SAT Area & Volume Question 45 | Free SAT Prep | Aniko

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Question 45·Hard·Area and Volume

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A vertical right circular cylindrical tank has an interior radius of 5 feet and initially contains water to a depth of 8 feet. A solid metal sphere with radius 3 feet is then completely submerged in the tank, and no water spills out.

By how many feet does the water level in the tank rise?

For volume-displacement problems, first recognize that the object’s volume equals the volume of fluid it displaces. Compute the object’s volume using the appropriate formula, then write the fluid’s volume in a simple form like (base area) × (height change). Set these two volumes equal, cancel common factors such as $\pi$ to simplify, and solve the resulting one-step equation quickly for the unknown height or dimension.

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Relate the sphere to the water

When you submerge the metal sphere, how is the amount of water that moves (is displaced) related to the sphere’s volume?

Think about the shape of the extra water

The water level rises uniformly across the tank. What 3D shape does that extra layer of water form, and how do you write its volume in terms of the tank’s radius and the rise in height $h$ ?

Set up an equation with volumes

Write an equation that sets the volume of the sphere equal to the volume of the extra cylindrical layer of water, then solve for the unknown height $h$ .

Desmos Guide

Compute the sphere’s volume

In Desmos, type (4/3)*pi*3^3 to calculate the volume of the sphere. This gives the displaced water volume in cubic feet.

Compute the rise in water height

In a new Desmos line, enter ((4/3)*pi*3^3)/(25*pi) to divide the sphere’s volume by the cylinder’s base area $25\pi$ . The numeric result is the rise in the water level in feet; read that value from Desmos.

Step-by-step Explanation

Model the water rise as a cylinder

When the metal sphere is submerged, it displaces water equal to its own volume.

Let the rise in water level be $h$ feet. The extra water forms a cylindrical layer on top of the original water with:

radius $5$ feet (same as the tank)
height $h$ feet

So the volume of this extra cylindrical layer is the base area times the height, or $\text{(base area)} \times h$ .

Find the volume of the sphere

Use the volume formula for a sphere:

V_{\text{sphere}} = \frac{4}{3}\pi r^3

Here, $r = 3$ feet, so:

V_{\text{sphere}} = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi (27) = 36\pi

So the sphere’s volume (and thus the water displaced) is $36\pi$ cubic feet.

Express the volume of the risen water in the cylinder

The base of the tank is a circle of radius $5$ feet, so its area is:

A_{\text{base}} = \pi (5)^2 = 25\pi

If the water level rises by $h$ feet, the volume of this extra cylindrical layer of water is:

V_{\text{cylinder layer}} = 25\pi \cdot h

Set volumes equal and solve for the rise in height

The water displaced by the sphere equals the volume of the extra cylindrical layer:

25\pi h = 36\pi

Divide both sides by $\pi$ :

25h = 36

Now solve for $h$ :

h = \frac{36}{25}

So the water level in the tank rises by $\dfrac{36}{25}$ feet.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation