The radius of a sphere is increasing at a rate of 5 mm/s. how fast is the volume increasing when the diameter is 60 mm? evaluate your answer numerically. (round the answer to the nearest whole number.)

Answers

Answer 1

The diameter of the sphere is 60 mm which means its radius is 30mm. The radius of a sphere is increasing at a rate of 5 mm/s.
The formula for the radius when the time is x second would be:
r(t)= 30+ 5x
dr/dt= 5

The speed of volume increase of the sphere volume would be derived from sphere volume:
V(r)=4/3* pi * r^3
dV/dt= 4/3 * pi * 3r^2 * (dr/dt) =4/3* pi * 3(30)^2 * 5 = 18000 pi= 56571 mm^3/s

Answer 2

The rate of change of volume of the sphere is [tex]56520\;\rm{mm/sec}[/tex].

According to the question, the radius of a sphere is increasing at a rate of [tex]5 mm/s[/tex] and the diameter is [tex]60 mm[/tex].

The volume of the sphere is, [tex]V=\dfrac{4}{3}\pi r^3[/tex] where, [tex]r[/tex] is the radius of the sphere.

Differentiate the volume of the sphere both sides with respect to [tex]t[/tex] as-

[tex]\dfrac{dV}{dt}=\dfrac{4\pi}{3}\dfrac{dr^3}{dt}\\\dfrac{dV}{dt}=\dfrac{4\pi}{3}\times 3r^2\dfrac{dr}{dt}\\\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}[/tex]

The radius of the sphere is-

[tex]r=\dfrac{60}{2}\\r=30mm[/tex]

Substitute the value of the parameter as-

[tex]\dfrac{dV}{dt}=4\pi\times (30)^2\times 5\\\dfrac{dV}{dt}=62.8\times 900\\\dfrac{dV}{dt}=56520\;\rm{mm/sec}[/tex]

Hence, the rate of change of volume of the sphere is [tex]56520\;\rm{mm/sec}[/tex].

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Related Questions

a pyramid has a square base. Its volume is 48 cubic centimeters and its height 4 centimeters. What is the length of one side og the base?

Answers

V = 1/3 B * h
B = ???
h = 4 cm
V = 48 cm^3
48 = 1/3 * B * 4 Multiply both sides by 3
48 * 3 = B * 4
144 = B * 4 Divide by 4
144/4 = B
36 = B
Since the base is a square the area of the base = s^2
s^2 = 36
Sqrt(s^2) = sqrt(36)
s = 6 cm <<<<==== answer.

A square playground has an area of 175 m2. What is the approximate length of each side of the playground? Round your answer to the nearest meter

Answers

We know that the area of a square = side ²

Therefore, for a known area A, the side length is
side = √ (area)
= √ (175)
= 13.2288 (to 4 places of decimals).

If the area of a square playground is 175 [tex]m^{2}[/tex] then the approximate length is 13 meters.

What is square?

A square is a two dimensional figure having four vertices, four edges, four angles and all the sides are equal to each other. The perimeter is equal is equal to 4*side and the area is side*side.

How to find side of square?

We have been given the area of the square playground equal to 175 meter square. We know that the area of a square is side*side means [tex]side^{2}[/tex].

Put the value of 175 equal to side square and we will get the value of slide.

Area =[tex]side^{2}[/tex]

175=[tex]side^{2}[/tex]

side=[tex]\sqrt{175}[/tex]

Side=13.22m

If we round to nearest meter then it will be equal to 13m.

Hence if the area of a square playground is 175 meter square then the side will be equal to 13m.

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A 14.5-gallon gasoline tank is 3434 full. How many gallons will it take to fill the tank? Write your answer as a mixed number.

Answers

This is a multiplication of fractions. If the tank is 3/4 full, and the tank holds 14.5 gal, then we need to multiply 14.5 * 3/4 First, convert 14.5 to 14 1/2 = 29/2 as an improper fraction Now, we have 29/2 * 3/4 = 87/8 = 10 7/8 gal That's how much gas is in the tank now. However, we want to know how much is needed to fill the tank. That can be determined by subtracting 10 7/8 from the amount that the tank can hold when full, which is 14.5 or 14 1/2. 14 1/2 - 10 7/8 can be re-written using improper fractions 29/2 - 87/8 We need a common denominator, which is 8 116/8 - 87/8 = 29/8 = 3 5/8 gal That's how much is needed to fill the tank

and are similar. Find the value of x. A. 5 B. 15 C. 60 D. 240

Answers

Dude we need to actualy see the question all we see is find the value of x but we don't know our other value?

