total number of students.
To calculate P(G|J), which is the probability of a student being a girl given that they are in grade 11, we need to use the numbers from the table.
From the table, we can see that there are 150 girls in grade 11. To determine the total number of students in grade 11, we add the number of girls and boys, which gives us 360.
Therefore, P(G|J) = Number of girls in grade 11 / Total number of students in grade 11 = 150 / 360 ≈ 0.4167
Hence, the probability of a student being a girl given that they are in grade 11 is approximately 0.4167 or 41.67%.
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Select the answer which is equivalent to the given expression using your calculator.
The equivalent expression to the sine of 2A is the third option:
2160/2601
How to find the value of sin(2A)?Here we start by knowing the equation:
Cos(A)= 45/53
And that angle A is on quadrant 1.
If we use the inverse cosine function, then we will get:
A = Acos(45/51)
A = 28.07°
Now we want to evaluate the sine function in 2A, then we will get:
Sin(2A) = Sin(2*28.07°) = 0.83
From the given options, the one that is equivalent to this is the third option:
2160/2601
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Proceed as in this example to find a solution of the given initial-value problem. xy" – 2xy' + 2y = x In(x), y(1) = 1, y'(1) = 0 y(x) =
The solution to the initial-value problem is: y(x) = (1/2) x ln(x) + (1/4) x
To solve this initial-value problem, we will use the method of undetermined coefficients.
First, we assume that the solution has the form:
y(x) = axln(x) + bx + c
where a, b, and c are constants to be determined. We differentiate this equation twice to obtain:
y'(x) = a(ln(x) + 1) + b
y''(x) = a/x
Substituting these expressions into the differential equation, we get:
x(a/x) - 2x(a(ln(x) + 1) + b) + 2(axln(x) + bx + c) = x ln(x)
Simplifying this equation, we get:
(a - 2b + 2c) xln(x) + (-2a + 2b) x + 2c = x ln(x)
Equating the coefficients of x ln(x), x, and the constant term, we get the following system of equations:
a - 2b + 2c = 1
-2a + 2b = 0
2c = 0
Solving for a, b, and c, we get:
a = 1/2
b = 1/4
c = 0
Therefore, the solution to the initial-value problem is:
y(x) = (1/2) x ln(x) + (1/4) x + 0
To verify that this solution satisfies the differential equation and the initial conditions, we differentiate y(x) and substitute it into the differential equation:
y'(x) = (1/2) ln(x) + (1/4)
y''(x) = 1/(2x)
xy''(x) - 2xy'(x) + 2y(x) = x ln(x)
So the differential equation is satisfied. Finally, we substitute x = 1 into y(x) and y'(x) to get:
y(1) = (1/2) + (1/4) + 0 = 3/4
y'(1) = (1/2)(0) + (1/4) = 1/4
Therefore, the solution to the initial-value problem is:
y(x) = (1/2) x ln(x) + (1/4) x
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a sequence (xn) of irrational numbers having a limit lim xn that is a rational number
An example of a sequence (xn) of irrational numbers having a limit lim xn that is a rational number is xn = 3 + (-1)^n * 1/n.
This sequence alternates between the irrational numbers 3 - 1/1, 3 + 1/2, 3 - 1/3, 3 + 1/4, etc. The limit of this sequence is the rational number 3, which can be shown using the squeeze theorem. To prove this, we need to show that the sequence is bounded above and below by two convergent sequences that have the same limit of 3. Let a_n = 3 - 1/n and b_n = 3 + 1/n. It can be shown that a_n ≤ x_n ≤ b_n for all n, and that lim a_n = lim b_n = 3. Therefore, by the squeeze theorem, lim x_n = 3.
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A scale is tested by repeatedly weighing a standard 9.0 kg weight. The weights for 10 measurements are
9.1,8.9,9.5,9.3,8.9,9.4,9.3,8.9,9.4,8.39.1,8.9,9.5,9.3,8.9,9.4,9.3,8.9,9.4,8.3
Determine the mean weight. Give your answer precise to one decimal place.
mean:
The value of the mean weigh is,
⇒ 9.1
We have to given that;
A scale is tested by repeatedly weighing a standard 9.0 kg weight. The weights for 10 measurements are
⇒ 9.1,8.9,9.5,9.3,8.9,9.4,9.3,8.9,9.4,8.3
Now, We can find the mean as;
⇒ 9.1 + 8.9 + 9.5 + 9.3 + 8.9 + 9.4 + 9.3 + 8.9 + 9.4 + 8.3 / 10
⇒ 91/10
⇒ 9.1
Thus, The value of the mean weigh is,
⇒ 9.1
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