A container in the shape of a right circular cylinder is 1/2

Question

A container in the shape of a right circular cylinder is 1/2 full of water. If the volume of water in the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of the base of the cylinder, in inches?

(A) \(\frac{16}{9\pi}\)

(B) \(\frac{4}{\pi}\)

(C) \(\frac{12}{\pi}\)

(D) \(\sqrt{\frac{2}{\pi}}\)

(E) \(4\sqrt{\frac{2}{\pi}}\)

(A)  \frac{16}{9\pi}

(B)  \frac{4}{\pi}

(C)  \frac{12}{\pi}

(D)  \sqrt{\frac{2}{\pi}}

(E)  4\sqrt{\frac{2}{\pi}}

Bunuel · Accepted Answer

SOLUTION

A container in the shape of a right circular cylinder is 1/2 full of water. If the volume of water in the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of the base of the cylinder, in inches?

(A) \(\frac{16}{9\pi}\)

(B) \(\frac{4}{\pi}\)

(C) \(\frac{12}{\pi}\)

(D) \(\sqrt{\frac{2}{\pi}}\)

(E) \(4\sqrt{\frac{2}{\pi}}\)

(A)  \frac{16}{9\pi}

(B)  \frac{4}{\pi}

(C)  \frac{12}{\pi}

(D)  \sqrt{\frac{2}{\pi}}

(E)  4\sqrt{\frac{2}{\pi}}

Since 36 cubic inches of water occupy 1/2 of the cylinder, then the volume of the cylinder is 72 cubic inches.

So, we have that  volume_{cylinder}=\pi*{r^2}*h=72  -->  \pi*{r^2}*9=72  -->  r^2=\frac{8}{\pi}  -->  r=\sqrt{\frac{8}{\pi}}=2*\sqrt{\frac{2}{\pi}} . Hence the diameter equals  2*(2*\sqrt{\frac{2}{\pi}})=4*\sqrt{\frac{2}{\pi}} .

Answer: E.

Hope it's clear.

TechWithNoExp · Answer

the answer is E.

The volume of the cylinder is = hr^2(Pie).
If Half of the volume is equal to 36 than the full volume is 72.
since h =9 divide 72/9(pie) u get r^2=8/Pie or r^2=(4*2)/pie.
so r= 2*Sqrt(2/pie)
Since 2r = diameter => diameter = 4*Sqrt(2/pie)

DonCarter · Answer

Hi

Could someone simplify it. I don't get it.

What I understand so far is that the container is half full at 36 cubic = full 72 cubic
Height of container is 9 inches = so container with half full is at 4.5 inches height...

72 = 9
36 = 4.5

after this I get stuck with the explanations that have been given....

Thanks

Bunuel · Answer

You don't need to for half-full container after you get that the volume of the whole container is 72 cubic inches.

We have that the volume is 72 cubic inches and the height is 9 inches -->  volume_{cylinder}=\pi*{r^2}*h=72  -->  \pi*{r^2}*9=72  -->  r^2=\frac{8}{\pi}  -->  r=\sqrt{\frac{8}{\pi}}=2*\sqrt{\frac{2}{\pi}} . Hence the diameter equals  2*(2*\sqrt{\frac{2}{\pi}})=4*\sqrt{\frac{2}{\pi}} .

Hope it's clear.

Bunuel · Answer

Responding to a pm:

As I wrote in my post "you don't need to go back to half-full container after you get that the volume of the whole container is 72 cubic inches".

The volume of a cylinder is given by:  volume_{cylinder}=\pi*{r^2}*h . We know that it equals to 72 and its height is 9. Substitute these values to get  \pi*{r^2}*9=72  --> divide by 9:  \pi*{r^2}=8  --> divide by  \pi :  r^2=\frac{8}{\pi}  --> take the square root:  r=\sqrt{\frac{8}{\pi}}=2*\sqrt{\frac{2}{\pi}} . Diameter is twice the radius, thus  d=2*(2*\sqrt{\frac{2}{\pi}})=4*\sqrt{\frac{2}{\pi}} .

Hope it helps.

kck · Answer

How do you reach the square root of 8 over pie being = to 2 multiplied the square root of 2 over pie?

