Answer:
[tex]T_{2} = \frac{P_{2} V_{2} T_{1}} {P_{1} V_{1}}[/tex]
Step-by-step explanation:
The Combined Gas Law expression is:
P₁V₁ / T₁ = P₂V₂ / T₂
To solve for T₂, you need to.....
[tex]\frac{P_{1} V_{1} }{T_{1} } = \frac{P_{2} V_{2} }{T_{2} }[/tex] <------ Original expression
[tex]\frac{T_{2} P_{1} V_{1} }{T_{1} } = {P_{2} V_{2} }[/tex] <----- Multiply both sides by T₂
[tex]{T_{2} P_{1} V_{1} } = {P_{2} V_{2} T_{1}}[/tex] <----- Multiply both sides by T₁
[tex]T_{2} = \frac{P_{2} V_{2} T_{1}} {P_{1} V_{1}}[/tex] <----- Divide both sides by P₁V₁
Express x²-8x+5 in form of (x-a)^2 -b
Answer:
a=4, b=11
Step-by-step explanation:
You have to complete the square.
x²-8x+5 = (x-4)²-16 +5 = (x-4)² - 11
Sam the snail crawls at a rate of 2. 64 ft. /minute. What is Sam’s rate in miles per hour? State your answer to the nearest hundredth. (1 miles = 5280 feeet)
Sam the snail's rate is approximately 0.03 miles per hour.
To find Sam's rate in miles per hour, we need to convert his speed from feet per minute to miles per hour.
We know that 1 mile is equal to 5280 feet. First, we can convert Sam's speed from feet per minute to feet per hour by multiplying it by 60 since there are 60 minutes in an hour.
Therefore, Sam's speed in feet per hour is 2.64 ft/min * 60 min/hr = 158.4 ft/hr.
Next, we can convert Sam's speed from feet per hour to miles per hour. Since 1 mile is equal to 5280 feet, we can divide Sam's speed in feet per hour by 5280 to get his speed in miles per hour.
Therefore, Sam's speed in miles per hour is 158.4 ft/hr / 5280 ft/mi = 0.03 mi/hr.
Therefore, Sam the snail crawls at a rate of approximately 0.03 miles per hour.
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Solve the equation by completing the square
a^2+14a-51=0
Answer:
a = 3, -17
Step-by-step explanation:
a ² + 14a - 51 = 0
1) put the a, not a ², in parenthesis.
2) half the coefficient (14) of a. that is 7. Put that into same parenthesis.
3) we have (a + 7)
4) square this and multiply out. (a + 7) ² = a ² + 7a + 7a +49 = a ² +14a + 49
5) this looks just like the original equation except for +49. What do we have to do to get back to original? 49 – (-51) = 49 + 51 = 100. We have to subtract 100
6) now we have (a + 7) ² – 100 =0
7) (a + 7) ² = 100
8) (a + 7) = ± √100
9) a = ± √100 - 7
a = ±10 - 7
= -17 and 3
sketch the region enclosed by the given curves. y = 2 x , y = 8x, y = 1 8 x, x > 0
The sketched region enclosed by the given curves, y = 2/x, y = 8x, and y = x/8 is given below.
To sketch the region enclosed by the given curves, we'll first plot each curve separately and then identify the region between them. The curves are:
y = 2/x
y = 8x
y = x/8
Let's start by plotting these curves one by one:
y = 2/x:
Since x > 0, the curve y = 2/x is a hyperbola with the y-axis as an asymptote and passes through the point (1, 2).
y = 8x:
This is a straight line passing through the origin (0, 0) with a slope of 8. The line goes through the first quadrant.
y = x/8:
This is another straight line with a slope of 1/8. It passes through the origin (0, 0) and also goes through the first quadrant.
Therefore, the final graph is given below.
