Question 142·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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A solid steel object is formed by drilling a cylindrical hole through the center of a larger right circular cylinder.

The larger cylinder has radius 4 centimeters and height 15 centimeters.
The drilled hole is a right circular cylinder with radius 2 centimeters and the same height.
The density of steel is 7,800 kilograms per cubic meter.

To the nearest whole number, what is the mass of the object, in kilograms?

A drilled hole signals composite volume: find the larger cylinder’s volume, then subtract the volume removed. Use $V=\pi r^2h$ and let Desmos evaluate the difference. Because the density is in kilograms per cubic meter, convert the remaining volume to cubic meters before multiplying. Using a length conversion only once makes the mass far too large.

Hints

Hint 1
A drilled hole removes material. Find each cylinder’s volume with $V=\pi r^2h$ , then subtract the smaller volume from the larger. Both cylinders have the same height; which radius belongs in each term?
Hint 2
The dimensions are in centimeters, so the volume you find will be in cubic centimeters. A cubic meter is $100$ centimeters long in each of three directions. How many cubic centimeters are in one cubic meter?
Hint 3
Density means mass per unit of volume. Once the steel’s volume is in cubic meters, multiply by kilograms per cubic meter so the volume units cancel. Keep the full calculator value and round the mass last.

Step-by-step

Subtract, convert, then multiply by density
Step 1
Write the volume of steel
The hole takes steel away. A cylinder’s volume, the space it occupies, is $V=\pi r^2h$ , so use the outer radius for the large cylinder and the hole’s radius for what was removed:
$V_{\text{steel}}=\pi(4)^2(15)-\pi(2)^2(15)\text{ cm}^3.$
Step 2
Calculate the remaining volume
Type $v=\pi(4)^2(15)-\pi(2)^2(15)$ , where $v$ is the steel volume in cubic centimeters. Desmos shows $v\approx565.487$ . This is the volume left after drilling, not the volume of the hole.
Step 3
Find the cubic-unit conversion
One meter is $100$ centimeters. A cubic meter has that length in three directions, so cube the length conversion for volume:
$1\text{ m}^3=100^3\text{ cm}^3=1{,}000{,}000\text{ cm}^3.$
Step 4
Convert the steel volume to cubic meters
Since one cubic meter contains $100^3$ cubic centimeters, divide the steel volume by $100^3$ . Add $w=\frac{v}{100^3}$ in Desmos, where $w$ is the volume in cubic meters. It shows $w\approx0.000565487$ .
Step 5
Find and round the mass
Density is mass per unit of volume. Multiply $w$ cubic meters by $7{,}800$ kilograms per cubic meter; the cubic meters cancel. Type $p=7800w$ in Desmos. It shows $p\approx4.410796$ kilograms, so round only now. The object’s mass is $4$ kilograms. Choice B.

Question 142·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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