A storage tank contains 1,500 gallons of water before draining begins. The tank is drained at a constant rate of at least 12 gallons per minute and no more than 15 gallons per minute for 40 minutes.
Which inequality represents all possible remaining volumes , in gallons, of water in the tank after the 40 minutes?
For inequality questions involving a starting amount and a rate range, first write an inequality for the rate (using phrases like "at least" and "no more than" to decide on inclusive symbols and ), then multiply by time to get the minimum and maximum total change. Next, convert that range into a range for the final quantity (here, remaining water) by adding or subtracting the inequality from the starting value, being careful to perform the operation on all three parts of the compound inequality. Finally, match your resulting inequality—including the correct endpoints and inequality symbols—to the answer choices.
Hints
Start with how much water is drained
Instead of thinking about the water left in the tank right away, first find the smallest and largest amounts of water that could be drained in 40 minutes.
Use the rate inequality
Write an inequality for the draining rate using "at least 12" and "no more than 15," then multiply every part of that inequality by 40 to get an inequality for the total amount drained .
Connect drained water to remaining water
Once you know the range of the drained amount , remember that the remaining volume is . How does subtracting from 1,500 affect the inequality signs and the endpoints?
Desmos Guide
Use Desmos to compute the minimum and maximum remaining volume
In two separate lines, type 1500 - 12*40 and 1500 - 15*40. These correspond to using the smallest and largest possible draining rates for the full 40 minutes.
Interpret the Desmos outputs as a range for V
Look at the two numerical results from Desmos: the smaller value is the minimum possible remaining volume, and the larger value is the maximum possible remaining volume. Write an inequality of the form min_value <= V <= max_value, using "<=" signs because the extreme rates (12 and 15 gallons per minute) are allowed.
Match the inequality to an answer choice
Compare the inequality you wrote from the Desmos outputs with the four answer options and select the one whose lower and upper bounds, and inequality symbols, match your result.
Step-by-step Explanation
Express the draining rate as an inequality
The problem says the tank is drained at at least 12 gallons per minute and no more than 15 gallons per minute.
That means the draining rate (in gallons per minute) satisfies
The tank drains for 40 minutes, and the total amount drained is given by .
Find the range of total water drained
Multiply the rate inequality by 40 to get the total drained in 40 minutes:
So the total amount of water drained satisfies
This means between 480 and 600 gallons of water are drained from the tank.
Translate to the remaining volume and write the inequality for V
The tank starts with 1,500 gallons. If gallons are drained, the remaining volume is
We know . Subtract this entire inequality from 1,500:
This simplifies to
So the possible remaining volumes after 40 minutes are given by the inequality , which corresponds to choice B.