Find a vector equation and parametric equations for the line segment that joins p to q. p(1, −1, 7), q(7, 6, 1) vector equation r(t) = <1+6t,−1+7t,7−6t> parametric equations (x(t), y(t), z(t)) =

Answers

Answers:

- vector equation: r(t) = <1 + 6t, -1 + 7t, 7 - 6t>
- parametric equations:
x = 1 + 6t
y = -1 + 7t
z = 7 - 6t

Explanation:

To obtain the vector equation, we first get a vector v that is parallel to the line. To get the vector v, we subtract p from q. So,

v = q - p
= (7,6,1) - (1,-1,7)
v = (6, 7, -6)

The vector equation of the line is given by

[tex]r(t) = v_0 + tv[/tex]

Where

[tex]v_0[/tex] = a point in the line (we choose point p(1,-1,7))

So, the equation of the line joining p and q is given by

[tex]r(t) = v_0 + tv \\ \indent = \left \langle 1, -1, 7 \right \rangle + t\left \langle 6, 7, -6 \right \rangle \\ \indent = \left \langle 1, -1, 7 \right \rangle + \left \langle 6t, 7t, -6t \right \rangle \\ \indent \boxed{r(t) = \left \langle 1 + 6t, 1 + 7t, 7 - 6t \right \rangle }[/tex]

In the parametric equation of the line, we just need to get the x, y and z coordinates in the vector equation.

Since the vector equation is given by

[tex]r(t) = \left \langle 1 + 6t, 1 + 7t, 7 - 6t \right \rangle[/tex]

The parametric equations of the line are given by:

[tex]x(t) = 1 + 6t \\y(t) = 1 + 7t \\z(t) = 7 - 6t[/tex]

Final answer:

The vector equation for the line segment joining the points p(1, -1, 7) and q(7, 6, 1) is r(t) = <1+6t, -1+7t, 7-6t> and the corresponding parametric equations are x(t) = 1 + 6t, y(t) = -1 + 7t, z(t) = 7 - 6t.

Explanation:

To find the vector and parametric equations for the line segment joining two points p(1, -1, 7) and q(7, 6, 1), we first need to understand that the vector equation for a line segment in space is given by r(t) = p + t (q - p), where 0 ≤ t ≤ 1, and p and q are the coordinates of the points. The parametric equations are obtained by expressing the x, y, and z coordinates of r(t) as individual functions of t.

Substituting the given points into the vector equation we get: r(t) = <1+6t, -1+7t, 7-6t>. Then, the corresponding parametric equations will be x(t) = 1 + 6t, y(t) = -1 + 7t, and z(t) = 7 - 6t.

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can someone please help me with this equation? thank you.

Answers

I found it simplest to start with the given line in standard form. That is
.. x +2y = 3 . . . . . . . . divide the given equation by 3

Now, the perpendicular line is easily written as
.. 2(x -(-4)) -(y -2) = 0 . . . . . swap x, y coefficients, negate one. Translate to the given point

Simplifying, we have
.. 2x -y +10 = 0
Solving for y gives
.. y = 2x +10 . . . . . . . . . . . . corresponds to selection B

Can somebody please help me out ?

Answers

The area of a triangle is:
A = (1/2) * b * h
Where
b: base
h: height.
For the lower triangle we have:
A = (1/2) * (10) * (root ((8) ^ 2- (5) ^ 2))
A = 31 in ^ 2
We add this area with the area of the upper triangle:
31 in ^ 2 + 17 in ^ 2 = 48 in ^ 2
Answer:
The quadrilateral area will be:
48 in ^ 2 (option 3)

for what values of x is the rational expression below undefined? x-7/x^2-7x-8
check all that apply
a. 1
b. 7
c. 8
d. -8
e. -7
f. -1

Answers

we have

f(x)=(x-7)/(x²-7x-8)

resolving the quadratic equation
using a graph tool-------- > see the attached figure
x=-1 x=8
therefore
(x-7)/(x²-7x-8)=(x-7)/[(x+1)*(x-8)

the answer is -1 and 8

The Bureau of Justice Statistics reports that the number of Americans on probation increased 76% from 1980 to 1995 and that 3.09 million Americans were on probation in 1995. If the rate of increase continues in the same way for the next 15-year period, what might be the number of Americans on probation in 2010?

Answers

For this case, what we are going to do is first define variables.
x: current number of Americans.
y: number of Americans after the next 15 years.
We now write the expression to model the problem, assuming:
"the rate of increase continues in the same way for the next 15-year period".
We have then:
y = 1.76 * x
y = 1.76 * (3.09)
y = 5.44 million Americans
Answer:
the number of Americans on probation in 2010 might be:
y = 5.44 million Americans

12x3+4x3+3x+1 factor by grouping

Answers

(3x2 + 1) • (4x - 3)

I believe that is the answer, but please remember to try the question yourself!

EXPERTS/ACE/GENIUSES

Answers

the answer would be D

3^2•3^n simplifies to 3^20 what is the value of the exponent n?

Answers

Given that:
[tex]3^2 \times 3^n = 3^20 [/tex]
Solving for n.
Divide both sides by 3²
[tex] \frac{3^2 \times 3^n}{3^2} = \frac{3^{20}}{3^2} [/tex]
Simplify
[tex]3^n= 3^{20-2} \\ 3^n=3^{18} \\[/tex]
[tex]if \ a^{f(x)}=a^{g(x)} , \ then \ f(x)=g(x)[/tex]
So, n=18

Answer: n=18

how much difference is there between investing $5,000 at 4% simple interest for 5 years and investing that same amount at 4% compounded quarterly

Answers

To find the difference between the tow investments we are going to use tow formulas. Simple interest formula for our simple interest investment, and compound interest formula for our compound interest investment.
- Simple interest formula: [tex]A=P(1+rt)[/tex]
where
  [tex]A[/tex] is the final investment value
  [tex]P[/tex] is the initial investment
  [tex]r[/tex] is the interest rate in decimal form
  [tex]t[/tex] is the time in years
For our problem we know that [tex]P=5000[/tex], [tex]r= \frac{4}{100} =0.04[/tex], and [tex]t=5[/tex], so lets replace those values in our simple interest formula to find [tex]A[/tex]:
[tex]A=5000(1+(0.04)(5))[/tex]
[tex]A=5000(1.2)[/tex]
[tex]A=6000[/tex]

Now that we know the final investment value of our simple interest investment, lets use the compound interest formula to find the final investment value of the other one:
- Compound interest formula: [tex]A=P(1+ \frac{r}{n})^{nt} [/tex]
where
  [tex]A[/tex] is the final investment value
  [tex]P[/tex] is the initial investment
  [tex]r[/tex] is the interest rate in decimal form
  [tex]n[/tex] is the number of times the interest is compounded per year
  [tex]t[/tex] is the time in years
For our problem we know that [tex]P=5000[/tex], [tex]r= \frac{4}{100} =0.04[/tex], and [tex]t=5[/tex]. Since we know that the interest is compounded quarterly (each 4 months), it mean that is compounded [tex] \frac{12months}{4months} =3[/tex] times per year, so [tex]n=3[/tex]. Now that we have all the vales, lets replace them in our formula:
[tex]A=5000(1+ \frac{0.04}{3} )^{(3)(5)} [/tex]
[tex]A=5000(1+ \frac{0.04}{3} )^{15} [/tex]
[tex]A=6098.95[/tex]

now that we know the final amounts of our investments, lets find how much difference is between them:
[tex]6098.95-6000=98.95[/tex]

We can conclude that the difference between invest $5000 in a compound interest investment vs a simple interest investment is $98.95.

simplify the complex fraction

Answers

pl see the attachment
the final answer is (y+z)/(y-z)

As surface area of a cell increases, the surface area volume ratio A) increases. B) decreases. C) remains the same. D) remains constant, then increases exponentially.

Answers

as the surface area of a cell increases the surface are volume ratio decreases so the best answer is B

B) decreases is the answer

Derek ran 3/8 of a race before he had to stop for a break. If the race is 6 miles long, did Derek run more than, less than, or equal to 6 miles before he took a break?

Answers

Derek ran 2.25 miles, less than half of the race.

The sixth grade art students are making a mosaic using tiles in the shape of right triangles. Each tile has leg measures of 3 centimeters and 5 centimeters. If there are 200 tiles in the mosaic, what is the are of the mosaic? Explain. PLEASE HELP!!!

Answers

1,600
3 TIMES 200=600
5 TIMES 200=1000
1000+600=1600

Rob cuts a 15-foot wire into 8 equal pieces. about how long is each piece?

Answers

Divide 15 with 8

15/8 = 1.875

Each piece is 1.875 foot long

hope this helps

To find out how long each piece is, you would need to divide 15 by 8 to get an answer of 1.875 feet.
I hope this helped! :-)

A zoo has three panda bears and five giraffes. What is the ratio of panda bears to giraffes at that zoo?

Answers

Hey there! :D

This is a ratio:

x:y

The ratio asks for panda bears to giraffes.

panda bears= 3

giraffes= 5

3:5 <== the ratio

I hope this helps!
~kaikers

Hey there!

The answer is 3:5

Hope this helps!

Always remember, you are a Work Of Art!
- Nicole <3 :)

The prism is completely filled with 1750 cubes that have edge length of 1/5 feet. What is the volume of the prism,

Answers

we know that
the volume one cube=(1/5)*(1/5)*(1/5)=(1/125) ft³
the volume of prism=1750*(1/125)=14 ft³

the answer is 14 ft³

A manager of a grocery store wants to determine if consumers are spending more than the national average. The national average is $150.00 with a standard deviation of $30.20. The manager collects 40 random receipts and finds that the average is $160. Complete a hypothesis test with a significance level of 2.5% to determine if the average customer spends more in his store than the national average. Which of the following is a valid conclusion for the manager based on this test?

Answers

Answer:

We conclude that consumers are spending more than the national average.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = $150.00

Sample mean, [tex]\bar{x}[/tex] = $160

Sample size, n = 40

Alpha, α = 0.025

Population standard deviation, σ = $30.20

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 150\text{ dollars}\\H_A: \mu > 150\text{ dollars}[/tex]

We use One-tailed z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{160 - 150}{\frac{30.20}{\sqrt{40}} } = 2.0942[/tex]

Now, [tex]z_{critical} \text{ at 0.025 level of significance for one tailed test } = 1.96[/tex]

Since,

[tex]z_{stat} > z_{critical}[/tex]

We reject the null hypothesis and accept the alternate hypothesis. We conclude that consumers are spending more than the national average.

Consider a binomial experiment with n = 7 trials where the probability of success on a single trial is p = 0.35. (round your answers to three decimal places.) (a) find p(r = 0). (b) find p(r ≥ 1) by using the complement rule.

Answers

a) If the probability of success on a single trial is 0.35, then the probability of failure in a single trial is 1 - 0.35 = 0.65. Out of the 7 trials, P(r = 0), the probability that there will be 0 successes out of the 7 attempts, is calculated by raising (0.65)^7 = 0.0490.
b) If the result does NOT have zero successes, then we automatically know that there must have been at least 1 successful result (it could be anything from 1 to 7). Therefore if the probability of having 0 successes is 0.0490, then the probability of having at least 1 is 1 - 0.0490 = 0.951.

Answer:

yes

Step-by-step explanation:

the family arcade center charges an entry fee of 11$ plus an additional 1$ per game played

Answers

The answer would be your first option C(x) = $11x + $1÷x

Answer:

Option (c) is correct.

Average expense is given by function [tex]C(x)=\frac{\$11+\$1x}{x}[/tex]

Step-by-step explanation:

Given : The family arcade center charges an entry fee of 11$ plus an additional $1 per game played.

We have to choose from the given options a function that describes the average expense for a player per game played.

Let x be the number of game played.

So, The entry fee will be same for all that is $ 11

And in addition $1 is charged per game played.

when they play x games the charge will be $1 × x = $ x

Thus,

Average expense is defined as the total charge divided by number of player per game played.

That is Total charge per person = $ 11+ $1x

Since, there are x game played.

