Question 31·Easy·Linear Equations in Two Variables
Characteristics for Rock Types
| Rock type | Weight per volume (lb/ft³) | 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. All four rock types in the table will be used. The total cost, in dollars, of the rocks used in the project is modeled by the equation
where is the number of cubic feet of limestone and is the number of cubic feet of basalt.
Which of the following is the best interpretation of in this context?
For questions that ask you to interpret a number or term in an equation, first identify what the whole equation represents (often a total). Then, match each algebraic term to quantities in the table or description: variable terms usually represent parts that change with amount, and constant terms represent fixed amounts or categories not tied to the given variables. Finally, see which real-world items are already accounted for by the variable terms and assign the remaining constant to what’s left; this process is much faster and safer than guessing based on the size of the number alone.
Hints
Understand the whole equation
Look at the entire equation and decide what the right-hand side, , represents in the context of the project.
Match each variable term to a rock type
Use the table: which rock type has lb/ft³ and costs $0.03 per pound, and which has lb/ft³ and costs $0.18 per pound? Match those to and .
Focus on what is left over
Once you know which rock types are represented by the variable terms, ask: what rock types are not in those terms? The constant must be the cost associated with those remaining rock types.
Desmos Guide
Enter the cost expression
Type 0.03*120*w + 0.18*180*z + 3385.8 into Desmos as an expression. Look at which parts of this expression include variables ( or ) and which part is just a number.
Identify the constant part of the cost
Notice that only the first two terms change with or . The number 3385.8 is a constant term, meaning it is the part of the total cost that does not depend on the amounts of limestone () or basalt (). Use the table and the problem statement to decide which rock types this fixed cost must correspond to.
Step-by-step Explanation
Identify what the equation is modeling
The equation
is modeling the total cost of all rocks used in the project. The right side, , is the total cost in dollars. Each term on the left side represents a part of that total cost.
Interpret the term involving limestone
From the table, limestone has:
- Weight per volume: lb/ft³
- Cost per pound: $0.03
There are cubic feet of limestone. That means:
- Total weight of limestone: pounds
- Cost of that limestone: dollars
So is the cost of the limestone used in the project.
Interpret the term involving basalt
From the table, basalt has:
- Weight per volume: lb/ft³
- Cost per pound: $0.18
There are cubic feet of basalt. That means:
- Total weight of basalt: pounds
- Cost of that basalt: dollars
So is the cost of the basalt used in the project.
Figure out what the constant 3,385.80 must represent
The left side of the equation is:
- = cost of limestone
- = cost of basalt
- = another part of the total cost that does not depend on or
Because the problem states that all four rock types will be used and the variable terms already cover limestone and basalt, the remaining cost must be for the other two rock types listed in the table: granite and sandstone. Therefore, represents the cost of the granite and sandstone needed for the project (choice C).