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The complete question:
Sketch the region enclosed by the given curves.
y = 2/x, y = 8x, y = x/8, x>0
Solve the simultaneous equations
x^2 +y^2 =9
X+y=2
The given simultaneous equations are x² + y² = 9 ...............(1)
x + y = 2 ...............(2)
Equation (2) is solved for y by taking x as the subject:
y = 2 - x
Substitute this value of y in the equation (1):
x² + y² = 9x² + (2 - x)² = 9x² + 4 - 4x + x² = 9
Rearrange the above equation in the standard quadratic form by bringing all terms to one side of the equation:
x² + x² - 4x - 5 = 02
x² - 4x - 5 = 0
This equation is a quadratic equation and can be solved by using the quadratic formula:
x = [-(-4) ± √(-4)² - 4(2)(-5)]/2(2)
x = [4 ± √56]/4
x = [4 ± 2√14]/4
x = [2 ± √14]/2
Substitute these values of x in equation (2) to find the corresponding values of y:
For x = [2 + √14]/2,
y = 2 - [2 + √14]/2
y = (4 - [2 + √14])/2
y = (2 - √14)/2
For x = [2 - √14]/2,
y = 2 - [2 - √14]/2
y = (4 - [2 - √14])/2
y = (2 + √14)/2
Therefore, the solution of the given simultaneous equations is
x = [2 + √14]/2,
y = (2 - √14)/2
OR
x = [2 - √14]/2,
y = (2 + √14)/2
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If K = -1, which Dilation would it be?
A - Enlargement
B - Reduction
C - Congruence Transformation
If K = -1, the dilation would be a reduction. Dilation is a geometric transformation that either enlarges or reduces the size of an object. Which can be positive or negative.
When the scale factor, K, is positive, the dilation is an enlargement. This means that the image of the object is larger than the original. The positive scale factor indicates that the object is being stretched or magnified.
However, when the scale factor, K, is negative, the dilation is a reduction. In this case, the image of the object is smaller than the original. The negative scale factor indicates that the object is being compressed or diminished.
Therefore, if K = -1, it signifies that the dilation is a reduction. The object will be transformed into a smaller version of itself. It is important to note that the absolute value of the scale factor determines the magnitude of the reduction, with a larger absolute value resulting in a greater reduction in size.
In summary, if K = -1, the dilation is a reduction of the object.
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Complete each question. Make sure to show work whenever possible.
1. Find the value of x.
The value of x in the figure of similar triangles is
13.5
What are similar triangles?This is a term used in geometry to mean that the respective sides of the triangles are proportional and the corresponding angles of the triangles are congruent
Examining the figure shows that pair of proportional sides are
22 and 11, then 27 and x
The solution is worked out below
22 / 11 = 27 / x
22x =11 * 27
x = 11 * 27 / 22
x = 13.5
hence side x = 13.5
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If the average value of the function f on the interval 1≤x≤4 is 8, what is the value of ∫41(3f(x) 2x)dx ?
According to question the value of ∫41(3f(x) 2x)dx is 73.
We know that the average value of the function f on the interval [1,4] is 8. This means that:
(1/3) * ∫1^4 f(x) dx = 8
Multiplying both sides by 3, we get:
∫1^4 f(x) dx = 24
Now, we need to find the value of ∫4^1 (3f(x) 2x) dx. We can simplify this expression as follows:
∫1^4 (3f(x) 2x) dx = 3 * ∫1^4 f(x) dx + 2 * ∫1^4 x dx
Using the average value of f, we can substitute the first integral with 24:
∫1^4 (3f(x) 2x) dx = 3 * 24 + 2 * ∫1^4 x dx
Evaluating the second integral, we get:
∫1^4 x dx = [x^2/2]1^4 = 8.5
Substituting this value back into the equation, we get:
∫1^4 (3f(x) 2x) dx = 3 * 24 + 2 * 8.5 = 73
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What is the surface area of a square-based pyramid with a base length of 3 in, height of 7 in, and a slant height of 5 in?
Answer:
Surface area of a square pyramid = a 2 + 2al Where, a denotes base length of a square pyramid and, l denotes the slant height or the height of each side face.