Thus, Average expense is given by function

[tex]C(x)=\frac{\$11+\$1x}{x}[/tex]

6 Bands were going to play at a concert. How many ways can the concert manager send them on stage?

Answers

Try this option:
there are two ways to resolve this task:
1. the 1st in this sequence is for 6 bands, the 2d in this sequence is for 5 rest bands, the 3d in this sequence is for 4 rest bands, the 4th in this sequence is for rest 3 bands, the 5th in this sequence is for rest 2 bands and the 6th in this sequence is for the last 1 band. Using the logical rule 'AND': A=6*5*4*3*2*1=720.
2. According to arrangement rules it is the formula: A=n!, where n - number of bands.
A=6!=720.

answer: 720.

A math test took 50 minutes to complete. The test ended at 3:55. What time did the test begin?

Answers

The test started at 3:05 if it ended 50 minutes later at 3:55

A math test ended at 3:55 and it took 50 minutes to complete. What time did the math test start?

The math test started at 3:05.

If we know that the math test ended at 3:55 and lasted 50 minutes, we can subtract 50 minutes from 3:55 and we will get an answer of 3:05.

Therefore, the math test started at 3:05.

find the product 3012 and 4

Answers

product is multiplying, so it is 3012*4 which equals 12048

753. Its hard to show long division with out a paper, but if you remember divide, multiply and subtract, bring it all down and bring it all back, you'll be good. There is a really good video online to help remember it, but with this specific problem, 4 ends up going into 3012, 753 times.

Shelly has a photo that is 7 1/3 inches tall. she wants to shrink it down to fit in a picture frame that is only 5 1/3 inches tall. the photo shop can only reduce photos by certin fractions. they can reduce it to 2/3 the original size, 3/4 the original size, or 5/9 the original size. which reduction should she use so the picture fills as much of the frame as possible, without being too large?

Answers

we know that

the photo is 7 1/3 inches tall-----> 7 1/3=(7*3+1)/3=22/3 in ----- > (7.33 in)
the picture frame is 5 1/3 inches tall--> 5 1/3=(5*3+1)/3=16/3 in----> (5.33 in)

a) case A reduce 2/3 the original size------> (22/3)*(2/3)=44/9 in-----> (4.89 in)

b) case B reduce 3/4 the original size---> (22/3)*(3/4)=66/12 in-----> (5.5 in)
this is not solution because 66/12 (5.5 ) in >16/3 (5.33) in ( picture frame)

c) case C reduce 5/9 the original size---> (22/3)*(5/9)=110/27 in-----> (4.07 in)

the answer is the case A reduce 2/3 the original size----> 44/9 in (4.89 in)

plz help ill give u branlist

Answers

Remember complementary sum is 90
so...
48 + 7(x + 1) = 90
7(x + 1) = 42
x + 1 = 6
x = 5

Suppose that a weight on a spring has initial position s(0) and period P. s(0)= 1in ;p=0.8

Answers

The question is to find a s function given by s(t) = a cos(ωt) that models the displacement of the weight.

Answer:
The function is given as:
s(t) = a cos(ωt)
This means that to get the exact function modelling the given situation, we will have to get values for ω and a.

1- getting the value of ω:
period = 0.8 sec
frequency = 1/period = 1/0.8 = 5/4 hz
ω = 2πf = 2*π*5/4 = 2.5π rad/sec
The function now became:
s(t) = a cos(2.5πt)

2- getting the value of a:
We are given that:
s(0) = 1 in
This means that:
s(t) = a cos(2.5πt)
1 = a cos(2.5π*0)
1 = a cos(0)
a = 1

Therefore, the function is:
s(t) = cos(2.5πt)

To get the value of the function at any instance (t), substitute with the value of "t" in the above function and get the corresponding s(t)

Hope this helps :)

How many different possible outcomes are there when Jillana spins the spinner below and then flips the coin?

The spinner is from 1-6.

Answers

the answer is 12 because for each number on the spinner you have 2 options which the coin may land, heads or tails, so 6x2 is 12

Answer:

There are total 12 outcomes

Step-by-step explanation:

The number of outcomes in spinning a spinner is 6 (getting a number from 1 to 6)

The number of outcomes in flipping a coin=2 (head or tail)

Hence, total 12 outcomes are possible which are:

(1,H) (2,H) (3,H) (4,H) (5,H) (6,H)

(1,T) (2,T) (3,T) (4,T) (5,T) (6,T)

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