Sure! Let's go through this step-by-step.
1. Finding the Slope of the Linear Function and Its Units:
The given cost function [tex]\( C(x) \)[/tex] is of the form:
[tex]\[ C(x) = 24 + 0.35x \][/tex]
In this equation:
- The coefficient of [tex]\( x \)[/tex] (0.35) represents the rate of change of cost with respect to the number of miles driven.
So, the slope of the linear function is [tex]\( 0.35 \)[/tex].
Units of the Slope:
- The cost increases by [tex]\( \$0.35 \)[/tex] for each additional mile driven.
Therefore, the units of the slope are dollars per mile.
Answer:
- Slope: [tex]\( 0.35 \)[/tex]
- Units: dollars per mile
2. Finding the [tex]\( y \)[/tex]-Intercept and Its Units:
In the cost function [tex]\( C(x) = 24 + 0.35x \)[/tex]:
- The constant term (24) represents the fixed cost when no miles are driven (i.e., [tex]\( x = 0 \)[/tex]).
So, the [tex]\( y \)[/tex]-intercept is [tex]\( 24 \)[/tex].
Units of the [tex]\( y \)[/tex]-Intercept:
- This is simply the initial fixed cost in dollars.
Therefore, the units of the [tex]\( y \)[/tex]-intercept are dollars.
Answer:
- [tex]\( y \)[/tex]-Intercept: [tex]\( 24 \)[/tex]
- Units: dollars
3. Formulating the Linear Function, [tex]\( C(x) \)[/tex]:
Based on the problem statement, the linear cost function is written as:
[tex]\[ C(x) = 24 + 0.35x \][/tex]
In this function:
- [tex]\( C(x) \)[/tex] is the total cost in dollars
- [tex]\( x \)[/tex] is the number of miles driven
Answer:
- Linear function, [tex]\( C(x) \)[/tex]: [tex]\( C(x) = 24 + 0.35x \)[/tex]
Putting it all together:
- Slope: [tex]\( 0.35 \)[/tex] (dollars per mile)
- [tex]\( y \)[/tex]-Intercept: [tex]\( 24 \)[/tex] (dollars)
- Linear Function: [tex]\( C(x) = 24 + 0.35x \)[/tex]