Step-by-step explanation:
I used to do stuff like this but I haven't in a long time so I believe you have to add up all the numbers or multiply them.
a lawn roller in the shape of a right circular cylinder has a diameter of 18in and a length of 4 ft find the area rolled during onle complete relvutitopn of the roller
During one complete revolution, the lawn roller covers approximately 2713.72 square inches of area.
A lawn roller in the shape of a right circular cylinder has a diameter of 18 inches and a length of 4 feet.
To find the area rolled during one complete revolution of the roller, we need to calculate the lateral surface area of the cylinder.
First, let's convert the length to inches: 4 feet = 48 inches.
The formula for the lateral surface area of a cylinder is 2πrh, where r is the radius and h is the height (length).
Since the diameter is 18 inches, the radius is 9 inches (18/2).
Plugging in the values, we get:
2π(9)(48) = 2π(432) ≈ 2713.72 square inches.
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Use the Ratio Test to determine whether the series is convergent or divergent.
[infinity] n
6n
n = 1
Identify
an.
Evaluate the following limit.
lim n → [infinity]
an + 1
an
the series ∑(n=1 to infinity) [tex]n^{6}[/tex] / n! is convergent by using ratio test.
To apply the Ratio Test, we need to evaluate the limit of the ratio of consecutive terms, lim(n→∞) (a(n+1) / a(n)).
In this case, a(n) = [tex]n^{6}[/tex] / n! and a(n+1) =[tex](n+1)^{6}[/tex] / (n+1)!.
Taking the limit, we have:
lim(n→∞) [[tex](n+1)^{6}[/tex] / (n+1)!] / [[tex]n^{6}[/tex] / n!]
= lim(n→∞) [[tex](n+1)^{6}[/tex] / [tex]n^{6}[/tex]] * [n! / (n+1)!]
= lim(n→∞) [[tex](n+1)^{6}[/tex] / [tex]n^{6}[/tex]] * [1 / (n+1)]
= 1 * 0 = 0.
Since the limit of the ratio of consecutive terms is 0, which is less than 1, the series converges by the Ratio Test.
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Just having a rough time with this please help. Thank you
Answer:
The formatting is a bit off but assuming that -x + 2y = 6 and -3x + y = -2 are the two separate equations, the solution to your system of equations is (2,4) or x = 2, y = 4.
Step-by-step explanation:
Here is how you could solve this system of equations using the elimination method:
1. The first step is to find a variable you can eliminate, such as y.
-x+2y=6
-3x+y=-2
(multiply the second equation by -2)
−x+2y=6
6x-2y=4
This is your new set
2. Next, "add" your set together by lining it up and combining like terms.
-x+2y=6
+. 6x-2y=4
——————
5x = 10
3. Solve for x by dividing by 5
5x=10
10÷5=2
x=2
4. Now that you have your x, find y by substituting 2 for x in any of your original set's equations. We'll do the first equation, −x+2y=6.
−x+2y=6
-2+2y=6 ---> add 2 on both sides to remove -2
2y=8 ---> divide by 2 on both sides to remove the 2 from y
y=4
5. Set your answers up as an ordered pair like this ( ___ , ___ )
x=2 , y=4
(2, 4)
Hope this helps!
Describe what each variable does to transform the basic function.
+ d
.
g(x) = a - 2b(x-c)
)
c:
a:
d:
b:
Main answer: Transformations of basic functions depend on the changes made to their variables.
Supporting answer :Functions can be transformed in different ways. The variable a modifies the vertical stretch or compression of a function. A negative value of a produces a reflection over the x-axis. The variable b is used to modify the horizontal stretch or compression of the function. A negative value of b produces a reflection over the y-axis. The variable h translates the graph to the left (h > 0) or to the right (h < 0). Lastly, the variable k translates the graph up (k > 0) or down (k < 0).
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The total weight of the raw material will not be less than 1,500 tons. The factory manager plans to use two different trucking firms. Big Red has heavy-duty trucks that can transport 200 tons at a cost of $50 per truckload. Common Joe is a more economical firm, costing only $20 per load, but its trucks can transport only 90 tons. The factory manager does not wish to spend more than $450 on transportation. The availability of trucks is the same for both firms
The total cost of transporting the raw materials is $890, which is less than the $450 budget of the factory manager.
