Question 93·Hard·Linear Equations in One Variable
A factory begins the day with 10,000 kilograms of raw material in a hopper. The production line consumes the material at a constant rate. After 2.5 hours of operation, 9,250 kilograms of material remain in the hopper. If the production line continues to consume material at this rate, how many hours after it began will only 7,900 kilograms of raw material remain in the hopper?
For word problems about a quantity decreasing at a constant rate, first compute the rate by dividing the change in the quantity by the change in time. Then write a linear equation in the form amount = initial ± rate × time, plug in the target amount, and solve for the time. Keeping the structure "start ± rate × time" in mind helps you quickly translate the words into an equation and avoid guessing from the answer choices.
Hints
Use the information after 2.5 hours
First figure out how much material was used during the first 2.5 hours by comparing the starting amount and the amount remaining.
Find the rate per hour
You know how much was used in 2.5 hours. Divide that amount by 2.5 to get the constant rate of consumption per hour.
Write a linear equation
Express the amount remaining as "starting amount minus (rate × time)." Then set this equal to 7,900 and solve for the time.
Solve the equation carefully
When you solve the equation, pay attention to subtracting correctly and dividing without arithmetic mistakes.
Desmos Guide
Enter the linear model for remaining material
In Desmos, type y = 10000 - 300x to represent the amount of material remaining after hours.
Mark the target remaining amount
On a new line, type y = 7900 to draw a horizontal line showing when the hopper has 7,900 kilograms left.
Find the intersection
Look at the point where the two lines intersect. The -coordinate of this intersection is the number of hours after the start when the hopper will have 7,900 kilograms remaining.
Step-by-step Explanation
Find how much material was used in 2.5 hours
The hopper starts with 10,000 kilograms and has 9,250 kilograms after 2.5 hours.
So the amount used in 2.5 hours is:
This means 750 kilograms were consumed in 2.5 hours.
Compute the constant rate of consumption
Use the formula .
So the production line uses 300 kilograms of material each hour.
Write an equation for the amount remaining after t hours
Let be the number of hours after the production line begins.
- Initial amount: 10,000 kilograms
- Amount used after hours: kilograms (because the rate is 300 kg/hour)
So the amount remaining after hours is:
We want to find when the remaining amount is 7,900 kilograms.
Set the remaining amount equal to 7,900 and solve for t
Set up the equation:
Subtract 7,900 from both sides (or move terms to isolate ):
Divide both sides by 300:
So it will take 7 hours after the production line begins for only 7,900 kilograms of material to remain in the hopper.