Question 45·Hard·Linear Inequalities in One or Two Variables
A cylindrical water tank holds a maximum of 500 gallons. The tank currently contains 120 gallons. A pump adds water at a rate of 15 gallons per minute while a drain removes water at a rate of 8 gallons per minute.
If both the pump and the drain operate at the same time for minutes, which of the following inequalities gives the values of for which the tank will overflow?
For word problems involving inflow and outflow, first compute the net rate (inflow minus outflow). Then write a linear expression in the form initial amount + (net rate)·t to represent the quantity after units of time. Finally, translate the verbal condition (“overflow,” “at least,” “no more than,” etc.) into an inequality comparing that expression to the given limit, making sure the inequality sign correctly reflects whether the amount must be greater than, less than, or equal to the limit.
Hints
Combine the pump and drain rates
The pump adds 15 gallons per minute, and the drain removes 8 gallons per minute. What single number tells you how many gallons are added to the tank each minute overall?
Write the expression for the water amount
Start from the initial 120 gallons and add the net number of gallons gained each minute times . What expression represents the total gallons after minutes?
Think about what “overflow” means
If the tank can hold at most 500 gallons, should the water amount in the tank be greater than, less than, or equal to 500 gallons when it overflows? Use that to decide which inequality symbol to use with your expression for the water amount and 500.
Desmos Guide
Graph the water amount as a function of time
In Desmos, type the equation y = 120 + 7x. This line represents the amount of water in the tank (in gallons) after minutes.
Graph the tank’s capacity
Type the equation y = 500. This is a horizontal line showing the maximum capacity of the tank.
Use the graph to interpret the inequality
Find the point where the two lines intersect; note the -value there. For -values to the right of that point, the line is above , meaning the water amount is greater than 500 gallons. Choose the answer that compares the water amount expression on the left and 500 on the right with an inequality sign that matches this situation (water amount greater than capacity).
Step-by-step Explanation
Find the net rate of water going into the tank
The pump adds 15 gallons per minute and the drain removes 8 gallons per minute.
Net change in water each minute is:
So, overall, the amount of water in the tank increases by 7 gallons every minute.
Write an expression for the amount of water after minutes
The tank starts with 120 gallons. Each minute, it gains 7 gallons.
Amount of water after minutes:
This expression gives the number of gallons in the tank after minutes when both pump and drain are running.
Translate “overflow” into an inequality and match it
The tank’s capacity is 500 gallons. The tank will overflow when the amount of water in it is greater than 500 gallons.
So we want:
This matches answer choice D) .