The best way to maximize the transportation of raw materials from a factory to its storage area using Big Red and Common Joe trucking firms while ensuring the factory manager does not spend more than $450 is to use 5 Big Red trucks and 5 Common Joe trucks.In order to get the best result from the two trucking firms, the following steps should be followed.
Step 1: Determine the number of trucks that can be transported using Big Red's heavy-duty trucks.
$200 per truck is the cost of transporting 200 tons by Big Red.
The formula for calculating the number of trucks that can be used is as follows:
$450/$50 = 9 truckloads
Step 2: Determine the number of trucks that can be transported using Common Joe trucks.
$20 per truck is the cost of transporting 90 tons by Common Joe.
The formula for calculating the number of trucks that can be used is as follows:
$450/$20 = 22.5 truckloads
The number of trucks that can be used is 22, but since it is not an integer, it will be rounded down to 22.The total number of tons that can be transported using the two trucking firms is calculated as follows:
5 * 200 = 1000 tons of raw materials can be transported by Big Red
5 * 90 = 450 tons of raw materials can be transported by Common Joe
The total tons of raw materials that can be transported is therefore 1,450 tons.
Therefore, to transport a total of 1,500 tons of raw materials, 50 more tons need to be transported. 10 more truckloads of Big Red will transport these additional tons.
Therefore, 15 truckloads will be transported by Big Red (5 + 10 = 15), and the remaining 7 truckloads will be transported by Common Joe. (22 - 15 = 7).
As a result, the total cost of transporting the raw materials is:
$50 * 15 + $20 * 7 = $750 + $140
= $890, which is less than the $450 budget of the factory manager.
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let s = {3, 8, 13, 18, 23, 28}, e = {8, 18, 28}, f = {3, 13, 23}, and g = {23, 28}. (enter ∅ for the empty set.) find the event (e ∩ f ∩ g)c.
The event (e ∩ f ∩ g)c is equal to the set {3, 8, 13, 18}.
To find the complement of the intersection of sets e, f, and g, denoted as (e ∩ f ∩ g)c, we first need to determine the intersection of sets e, f, and g.
The intersection of sets e, f, and g is the set of elements that are present in all three sets. In this case:
e ∩ f ∩ g = {23, 28}
To find the complement of this intersection, we need to consider all the elements that are not in the set {23, 28}.
Given that the original set s = {3, 8, 13, 18, 23, 28}, the complement of the intersection can be found by subtracting {23, 28} from set s:
(e ∩ f ∩ g)c = s - {23, 28}
Calculating this, we have:
(e ∩ f ∩ g)c = {3, 8, 13, 18}
Therefore, the event (e ∩ f ∩ g)c is equal to the set {3, 8, 13, 18}.
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Suppose you walk 18. 2 m straight west and then 27. 8 m straight north. What vector angle describes your
direction from the forward direction (east)?
Add your answer
Given that a person walks 18.2 m straight towards the west and then 27.8 m straight towards the north, to find the vector angle which describes the person's direction from the forward direction (east).
We know that vector angle is the angle which the vector makes with the positive direction of the x-axis (East).
Therefore, the vector angle which describes the person's direction from the forward direction (east) can be calculated as follows:
Step 1: Calculate the resultant [tex]vectorR = √(18.2² + 27.8²)R = √(331.24)R = 18.185 m ([/tex]rounded to 3 decimal places)
Step 2: Calculate the angleθ = tan⁻¹ (opposite/adjacent)where,opposite side is 18.2 mandadjacent side is [tex]27.8 mθ = tan⁻¹ (18.2/27.8)θ = 35.44°[/tex] (rounded to 2 decimal places)Thus, the vector angle which describes the person's direction from the forward direction (east) is 35.44° (rounded to 2 decimal places).
Hence, the correct option is 35.44°.
