Question

9. Julia is driving toward Denver at a constant speed. After driving for an hour, she was 520 miles from Denver. After driving for three hours, she was 390 miles from Denver. Be sure to show work supporting your answers. a. How fast was Julia driving? b. How many miles from Denver was Julia when she started driving? c. Let f be a function that determines Julia's distance from Denver in miles, d, after having driven for t hours. Determine a rule for f. d. What does f^−1(125) represent in this context? (No need to evaluate) e. After how many hours of driving does Julia reach Denver

Answer 1

Julia was driving at a constant speed of 130 miles per hour. She was 390 miles away from Denver when she started driving. The function f(t) = 130t + 390 represents Julia's distance from Denver after having driven for t hours. f^−1(125) represents the number of hours Julia has been driving when she is 125 miles away from Denver. Julia reaches Denver after 3 hours of driving.

Step-by-step explanation:

a. How fast was Julia driving?

Let's first calculate the distance Julia covered in one hour. We can subtract the distance she was from Denver after driving for three hours from the distance after driving for one hour.

Distance covered in 1 hour = 520 miles - 390 miles = 130 miles

Julia was driving at a constant speed of 130 miles per hour.

b. How many miles from Denver was Julia when she started driving?

Since Julia was 520 miles from Denver after driving for one hour, we can subtract the distance covered in one hour from 520 miles.

Miles from Denver when Julia started driving = 520 miles - 130 miles = 390 miles

Julia was 390 miles away from Denver when she started driving.

c. Determine a rule for f, the function that determines Julia's distance from Denver in miles, d, after having driven for t hours.

The distance d can be represented by the equation f(t) = mt + b, where m is the constant speed at which Julia is driving and b is the initial distance from Denver when Julia started driving. From part a and b, we know that m = 130 and b = 390. Therefore, the rule for f is f(t) = 130t + 390.

d. What does f−1(125) represent in this context? (No need to evaluate)

In this context, f−1(125) represents the number of hours Julia has been driving when she is 125 miles away from Denver. It gives the inverse of the function f(t) = 130t + 390.

e. After how many hours of driving does Julia reach Denver?

We can set the distance from Denver, d, to 0 and solve the equation f(t) = 130t + 390 = 0 for t.

130t = -390

t = -3

Since time cannot be negative, Julia reaches Denver after 3 hours of driving.

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