Bunuel · Answer

Welcome to the club!

r=\sqrt{\frac{8}{\pi}}=\sqrt{\frac{4*2}{\pi}}=2*\sqrt{\frac{2}{\pi}} .

Does this make sense?

kck · Answer

Thank you for the welcome and yes it does! Now it seems so simple. Thanks again buddy!

aeglorre · Answer

They give us cubic inches ----> 36in^3 and then want us to find find a value that is in inches ---> in^1 .. Why are they not consistent? If they talk about cubic inches then they should also take the cubic root out on both sides... I don't get it.

Bunuel · Answer

Volume is in cubic inches and the length is in inches. How can length be in cubic inches?

aeglorre · Answer

Wow, that is embarassing. Thanks for the heads up, Ill just blame it on mind fatigue!

WholeLottaLove · Answer

A container in the shape of a right circular cylinder is 1/2 full of water. If the volume of water in the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of the base of the cylinder, in inches?

I've gotten this question wrong before and I'm still not 100% sure why it's solved the way it is.

If the container is 1/2 full of water and the volume of water is 36 when the total volume is twice that, or 72 inches.

V = pi * r^2 * h
72 = pi * r^2 * (9)
8 = pi * r^2
8/pi = r^2
√8 / √pi = r
d = 2r
d = 2(√8 / √pi)
d = 2√8 / √pi
d = (2 * √8 * √pi) / pi

:oops:

Bunuel · Answer

The red part can simplified:  r=\sqrt{\frac{8}{\pi}}=\sqrt{\frac{4*2}{\pi}}=2*\sqrt{\frac{2}{\pi}} .

WholeLottaLove · Answer

Ahh...simple mistake! Thanks!

quote="Bunuel"]

The red part can simplified:  r=\sqrt{\frac{8}{\pi}}=\sqrt{\frac{4*2}{\pi}}=2*\sqrt{\frac{2}{\pi}} .

zerosleep · Answer

I understand how it is done above. But can someone help me explain what is wrong with below -
Given is a right circular cylinder, which is half full.
V/2 = 36. Also height will be half i.e. 4.5 inches when the culinder is half full.
Hence, (pi*r^2*4.5)/2 = 36. This gives radius = 4/sqrt(pi). This is wrong though.

I am not quite sure what is wrong with this. Can someone explain ?

Bunuel · Answer

It should be  \pi{r^2}*4.5=36 , no need to divide by 2, since you already accounted for half of the volume by dividing the height by 2.

Hope it's clear.

ScottTargetTestPrep · Answer

Recall that the volume of a cylinder is:

volume = π(radius)^2(height)

Since half of the capacity of the cylinder is 36, the full capacity of the cylinder is 72; thus:

72 = πr^2(9)

8/π = r^2

√(8/π) = r

√8/√π = r

(2√2)/√π = r

2√(2/π) = r

The diameter is twice the radius. Thus, the diameter is 2 x 2√(2/π) = 4√(2/π).

Answer: E

dabaobao · Answer

Volume = 36 = 1/2π*r^2 * 9 
r = sqrt(4*2/π) = 2sqrt(2/π)

Diameter = 4sqrt(2/π)

ANSWER: E

EMPOWERgmatRichC · Answer

Hi All,

We’re told that a cylindrical tank (re: a cylinder/tube) is HALF full of water and contains 36 cubic inches of water (with the height of the CONTAINER equaling 9 inches). We’re asked for the DIAMETER of the base of the cylinder in inches.

Volume of a cylinder is (pi)(Radius^2)(Height), so we can use that formula – along with what we know about the water – to figure out the radius and diameter of the tank… It’s important to note that while the Height of the cylinder is 9 inches, the water reaches HALF of that height (meaning that the height of the water is 4.5 inches). Since the answers are all written as fractions, we should write that 4.5 as 9/2…

V = (pi)(R^2)(H) = 
36 = (pi)(R^2)(9/2)

We can ‘cancel out’ the 9/2 by multiplying both sides by 2/9…

72/9 = (pi)(R^2)
8/pi = R^2

R = √(8/pi), which can be rewritten as 2√(2/pi). Since the diameter is TWICE the radius, the diameter is 2(2√2/pi) = 4√(2/pi)

Final Answer:  E

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A container in the shape of a right circular cylinder is 1/2

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