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I pls need the answer
The equation of the line in the graph is
y = -3/2 x + 5.How to write the equation of the line in the graphFrom the graph the line passed through points (4,-1) and (0,5)
using the slope-intercept form of a line, which is y = mx + b,
where
m is the slope and
b is the y-intercept.
the slope of the line
m = (5 - (-1)) / (0 - 4) = 6 / -4 = -3/2
form the points the y intercept is 5
Therefore, the equation of the line is y = -3/2 x + 5.
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When Tom plays darts, he hits the
target 65% of the time. Find the
probability that he hits the target at
least four out of next six attempts.
A. 57.17%
B. 64.71%
C.42.83%
D. 35.29%
Option A is correct, 57.17% is the probability that he hits the target at least four out of next six attempts.
Let's calculate the probability of hitting the target exactly four times out of six attempts:
P(4 hits) = C(6, 4) × (0.65)⁴ × (1 - 0.65)⁶⁻⁴
The probability of hitting the target exactly five times out of six attempts:
P(5 hits) = C(6, 5) × (0.65)⁵ × (1 - 0.65)⁶⁻⁵
Now calculate the probability of hitting the target all six times:
P(6 hits) = (0.65)⁶
Now, we can find the probability that Tom hits the target at least four times by summing up the individual probabilities:
P(at least 4 hits) = P(4 hits) + P(5 hits) + P(6 hits)
P(at least 4 hits) = C(6, 4) × (0.65)⁴ × (1 - 0.65)⁶⁻⁴ + C(6, 5) × (0.65)⁵ × (1 - 0.65)⁶⁻⁵ + (0.65)⁶
=57.17%
Hence, 57.17% is the probability that he hits the target at least four out of next six attempts.
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the assembly time for a product is uniformly discributed between 6 to 10 minutes the standard deviaiton of assembly time in minutes is approximately
The assembly time for a product is uniformly distributed between 6 to 10 minutes the standard deviation of assembly time in minutes is approximately 1.155.
To find the standard deviation of assembly time for a product that is uniformly distributed between 6 to 10 minutes, we can use the following formula for a uniform distribution:
Standard Deviation (σ) = √((b - a)² / 12)
Here, 'a' is the lower limit (6 minutes) and 'b' is the upper limit (10 minutes).
Step 1: Calculate (b - a)²
(10 - 6)² = 4² = 16
Step 2: Divide by 12
16 / 12 = 1.3333
Step 3: Find the square root
√1.3333 ≈ 1.155
So, the standard deviation of assembly time for a product in minutes is approximately 1.155.
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linear polystyrene has phenyl groups that are attached to alternate not adjacent carbons of the polymer chain. Explain the mechanistic basis for this fact
The mechanistic basis for linear polystyrene having phenyl groups attached to alternate carbons of the polymer chain is due to the nature of the polymerization reaction, specifically free-radical polymerization.
1. Free-radical polymerization of styrene starts with the initiation step, where a free radical initiator generates a reactive radical site.
2. The reactive radical site reacts with the double bond of the styrene monomer, forming a new radical site on the styrene molecule.
3. This new radical site on the styrene molecule can now react with another styrene monomer, effectively joining them together.
4. As the radical site is always at the end of the growing polymer chain, the phenyl groups of each added styrene monomer will be attached to alternate carbons. This occurs because the reactive site is situated between the phenyl group and the double bond in the monomer, creating a zigzag pattern as the chain grows.
Conclusion:
The attachment of phenyl groups to alternate carbons of the polymer chain in linear polystyrene can be attributed to the free-radical polymerization mechanism. The reactive radical site, created during the polymerization, allows the phenyl groups to be connected in an alternating pattern along the chain.
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Evaluate the integral. integral_0^1 (x^17 + 17^x) dx x^18/18 + 17^x/log(17)
The value of the given integral is (1/18) + (17/ln(17)).
The integral is evaluated using the sum rule of integration, which states that the integral of the sum of two functions is equal to the sum of their integrals. Therefore, we can evaluate the integral of each term separately and then add them together.
For the first term, we use the power rule of integration, which states that the integral of [tex]x^n[/tex]is equal to[tex](x^(n+1))/(n+1)[/tex]. Therefore, the integral of [tex]x^17[/tex]is [tex](x^18)/18.[/tex]
For the second term, we use the exponential rule of integration, which states that the integral of [tex]a^x[/tex]is equal to [tex](a^x)/(ln(a))[/tex]. Therefore, the integral of [tex]17^x is (17^x)/(ln(17)).[/tex]
Adding these two integrals together gives us the final answer of (1/18) + [tex](17/ln(17)[/tex]).
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1. if a is an n × n matrix and x is a vector in rn, then the product ax is a linear combination of the columns of matrix a. True or false?
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It can be used to represent systems of linear equations, transformations in geometry, and a wide range of other mathematical concepts in a compact and organized form.
When you multiply a matrix A (n × n) by a vector x (in R^n), the resulting product Ax is a linear combination of the columns of matrix A.
Here's a step-by-step explanation:
1. Let A be an n × n matrix with columns C₁, C₂, ..., Cₙ, and x be a vector in R^n with elements [x₁, x₂, ..., xₙ]^T (transpose).
2. When you multiply the matrix A by the vector x, the resulting vector Ax can be represented as:
Ax = A * x = [C₁ C₂ ... Cₙ] * [x₁, x₂, ..., xₙ]^T
3. The multiplication of A and x results in a new vector, where each element is formed by taking the dot product of the corresponding row of A with the vector x:
Ax = [x₁*C₁ + x₂*C₂ + ... + xₙ*Cₙ]
4. In the resulting vector Ax, you can see that each column of matrix A is multiplied by its corresponding scalar from the vector x, forming a linear combination of the columns of matrix A.
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Calculate the net force these forces acts on a single object, 30n up 25n down 5n down 5n up
The net force acting on the object is 10N up
When multiple forces act on an object, the net force is the total force acting on the object. It determines the object's motion, including its direction and speed.
To calculate the net force, we need to add all the forces acting on the object. If the net force is zero, the object will remain at rest or move with a constant velocity, while if it is non-zero, the object's velocity will change, and it will accelerate in the direction of the net force.
In this scenario, there are four forces acting on the object, two pointing up and two pointing down. To calculate the net force, we need to add all the forces together, taking into account their direction and magnitude.
Since the forces pointing up and down are opposite in direction, we subtract the smaller force from the larger one to get the resultant force. In other words, we can cancel out the forces pointing in opposite directions, leaving us with a single net force acting on the object.
So, in this case, we have a 30N force pointing up, a 25N force pointing down, a 5N force pointing down, and a 5N force pointing up.
First, we'll cancel out the 5N force pointing down with the 5N force pointing up.
30N up - 25N down - 5N down + 5N up
= 30N - 25N - 5N + 5N
= 30N - 20N
= 10N up
Therefore, the net force acting on the object is 10N up. This means that the object will accelerate in the upward direction with a force of 10N
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Complete Question
Calculate the net force these forces acts on a single object, 30N [up],
25N [down], 5N [down] and 5N [up]
Sanjay’s closet is shaped like a rectangular prism. It measures feet high and has a base that measures feet long and feet wide. What is the volume of Sanjay’s closet?
The volume of Sanjay’s closet would be 82.875 ft³
It is known that a rectangular prism is a three-dimensional shape that has two at the top and bottom and four are lateral faces.
The volume of a rectangular prism=Length X Width X Height
Given parameters are;
4 1/4 ft long, 3 1/4 ft wide, and 6 ft tall.
V = Length X Width X Height
V = 3 1/4 x 4 1/4 x 6
V = 82. 7/8 ft³ or 82.875 ft³
The complete question is
Sanjay’s closet is shaped like a rectangular prism. It measures 4 1/4 ft long, 3 1/4 ft wide, and 6 ft tall. What is the volume of Sanjay’s closet?
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Test the polar equation for symmetry with respect to the polar axis, the pole, and the line θ = π 2 . (Select all that apply.) r = 3 + 6 cos(θ)
The polar equation r = 3+6cosθ is symmetric to the polar axis with respect to the polar axis.
