Question 761·Hard·Math

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A cylindrical water tank begins draining at a constant rate. At the start of draining, when $t = 0$ minutes, the tank contains 500 gallons of water. After 30 minutes, the tank contains 425 gallons. Which equation gives the volume $V$ , in gallons, of water remaining in the tank after $t$ minutes of draining?

The phrase constant rate points to a linear model: starting volume minus the same number of gallons each minute. Use the two volume readings to write an equation for the per-minute loss, then solve it with a Desmos regression. The total loss over the interval isn't the per-minute rate, and the tank's shape doesn't affect the calculation.

Hints

Hint 1
The initial value is the volume at time zero, before any water drains. Keep it separate from the amount lost as time passes.
Hint 2
A unit rate tells you how many gallons leave in one minute. If $r$ gallons leave each minute, multiply $r$ by the elapsed minutes to find the total loss.

Step-by-step

Find the per-minute loss
Step 1
Turn the readings into an equation
Let $r$ be the gallons drained each minute. At a constant rate, $30$ minutes of draining removes $30r$ gallons. The tank starts with $500$ gallons and then has $425$ , so $500-30r=425$ . The loss is subtracted because the tank is draining.
Step 2
Find the rate in Desmos
Replace $r$ with $x_1$ and $=$ with $\sim$ so Desmos finds the unknown rate. Type $500-30x_1\sim425$ . Under PARAMETERS, Desmos shows $x_1=2.5$ , meaning the tank loses $2.5$ gallons per minute. The loss across $30$ minutes is not the loss per minute.
Step 3
Write the volume after any number of minutes
Since $2.5=\frac{5}{2}$ , the tank loses $\frac{5}{2}t$ gallons in $t$ minutes. Subtract that from the starting volume: $V=500-\frac{5}{2}t$ . This gives the gallons remaining after $t$ minutes. Choice D.

Question 761·Hard·Math

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Step-by-step

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Strategy

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Step-by-step

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