Question 117·Hard·Linear Equations in One Variable
A highway maintenance truck begins spreading salt from a warehouse located at mile marker 0. The truck spreads the salt at a constant rate as it travels. When the truck passes mile marker 10, it has 10 tons of salt remaining in its hopper. When it reaches mile marker 22, only 7 tons remain. At what mile marker will the truck run out of salt completely?
For constant-rate word problems, translate the situation into a linear relationship: use the two data points to compute the rate of change (slope) as change in amount over change in distance. Then rewrite that rate in whichever form is most convenient (e.g., miles per ton instead of tons per mile), and use it to scale from the known amount to zero. Always pay attention to whether you are finding an extra distance or an actual mile marker, and add distances to the correct starting position to avoid off-by-constant mistakes.
Hints
Use the two data points
You know the truck has 10 tons at mile 10 and 7 tons at mile 22. How much salt is used over that distance, and how many miles is that?
Think in terms of rate
Once you know how many tons are used over 12 miles, find how many tons are used per mile. You are looking for a constant rate (like a slope).
Turn tons per mile into miles per ton
If you know how many tons are used each mile, you can flip that to figure out how many miles it takes to use 1 ton. Then think about how many tons are left when the truck is at mile 10.
Connect extra miles to the mile marker
After you find how many additional miles it takes to use up all remaining salt, remember to add that distance to the mile marker where the remaining amount was measured.
Desmos Guide
Model the salt amount as a linear function
In Desmos, enter a function like s(x) = 10 + (x - 10)*(7 - 10)/(22 - 10). This uses the point at mile 10 with 10 tons and the constant rate found from the two points.
View the graph and identify the x-intercept
Look at the graph of s(x). This shows the amount of salt (vertical axis) versus mile marker (horizontal axis). Use the cursor to find where the graph crosses the x-axis (where s(x) = 0); the x-coordinate of this intercept is the mile marker where the truck runs out of salt.
Step-by-step Explanation
Find how fast the salt is being used
From mile 10 to mile 22, the truck travels miles.
Over the same distance, the salt amount goes from 10 tons down to 7 tons, a change of tons.
So the rate of change (slope) is
This means the truck uses ton of salt per mile.
Rephrase the rate in a more useful way
If the truck uses ton per mile, that also means it uses 1 ton every 4 miles, because
So for each 1 ton of salt, the truck travels 4 miles.
Find how many more miles until the salt is gone
At mile 10, the truck still has 10 tons of salt.
If it uses 1 ton every 4 miles, then to use up those 10 tons, it must travel
This is how many miles it will travel after passing mile marker 10 before the salt runs out.
Find the mile marker where the salt runs out
The truck starts this final stretch at mile 10 and travels 40 more miles before the salt runs out.
So the mile marker where it runs out of salt is
Therefore, the correct answer is 50 (choice B).