To test the polar equation r = 3 + 6 cos(θ) for symmetry, we will consider each type of symmetry one by one:
1. Polar axis symmetry: Replace θ with -θ and check if the equation remains the same.
r = 3 + 6 cos(-θ) = 3 + 6 cos(θ) (since cosine is an even function)
Since the equation remains the same, the curve is symmetric with respect to the polar axis.
2. Pole symmetry: Replace r with -r and check if the equation remains the same.
-r = 3 + 6 cos(θ)
This equation is not equivalent to the original equation, so the curve is not symmetric with respect to the pole.
3. Line θ = π/2 symmetry: Replace θ with (π - θ) and check if the equation remains the same.
r = 3 + 6 cos(π - θ) = 3 - 6 cos(θ) (since cos(π - θ) = -cos(θ))
This equation is not equivalent to the original equation, so the curve is not symmetric with respect to the line θ = π/2.
In conclusion, the polar equation r = 3 + 6 cos(θ) is symmetric with respect to the polar axis, but not with respect to the pole or the line θ = π/2.
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Test the series for convergence or divergence. Σ (n^9 +1) / (n10 + 1) n = 1 a. convergent b. divergent
The given series is divergent.
We can use the limit comparison test to determine the convergence or divergence of the given series:
First, note that for all n ≥ 1, we have: [tex]\frac{(n^9 + 1) }{ (n^10 + 1)}[/tex] ≤ [tex]\frac{n^9 }{n^10} = \frac{1}{n}[/tex]
Therefore, we can compare the given series to the harmonic series ∑ 1/n, which is a well-known divergent series. Specifically, we can apply the limit comparison test with the general term [tex]a_n = \frac{(n^9 + 1)}{(n^{10} + 1)}[/tex] and the corresponding term [tex]b_n = \frac{1}{n}[/tex]:
lim (n → ∞) [tex]\frac{a_n }{ b_n}[/tex] = lim (n → ∞) [tex]\frac{\frac{(n^9 + 1)}{(n^10 + 1)} }{\frac{1}{n} }[/tex]
= lim (n → ∞) [tex]\frac{ n^{10} }{ (n^9 + 1)}[/tex]
= lim (n → ∞) [tex]\frac{n}{1+\frac{1}{n^{9} } }[/tex]
= ∞
Since the limit is positive and finite, the series ∑ [tex]\frac{(n^9 + 1) }{ (n^10 + 1) }[/tex] behaves in the same way as the harmonic series, which is divergent. Therefore, the given series is also divergent.
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Which expression is equivalent [a^8]^4
a^2
a^4
a^12
a^32
Answer:
a³²
Step-by-step explanation:
the law of exponents states that when raising a power to another power multiply the exponents
our answer will go like this:
(a⁸)⁴
a⁸*⁴
a³²
For the sequence an=(5+3n)^−3. Find a number k such that n^ka_n has a finite non-zero limit.
Answer:
n^3*a_n ≈ (1/27) * n^3 → non-zero limit
Step-by-step explanation:
We have the sequence given by a_n = (5+3n)^(-3), and we want to find a value of k such that n^k*a_n has a finite non-zero limit as n approaches infinity.
Let's simplify the expression n^k*a_n:
n^k*a_n = n^k*(5+3n)^(-3)
We can rewrite this as:
n^k*a_n = [n/(5+3n)]^3 * [1/(n^(-k))]
Using the fact that 1/(n^(-k)) = n^k, we can further simplify this to:
n^k*a_n = [n/(5+3n)]^3 * n^k
We want this expression to have a finite non-zero limit as n approaches infinity. For this to be true, we need the first factor, [n/(5+3n)]^3, to approach a finite non-zero constant as n approaches infinity.
To see why this is the case, note that as n gets large, the 3n term dominates the denominator and we have:
[n/(5+3n)]^3 ≈ [n/(3n)]^3 = (1/27) * n^(-3)
So we need k = 3 for n^k*a_n to have a finite non-zero limit. Specifically, as n approaches infinity, we have:
n^3*a_n ≈ (1/27) * n^3 → non-zero constant.
