a) The velocity vector of the particle is [2, -2t].
b) The equation of the tangent line at[tex](2t, 36 - t^2) is y - (36 - t^2) = -t(x - 2t).[/tex]
c) The x-coordinate of the point on the x-axis that the laser light hits is [tex]x = 2t + (36 - t^2)/t.[/tex]
d) The speed of the laser light along the x-axis at time t = 3 is 1, as it is the absolute value of the derivative of x with respect to t at t = 3.
(a) The velocity vector of the particle is the derivative of the position vector with respect to time:
v(t) = [x'(t), y'(t)] = [2, -2t]
(b) The slope of the tangent line is the derivative of y with respect to x:
dy/dx = (dy/dt)/(dx/dt) = (-2t)/(2) = -t
Using the point-slope form of the equation of a line, the tangent line at [tex](2t, 36 - t^2)[/tex] is:
[tex]y - (36 - t^2) = -t(x - 2t)[/tex]
(c) To find the x-coordinate of the point on the x-axis that the laser light hits, we need to find the intersection of the tangent line and the x-axis. Setting y = 0, we get:
[tex]-t(x - 2t) + (36 - t^2) = 0[/tex]
Solving for x, we get:
[tex]x = 2t + (36 - t^2)/t[/tex]
(d) The speed of the laser light along the x-axis is the absolute value of the derivative of x with respect to t:
[tex]|dx/dt| = |2 - (36 - t^2)/t^2|[/tex]
At time t = 3, we have:
|dx/dt| = |2 - (36 - 9)/9| = |2 - 3| = 1
Therefore, the speed of the laser light along the x-axis at time t = 3 is 1. The justification is that the absolute value of the derivative gives the magnitude of the rate of change of x with respect to time, which represents the speed.
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In a model of a construction project, 1 yard is represented as 0.25 inch. What would be the actual area of a garden in square yards, if it is represented by 20 square inches in the model?
uppose you want to test the value of the mean in a population. You collect a random sample of 60 which yields a sample mean and sample standard deviation. You only know sampled information to perform the test. What type of procedure is appropriate for your data
To test the value of the mean in a population when only sample mean and sample deviation are known to perform the test then it is appropriate to use z-test procedure.
What is a Z-test?
A statistical method of testing a hypothesis is the z-test where either:
We are aware of the population variance, which is equal to 2.Despite the fact that we are unsure of the population variance, our sample size, n 30, is sizable. In this instance, we estimate the population variance using the sample variance.A t-test unlike of z-test must be used if the sample size is less than 30 and the population variance is unknown.Additional prerequisites for using the z-test include the following:
Data must follow a normal distribution.Independent data points are required.The variances for each sample must be equal.To know more about the z-test..........
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Select the correct answer.
What is the zero of the function represented by this graph?
x = 5
x= -6
x=-5
x=6
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Here's the explanation ~
Zeros or roots of a function is the value of independent variable (usually x - coordinate or absicca) where the value of dependent variable (y - coordinate or ordinate) is 0.
That is :
The point where it cuts the x - axis, it has an x - coordinate = -6
Therefore, we can conclude that The zero of given function is -6
Answer:
x = -6.
Step-by-step explanation:
It is where the graph cuts the x-axis (where the function is zero).
That is -6.
What is Permutations and Combinations?
Answer:
See below
Step-by-step explanation:
Permutation is to select an object then arrange it and it cares about the orders while Combination is about only selecting an object without caring the orders.
Permutation can be expressed in math as:
[tex]\displaystyle{_n P _r = \dfrac{n!}{(n-r)!} \ \ \ (n \geq r) }[/tex]
where n is a number of total object and r is a number of selected object to arrange. Hence. n cannot be less than r.
Now let's see an example of permutation, suppose we have letter A, B and C. I'd like to know how many ways these words can be arranged:
Since there are 3 letters total and 3 selected letters to arrange then:
[tex]\displaystyle{_3 P _3 = \dfrac{3!}{(3-3)!}}\\\\\displaystyle{_3 P _3 = \dfrac{3 \times 2 \times 1}{0!}}\\\\\displaystyle{_3 P _3 = \dfrac{6}{1}}\\\\\displaystyle{_3 P _3 = 6}[/tex]
Therefore, there are 6 ways to arrange the letters - we can also demonstrate visually:
ABC - 1
ACB - 2
BAC - 3
BCA - 4
CAB - 5
CBA - 6
Notice that if you do visually, you'll get the same answer as the calculation of permutation!
