SAT Area & Volume Question 12 | Free SAT Prep | Aniko

00:00

Question 12·Hard·Area and Volume

0 / 50

A solid right circular cylinder has a radius of 8 cm and a height of 15 cm. A solid hemisphere with the same radius is attached to the top base of the cylinder so that their flat faces coincide, forming a single composite solid. The solid is completely submerged in a rectangular tank that has an interior base measuring 40 cm by 30 cm and is tall enough to accommodate the solid without overflowing.

By how many centimeters does the water level in the tank rise after the solid is submerged? Round your answer to the nearest tenth.

For composite-solid and water-displacement questions, first recognize that the water level rise equals the displaced volume divided by the base area of the tank. Break the solid into basic shapes (here, a cylinder and a hemisphere), write down the volume formula for each, and plug in the given radius and height carefully—paying attention to factors like $\frac{1}{2}$ for a hemisphere. Keep everything in terms of $\pi$ until the end to reduce rounding error, then add the volumes, divide by the base area, and round to the requested precision before choosing your answer.

Hints

Click me!

Identify the shapes and formulas

The solid is made of a right circular cylinder and a hemisphere. Recall: the cylinder volume is $\pi r^2 h$ , and the sphere volume is $\frac{4}{3}\pi r^3$ (a hemisphere is half of that).

Combine the volumes

Compute the volume of the cylinder and the volume of the hemisphere separately using radius $8$ cm, then add them to get the total volume of the solid that will be submerged.

Connect volume to water height

The tank has a rectangular base. How do you get the height of water added if you know the volume of water added and the base area ( $40 \times 30$ )? Write an equation relating volume, base area, and height.

Remember the final step: rounding

Once you find the height in centimeters, make sure to round your result to the nearest tenth before matching it to an answer choice.

Desmos Guide

Compute the total displaced volume

In the first expression line, type:

V = pi*8^2*15 + (2/3)*pi*8^3

This uses $\pi r^2 h$ for the cylinder and $\frac{2}{3}\pi r^3$ for the hemisphere. Desmos will show the numerical value of V, the total volume in cubic centimeters.

Convert volume to water height

In a new line, type:

h = V / (40*30)

This divides the displaced volume by the tank’s base area. The value of h is the water level rise in centimeters; read it from Desmos and round it to the nearest tenth to match the correct choice.

Step-by-step Explanation

Relate water level rise to displaced volume

When the solid is completely submerged, it pushes away (displaces) an amount of water equal to its own volume.

The tank has a rectangular base, so the extra water height $h$ satisfies

\text{displaced volume} = (\text{base area of tank}) \times h.

So,

h = \frac{\text{volume of solid}}{\text{base area of tank}}.

Our job is to find the volume of the composite solid, then divide by the tank’s base area.

Find the volume of the cylinder

The cylinder has radius $r = 8$ cm and height $h = 15$ cm.

Volume of a cylinder: $V = \pi r^2 h$ .

So,

V_{\text{cyl}} = \pi (8)^2 (15) = \pi \cdot 64 \cdot 15 = 960\pi \text{ cubic cm}.

Find the volume of the hemisphere and total solid

The hemisphere has the same radius $r = 8$ cm.

Volume of a sphere: $V = \frac{4}{3}\pi r^3$ .

A hemisphere is half a sphere, so its volume is

V_{\text{hemi}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3.

Substitute $r = 8$ :

V_{\text{hemi}} = \frac{2}{3}\pi (8)^3 = \frac{2}{3}\pi \cdot 512 = \frac{1024}{3}\pi \text{ cubic cm}.

Total volume of the solid is

V_{\text{total}} = V_{\text{cyl}} + V_{\text{hemi}} = 960\pi + \frac{1024}{3}\pi = \frac{3904}{3}\pi \approx 4088.3 \text{ cubic cm}.

Use the tank’s base area to find the water rise

The base of the tank is $40$ cm by $30$ cm, so its area is

A = 40 \times 30 = 1200 \text{ square cm}.

The water level rise $h$ is

h = \frac{V_{\text{total}}}{A} \approx \frac{4088.3}{1200} \approx 3.4069 \text{ cm}.

Rounded to the nearest tenth, the water level rises $3.4$ centimeters, which corresponds to choice B.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation