Question

sharon plans to travel from savannah, ga to hollywood, ca which is 2500 miles by car. she wants to stop and take in some of the sites along the way but wants to make sure that she travels 500 miles a day.

a. write a linear function f(d) to represent the distance left to travel for sharons trip after d days.

b. graph the function.

c. what is the domain and range of the function?

d. is the function linear or nonlinear?

e. how long will it take for sharon to reach hollywood?

Explanation:

Part a

Step1: Identify variables and rate

The total distance is 2500 miles. She travels 500 miles per day, so the distance traveled after \( d \) days is \( 500d \). The distance left \( f(d) \) is total distance minus distance traveled.
\( f(d)=2500 - 500d \)

Part b

Step1: Find intercepts

For the \( y \)-intercept (when \( d = 0 \)): \( f(0)=2500-500(0)=2500 \), so the point is \( (0, 2500) \).
For the \( d \)-intercept (when \( f(d)=0 \)): \( 0 = 2500-500d \), solving for \( d \) gives \( 500d = 2500 \), so \( d = 5 \). The point is \( (5, 0) \).

Step2: Draw the line

Plot the points \( (0, 2500) \) and \( (5, 0) \) on a coordinate system with \( d \) (days) on the x - axis and \( f(d) \) (distance left) on the y - axis, then draw a straight line through them.

Part c

Step1: Determine domain

The number of days \( d \) can't be negative, and she can't travel more than 2500 miles. From \( 0\leq d\leq5 \) (since when \( d = 5 \), she reaches her destination). So domain is \( [0, 5] \) (inclusive, as days are non - negative and up to when she arrives).

Step2: Determine range

When \( d = 0 \), \( f(d)=2500 \); when \( d = 5 \), \( f(d)=0 \). The function is decreasing, so the range is \( [0, 2500] \).

Part d

Answer:

Step1: Set distance left to 0

We want to find \( d \) when \( f(d)=0 \). From \( 0=2500 - 500d \), we solve for \( d \).
\( 500d=2500 \)
\( d=\frac{2500}{500}=5 \)

Part a Answer: \( f(d)=2500 - 500d \)

Part b Answer: Plot points \((0, 2500)\) and \((5, 0)\) and draw a line through them.

Part c Answer: Domain: \([0, 5]\), Range: \([0, 2500]\)

Part d Answer: Linear

Part e Answer: 5 days