----
Combination can be expressed mathematically as:
[tex]\displaystyle{_n C _r = \dfrac{n!}{(n-r)!r!} = \dfrac{_n P _r}{r!} \ \ \ (n \geq r) }[/tex]
The difference between permutation and combination is that you only find how many ways you can select object in combination. Therefore, no arrange and doesn't care about order, just ways to select.
Suppose we have same 3 letters: A, B and C. I want to find how many ways I can select these 3 letters:
Since there are 3 letters total and 3 selected letters:
[tex]\displaystyle{_3 C _3 = \dfrac{3!}{(3-3)!3!}}\\\\\displaystyle{_3 C _3 = \dfrac{3!}{0!3!}}\\\\\displaystyle{_3 C _3 = \dfrac{3!}{3!}}\\\\\displaystyle{_3 C _3 = 1}[/tex]
Hence, there is only one way to select 3 letters. This makes sense because if you have 3 letters then you can only select 3 letters only one way.
Permutations and combinations have many practical applications in various fields such as statistics, Probability theory, and computer science.
Permutations and combinations are two concepts in mathematics that deal with counting and arranging elements in a set.
Permutations refer to the number of ways in which a set of objects can be arranged or ordered. In other words, permutations involve counting the number of possible ways in which a group of objects can be arranged in a particular order. For example, if we have three objects A, B, and C, the possible permutations would be ABC, ACB, BAC, BCA, CAB, and CBA. The formula for permutations is nPr = n! / (n - r)!, where n is the total number of objects and r is the number of objects being arranged.
Combinations, on the other hand, refer to the number of ways in which a subset of objects can be selected from a larger set, without regard to the order in which they are selected. In other words, combinations involve counting the number of possible subsets that can be formed from a larger set. For example, if we have three objects A, B, and C, the possible combinations would be AB, AC, and BC. The formula for combinations is nCr = n! / (r! * (n - r)!), where n is the total number of objects and r is the number of objects being selected.
The key difference between permutations and combinations is that permutations involve ordering objects, while combinations do not. Another important distinction is that in permutations, each arrangement is considered distinct, whereas in combinations, different arrangements that contain the same objects are considered identical.
Permutations and combinations have many practical applications in various fields such as statistics, probability theory, and computer science. For example, they are commonly used in solving problems related to probability, combinations of locks, and permutations of passwords. Understanding these concepts is essential in order to effectively analyze and solve problems in these fields.
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(FIRST TO ANSWER GETS BRAINLIEST OR 5 STARS) For consecutive weeks, the manager of a local restaurant collected data on the number of cheese pizzas sold each week. The data is shown in the scatterplot below.
(scatterplot is below)
Can the given line be used to make reasonable predictions of the number of cheese pizzas that will be sold in upcoming weeks? Explain your reasoning.
Yes, the line can be used to make reasonable predictions of the number of cheese pizzas that would be sold in the upcoming weeks. This is because the line is the line of best fit
Line of best fitFrom the question, we are to determine if the line can be used to make reasonable predictions of the number of cheese pizzas that would be sold in the upcoming weeks
In the graph, we have a scatterplot.
The line drawn is the line of best fit
Hence,
Yes, the line can be used to make reasonable predictions of the number of cheese pizzas that would be sold in the upcoming weeks. This is because the line is the line of best fit.
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a carpool contains three kindergartners and five first-graders. if two children are ill, find the probability tyhat at lease one of them
The probability that at least one of them is a kindergartner; 0.643.
What is probability?Calculating or estimating how likely something is to occur is what probability is all about. The likelihood of an event occurring can be expressed using words like "certain," "impossible," or "probable."
Probabilities are always expressed in mathematics as fractions, decimals, or percentages with values ranging from 0 to 1.
Calculation for the probability of the one of them is a kindergarden;
The total number of children is 8.
The total number of kindergarden are 3.
