Question 17·Medium·Linear Equations in Two Variables
Characteristics for Rock Types
| Rock type | Weight per volume (lb/ft) | Cost per pound |
|---|---|---|
| Basalt | 180 | $0.18 |
| Granite | 165 | $0.09 |
| Limestone | 120 | $0.03 |
| Sandstone | 135 | $0.22 |
A city is planning to build a rock retaining wall, a monument, and a garden in a park. Only basalt, granite, and limestone will be used in the garden. The rocks in the garden will have a total weight of 1,000 pounds. If 330 pounds of granite is used, let be the volume of limestone and be the volume of basalt, both in ft. Which of the following equations shows the relationship between and ?
For word problems that ask for an equation from a data table, first identify what each variable represents and which table entries match those variables. Convert each variable’s quantity into an algebraic expression using “(rate from table) × (variable)”, include any fixed amounts separately, and then write a total equation like “sum of parts = overall total.” Finally, simplify the equation (for example, subtract known fixed weights from the total) and pick the choice that matches your simplified result, ignoring any extra information such as costs if the question is only about weight.
Hints
Focus on weight, not cost
Look only at the weight per volume (lb/ft) column for limestone and basalt. The cost column is a distraction for this question.
Turn volumes into weights
If limestone has a volume of ft and weighs 120 lb/ft, what is its weight in pounds? Do the same for basalt with 180 lb/ft and volume .
Account for the fixed 330 lb of granite
The total weight is 1,000 lb, and 330 lb of that is granite. How much weight is left over for just the limestone and basalt together? Think about subtracting.
Write the equation for limestone + basalt
Add the weight expression for limestone to the weight expression for basalt, and set that sum equal to the remaining total weight after you subtract the 330 lb of granite from 1,000 lb. Then see which answer choice matches that equation.
Desmos Guide
Use Desmos to find the remaining weight
In a Desmos expression line, type "1000 - 330" and press Enter. The result is the total number of pounds that limestone and basalt together must weigh.
Match that remaining weight to the equation form
From the table, the combined weight of limestone and basalt is . Use the number you just found in Desmos as the right side of the equation , then choose the answer option whose equation has this exact form.
Step-by-step Explanation
Identify which numbers from the table you need
The question defines as the volume of limestone and as the volume of basalt, both in ft.
From the table:
- Limestone weighs 120 lb/ft.
- Basalt weighs 180 lb/ft.
You can ignore the granite volume as a variable (its weight is fixed at 330 lb) and ignore the entire cost per pound column, since the problem only asks about weight.
Write expressions for the weights of limestone and basalt
Weight is .
- Limestone: volume is ft, weight per volume is 120 lb/ft, so its weight is pounds.
- Basalt: volume is ft, weight per volume is 180 lb/ft, so its weight is pounds.
So the total weight of limestone and basalt together is .
Use the total weight, including granite
You are told that:
- The total weight of all rocks in the garden is 1,000 pounds.
- There are 330 pounds of granite.
So the total weight comes from:
- Limestone:
- Basalt:
- Granite:
This gives the equation:
Isolate the terms with x and y
Subtract 330 from both sides to move the granite weight to the right:
120x + 180y = 670