PWN The SAT Book – Volume Chapter – Question 10

I am having hard time understanding how to differentiate if the question wants you to answer in feet or inches, since the last part asks : “How much does the water level rise, “in inches”, when Priya places the stone into the tank.

But the answer is 0.125 inches instead of 216 inches.

Shouldn’t the water displaced by a cube of sides 6 inches be 216 inches instead of 0.125 inches ?

Thanks

Here’s the question:

I think you’re getting confused between inches (a measure of length) and cubic inches (a measure of volume). A cube with 6 inch sides will have a volume of 6 inches × 6 inches × 6 inches = 216 cubic inches (also written \text{in}^3), but that doesn’t mean that placing a 6-inch-sided cube in a tank of water will make the water rise 216 inches (18 feet!).

To solve this question, you need to consider the volume of water in the tank and how much of that water will be displaced by the stone. Since the question is asking about inches of rise, let’s deal only in inches when we do our calculations, so instead of a 3-foot by 4-foot by 2-foot tank, we’re dealing with a 36-inch by 48-inch by 24-inch tank.

The tank begins with water filled up to the 12-inch mark, which means the volume of water in the tank to start is:

36\text{ in}\times 48\text{ in}\times 12\text{ in}=20,736\text{ in}^3

As we’ve already discussed, we’re displacing 216\text{ in}^3 when we add the stone. Since the cube will be completely submerged, the easier way to think about that is that now the volume in the tank is 20,736+216=20,952\text{ in}^3.

The width and length of the tank don’t change, so we can calculate the new height of the water in the tank, x, thusly:

36\text{ in}\times 48\text{ in}\times x\text{ in}=20,952\text{ in}^3

x\text{ in}=\dfrac{20,952\text{ in}^3}{36\text{ in}\times 48\text{ in}}

x\text{ in}=12.125\text{ in}

Since the height of the water before the stone was added was 12 inches, the water level only rises 0.125 inches.