The total number of first graders are 5.
Let 'P' be the probability that at least one of them is kindergarden.
P(at least one kindergartner) = 1 - P(no kindergartner) = 1 - P(all first-graders)
The probability that all are first-graders = (5/8)×(4/7)
= 5/14
Therefore, P(at least one kindergartner) = 1 - (5/14)
= 9/14
= 0.643
Therefore, if two children are ill then, the probability that at least one of them is a kindergartner is 0.643.
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The correct question is-
A carpool contains three kindergartners and five first-graders. if two children are ill, find the probability that at least one of them is a kindergartner.
Find the zeros of: [tex]f(x)=-\frac{1}{1500} (x^3+10x^2-275x-1500)[/tex], and the use them to find all of the linear factors of the polynomial function
The linear factors of the polynomial function are f(x) = - (1/1500) · (x + 20) · (x + 5) · (x - 15).
How to find the linear factors of a cubic equation
Cubic equations are polynomials of third grade, whose standard and factor forms are shown below:
Standard form
y = a · x³ + b · x² + c · x + d
Factor form
y = (x - r₁) · (x - r₂) · (x - r₃)
Where r₁, r₂, r₃ are the zeros of the cubic equation.
There are several approaches to find the roots of the polynomial, in this question we decided to use the graphical approach, which offers sufficiency and quickness for analysis. The roots of the polynomials are the points that pass through the x-axis.
In accordance with the image attached below, the polynomial has three real roots: r₁ = - 20, r₂ = - 5, r₃ = 15. Then, the linear factors of the polynomial function are f(x) = - (1/1500) · (x + 20) · (x + 5) · (x - 15).
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can someone please help me
Answer:
Cos (3pi/4) =0.999154554
Sin (3pi/4)=0.041111762
Step-by-step explanation:
On your calculator there should be SIN and COS next to each other
#1 tap COS
#2 put your given numbers, in this case
COS(3pi/4) = 0.999154554
same goes for SIN
SIN (3pi/4)=0.041111762
A science test, which is worth 100 points, consist of 24 questions. Each question is worth either 3 points or 5 points. If X is the number of 3 point questions and why is the number of 5 point questions, the system shown represents this situation. X+y=24, 3x+5y=100. What does the solution of this system indicate about the questions on the test?
Answer:
x= 10 y=14 : 10 3 point, 14 5 point
Step-by-step explanation:
Write your equation down vertically,
3x + 5y = 100
x + y = 24
Next find multiply bottom equation by one number from the tops opposite,
3x + 5y = 100
(-3)x +(-3)y = (-3)24
3x + 5y = 100
-3x - 3y= -72
Add the two functions to isolate y
2y= 28 y=14
Plug y back into original and solve
3x +5(14) =100 --> 3x + 70 =100 --> 3x = 30 x=10
twice a certain number is 58. 4 times that number will be
a) 4x58 b) 58 + 4 c) 58x2 d)8x58
Answer:
c) 58x2
Step-by-step explanation:
Twice the number 29 is 58
So, 29 is related to 58
Therefore 29*4 = 58*2
The number of Minghua's stamps is 1 1/2 times that of John's.
Step-by-step explanation:
Let the number of stamps of John be x.
Then number of Minghua's stamps:
1 1/2 * x3/2 * x(3/2)xJohn's: x
Minghua's: (3/2)x
For a certain bathtub, the hot water faucet can fill the tub in 12 minutes. The cold water faucet can fill the tub in 8 minutes. If both faucets are used together, how long will it take to fill the tub
If both faucets are used together, it time taken by them to fill the tub is 4.8 minutes.
What is the concept of time and work?Time and work share a fundamental idea in common with other arithmetic topics, namely the idea of proportionality.
When the amount of work is constant, efficiency is inversely proportional to the length of time required.
Important Time and Work Formula:
Work Done = Time Taken × Rate of Work.Rate of Work = 1 / Time Taken.Time Taken = 1 / Rate of Work.If a piece of work is done in x number of days, then the work done in one day = 1/x.Total Wok Done = Number of Days × Efficiency.Efficiency and Time are inversely proportional to each other.Calculation for the time taken by both faucet to fill the tub.