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Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = e4t cos 4t i + 3 j + e4t sin 4t k
The Reparametrized curve with respect to arc length is:
r(s) = (1/2) * sqrt(2) * e^(4t) cos(4t) i + 3 j + (1/2) * sqrt(2) * e^(4t) sin(4t) k
To reparametrize the curve with respect to arc length, we need to find the expression for the curve in terms of the arc length parameter s.
The arc length parameter s is given by the integral of the speed function |r'(t)| with respect to t:
s = ∫|r'(t)| dt
Let's calculate the speed function |r'(t)| first:
r(t) = e^(4t) cos(4t) i + 3 j + e^(4t) sin(4t) k
r'(t) = (4e^(4t) cos(4t) - 4e^(4t) sin(4t)) i + 0 j + (4e^(4t) sin(4t) + 4e^(4t) cos(4t)) k
|r'(t)| = sqrt((4e^(4t) cos(4t) - 4e^(4t) sin(4t))^2 + (4e^(4t) sin(4t) + 4e^(4t) cos(4t))^2)
= sqrt(16e^(8t) cos^2(4t) - 32e^(8t) cos(4t) sin(4t) + 16e^(8t) sin^2(4t) + 16e^(8t) sin^2(4t) + 32e^(8t) cos(4t) sin(4t) + 16e^(8t) cos^2(4t))
= sqrt(32e^(8t))
Now, we can express s in terms of t by integrating |r'(t)|:
s = ∫sqrt(32e^(8t)) dt
To find the integral, we can make a substitution u = 8t, du = 8 dt:
s = (1/8) ∫sqrt(32e^u) du
= (1/8) ∫2sqrt(2e^u) du
= (1/8) * 2 * sqrt(2) ∫e^(u/2) du
= (1/4) * sqrt(2) * ∫e^(u/2) du
= (1/4) * sqrt(2) * 2e^(u/2) + C
= (1/2) * sqrt(2) * e^(4t) + C
Therefore, the reparametrized curve with respect to arc length is:
r(s) = (1/2) * sqrt(2) * e^(4t) cos(4t) i + 3 j + (1/2) * sqrt(2) * e^(4t) sin(4t) k
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The curve reparametrized with respect to arc length is:
r(u) = e^(2u/√2) cos(2u/√2) i + 3j + e^(2u/√2) sin(2u/√2) k
We have the curve given by:
r(t) = e^(4t) cos(4t) i + 3j + e^(4t) sin(4t) k
The speed of the curve is:
|v(t)| = √( (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 )
= √( 16e^(8t) + 16e^(8t) )
= 4e^(4t) √2
Thus, the arc length from t = 0 to t = s is:
s = ∫0s |v(t)| dt
= ∫0s 4e^(4t) √2 dt
= √2 e^(4t) |_0^s
= √2 ( e^(4s) - 1 )
Solving for s, we get:
s = (1/4) ln( (s/√2) + 1 )
Let u be the parameter with respect to arc length, then we have:
u = ∫0t |v(t)| dt
= ∫0t 4e^(4t) √2 dt
= √2 e^(4t) |_0^t
= √2 ( e^(4t) - 1 )
Solving for t, we get:
t = (1/4) ln( (u/√2) + 1 )
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What is the equation of a line perpendicular to 4x+3y=15 that goes through the point (5,2)?
Answer:
y = (3/4)x - 7/4
Step-by-step explanation:
y – y1 = m (x – x1), where y1 and x1 are the coordinates of a given point.
4x + 3y = 15
3y = -4x + 15
y = -(4/3)x + 5.
the slope of this line is -4/3.
the slope of the perpendicular line is -1 / (-4/3) = +3/4.
equation of perpendicular line through (5, 2) is:
y - 2 = (3/4) (x -5) = (3/4)x - (15/4)
y = (3/4)x - (15/4) + 2
y = (3/4)x - 7/4