The unit should be tub per minute.
Then, do the reciprocal of the result to have minute per tub.
The hot water faucet can fill the tub in 12 minutes.
The cold water faucet can fill the tub in 8 minutes.
The filling rate for both faucets filling the tub at the same time is the sum of their rates.
Let 't' be the time taken by both faucets.
Then,
[tex]\frac{1}{12} +\frac{1}{8} =\frac{1}{t}[/tex]
[tex]\frac{8+12}{8*12} =\frac{1}{t}[/tex]
[tex]\frac{1}{t} =\frac{20}{96}[/tex]
[tex]\frac{t}{1} =\frac{96}{20}[/tex]
t = 4.8
Therefore, the time taken by both faucets to fill the tub is 4.8 minutes.
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Point W is on line segment \overline{VX} VX . Given VX=4x,VX=4x, VW=3x,VW=3x, and WX=5,WX=5, determine the numerical length of \overline{VX}. VX .
The numerical length of VX is undefined
How to determine the numerical length of VX?The given parameters are
VX = 4x
VW = 3x
WX = 5x
Point W is on line segment VX
So, we have
VX = VW + WX
This gives
4x = 3x + 5x
Evaluate
4x= 8x
The above equation is false
Hence, the numerical length of VX is undefined
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Maria is creating a round stained glass window like the one below. She needs to cut the dark pink glass to the correct size to fit into the frame. At what angle (x) should she cut the glass so that it fits if she knows the arc measure of a is 35º and the arc measure of b is 45º. What does x equal?
Applying the angle of intersecting chords theorem, the value of x is: 40º.
What is the Angle of Intersecting Chords Theorem?According to the angle of intersecting chords theorem the following equation is true:
x = 1/2(a + b).
Given the following:
a = 35º
b = 45º
Plug in the values
x = 1/2(35 + 45)
x = 1/2(80)
x = 40º
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The function f is defined by f(x)=2-x. If 2f(p)=8, what is f(2p)?
If function 'f' is defined by f(x)=2-x and 2f(p)=8 then the value of f(2p) is 6.
Given, f(x) = 2-x
Solve this problem by making a series of substitutions.
First, find the value of p.
Then use the definition of the function f to find the value of f(2p).
We are given
f(x) = 2 - x
Multiply both sides by 2, and simplify.
2f(x) = 2(-x +2)
2f(x) = -2x + 4
Let x = p.
2f(p) = -2(p) + 4
We are given that 2f(p) = 8.
Substitute 20 for 2f(p).
8 = -2p + 4
2p = -4
p = -2
Now we know that p = -2, so we can find f(2p).
f(x) = -x + 2
f(2p) = -2p + 2
Substitute -2 for p.
f(2p) = -2(-2) + 2
f(2p) = 4 +2
f(2p) = 6
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give the answer please
Using the domain concept, the only value of x that is not on the domain of [tex]f(x) = \frac{1}{3x + 5}[/tex] is given as follows:
[tex]x = -\frac{5}{3}[/tex]
What is the domain of a function?The domain of a function is the set that contains all possible input values for the function. For a rational function, for example, the zeros of the denominator are excluded from the domain. Another example is an even root, where the inside term has to be positive.
The rational function for this problem is given by the function f(x) presented below:
[tex]f(x) = \frac{1}{3x + 5}[/tex]
The denominator is 3x + 5, hence the restriction in the domain of f(x) is given as follows:
3x + 5 = 0
3x = -5
[tex]x = -\frac{5}{3}[/tex]
The solution to the equation above is the only value of x that is not part of the domain of f(x).
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URGENT WHOEVER ANSWERS CORRECTLY AND FAST GETS BRAINLY
40% of the house plants were watered today. 320 house plants were watered. Which equation can be used to find how many plants there were?
40x = 320
0.40x = 320
x
=
0.40
--------
320
x = 0.40(320)
Answer:
0.40x = 320
Step-by-step explanation:
If 40% of the total houseplants watered = 320
0.40 times x (total houseplants) = 320
Answer:
[tex]\frac{x}{0.40} = 320 Plants[/tex]
Step-by-step explanation:
[tex]\frac{x}{0.40} = 320 Plants[/tex]
[tex]X = (0.40)(320)Plants[/tex]
[tex]X = 128Plants[/tex]
A survey was conducted in the class about their favorite subjects. 35% of the class voted for Math and Science. 45% of the class voted for Math. Find the percent of the class who voted for Math also voted for Science.
The percent of the class who voted for Math also voted for Science is 38.8%
How to find the percent who voted math and also voted for science?Therefore,
1 = p(m) + p(s) - p(m ∩ s)
Hence,
p(m) = 45% = 0.45
p(s) = ?
p(m ∩ s) = 35% = 0.35
Therefore,
1 = 0.45 + p(s) - 0.35
1 - 0.45 + 0.35 = p(s)
p(s) = 0.9
Therefore,
p(m | s) = 0.35 / 0.9 = 0.38888888888 = 38.8%
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The programmer at a local public radio station is compiling data on listeners who stream podcasts. an executive for the national broadcaster has communicated that the overall population mean is 20,500 with a standard deviation of 2,180. the local programmer has a sample of 30 podcasts for her station. by the central limit theorem, which interval do about 99.7% of the sample means fall within?
The interval in which 99.7% of the sample means fall within 3 standard deviations of the mean is 19,306 to 21,694.
What is the central limit theorem?According to the central limit theorem,
Sample mean = Population mean = μ
Sample standard deviation = (Population standard deviation)/√n = σ/√n
where n is at least 30.
Calculation:It is given that,
The overall population mean = 20,500
Population standard deviation σ = 2180
Sample size n = 30
Since n = 30, the central limit theorem can be applied.
According to the Empirical rule(68-95-99), 99.7% of the sample means fall within the 3 standard deviations of the mean.
So, the interval is calculated by
Sample mean = Population mean = μ = 20500
Standard deviation(sample) = σ/√n
⇒ S = 2180/√30 =398.011 ≅ 398
Thus,
The lower bound = 20500 - 3 × 398 = 19,306
The upper bound = 20500 + 3 × 398 = 21,694
Therefore, the required interval is from 19,306 to 21,694.
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Jay went to an amusement park. the park charges an entrance fee of $10.50 and $4.50 for every ride. jay spent $46.50 on entrance fees and rides. which fuction can be used to find the number of rides he went on? f (x) = 4.5 x + 10.5 f (x) = 4.5 x f (x) = 4.5 x minus 10.5 f (x) = negative 4.5 x
The function that can be used to find the number of rides Jay went on is, f(x) = 4.5x + 10 = 46.5. Hence, the first option is the right choice.
Also, on solving the function we get that, Jay went on 8 rides.
We assume the number of rides Jay spent to be x.
The cost of each ride is given to be $4.50.
Thus, the total amount Jay spent on rides = $4.50x.
The entrance fee to the park is given to be $10.50.
Thus, the total expenditure of Jay can be shown as $(4.5x + 10.5).
But, in the question we are given that Jay spend $46.50 altogether.
Thus, the function to find the number of rides Jay went on can be shown as:
f(x) = 4.5x + 10 = 46.5.
Hence, the first option is the right choice.
To find the number of rides he went, we can solve this function as:
4.5x + 10 = 46.5,
or, 4.5x = 46.5 - 10.5 = 36,
or, x = 36/4.5 = 8.
Thus, Jay went on 8 rides.
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the cost of 4 ice cream cones from the snack bar was $10.40. if each ice cream costs the same amount, which model correctly illustrates the relationship? check all that apply
Solve 5x=125 using the one-to-one property of exponents.
A) X=In(125)
B) x = 1
C) x = 3
OD) x = 5
Anyone is here
Answer:
Step-by-step explanation:
hello :
A) X=In(125)
Consider the function f(x)=x2 bx c. if the minimum of this function is located at point (2,0), then what are the values of b and c?
The values of b and c are -4 and 4 respectively
How to determine the values of b and c?The function is given as:
f(x) = x^2 + bx + c
Differentiate f(x)
f'(x) = 2x + b
Set to 0
2x + b = 0
Solve for b
b = -2x
The minimum is (2, 0).
So, we have:
b = -2 * 2
b = -4
Substitute b = -4 in f(x) = x^2 + bx + c
f(x) = x^2 - 4x + c
Substitute (2, 0)
0 = (2)^2 - 4(2) + c
This gives
0 = 4 - 8 + c
Evaluate
c = 4
Hence, the values of b and c are -4 and 4 respectively
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Given: Circle X with Radius r and circle Y with radius s
Prove: Circle X is similar to Circle Y
It can be proved that Circles X and Y are similar.
How can we prove that the circles are similar?We were told that circle Y have a radius of s
And X have the Radius r
Going with the theorem that Similar shapes are proportional, we can find the scale factor from circle X to circle Y by making a division on the radii of both circles.
Then the scale factor is [tex]\frac{s}{r}[/tex] which implies that circle Y will gotten by dilation of circle Y.
Therefore, both circles are similar
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Helen had $12 but spent $3 of it. What per cent did she spend?
12%
25%
30%
33 1/3%
Answer:
25%
Step-by-step explanation:
3/12 = 0.25 = 25%
Answer:
25%
Step-by-step explanation:
$3 pent out of the original $12. We can ask what is the percentage of $12 that is equal to $3? This can be written as:
X($12) = $3, where X is the percentage.
X = (3/12)
X = 0.25 or 25%
Decide whether the triangles are similar. if so, determine the appropriate expression to solve for x. triangles abc and edf; triangle abc has angle a measuring 53 degrees, angle c measuring 62 degrees, side ac labeled as y, side ab labeled as w, and side bc labeled as x; triangle edf has angle d measuring 61 degrees, angle f measuring 53 degrees, side de labeled z, side ef labeled u, and side df labeled r.
Answer:
Step-by-step explanation:
Similar Triangle:
Two triangles are comparable if they have the same ratio of corresponding sides and equal pairs of corresponding angles.If more figures have the same shape, but their sizes are different, then such objects are referred to as similar figures. Consider a hula hoop and wheel of a cycle, the shapes of each of those gadgets are much like every other as their shapes are equal.
Triangle ABC has angles of 53 and 62 degrees....the sum is 115 degrees. The third angle must be 180 - 115 = 65 degrees.
A triangle with one angle = 61 degrees cannot be similar
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Answer: The triangles are not similar; no expression for x can be found.
5. The two rectangles are similar. Which is the correct proportion that can
be used to find the missing side?
12/4=x/20
12/4=x/8
12/8=x/4
4/12=x/8
Answer:
The second choice.
Step-by-step explanation:
Correspond sides are in the same ratio
12/4 = x/8
.
Given the series below, find S29.
{13+4+(−5)+(−14)+(−23)+...}
Select one:
a. -2,480
b. -2,752
c. -3,016
d. -3,277
Answer:
d
Step-by-step explanation:
there is a common difference between consecutive terms in the series, that is
4 - 13 = - 5 - 4 = - 14 - (- 5) = - 23 - (- 14) = - 9
this indicates the series is arithmetic with sum to n terms
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]
where a₁ is the first term and d the common difference
here a₁ = 13 and d = - 9 , then
S₂₉ = [tex]\frac{29}{2}[/tex] [ (2 × 13) + (28 × - 9) ]
= 14.4(26 + (- 252))
= 14.5(26 - 252)
= 14.5 × - 226
= - 3,277
find the volume of a sphere with a raduis of 3 in.
Answer:
V≈113.1in³.
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The graph of the piecewise function f(x) is shown.
On a coordinate plane, a piecewise function has 2 connecting lines. The first line has a closed circle at (negative 2, negative 5) and goes up to a closed circle at (2, negative 1). The second line has a closed circle at (2, negative 1) and goes down to an open circle at (4, negative 2).
What is the range of f(x)?
{x | −2 ≤ x < 4}
{x | −2 < x ≤ 4}
{y | −5 < y < −1}
{y | −5 ≤ y ≤ −1}
Answer: {y | −5 ≤ y ≤ −1}
Step-by-step explanation:
The extreme values are -5 and -1.
There is a closed circle at (-2, -5), so -5 is in the range.There is a closed circle at (2, -1), so -1 is in the range.So, the answer is {y | −5 ≤ y ≤ −1}.