Question

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car r hours after 7 A.M. on Friday morning and let g(t) be your distance from the car t hours after 7 A.M. on Sunday morning.

a. Evaluate f(0), f(2), g(0), and g(2).
b. Let h(t) = f(t) - g(t). Find h(0) and h(2).
c. Use the Intermediate Value Theorem to show that there is some point along the trail that you will pass at exactly the same time of morning on both days.

Answer 1

The answer is given below

Explanation:

The distance from the lake to the car is 3 miles and it takes 2 hours. The velocity of the hiker is 3/2 mi/hr. Therefore f(t) which is the distance from the car t hours after 7 A.M is given by:

`f(t)=(3)/(2)t`

When coming back on Sunday morning, the distance g(t) from the car at any point in time is given by:

`g(t)=3-(3)/(2)t`

a)

`f(t)=(3)/(2)t\\f(0)=(3)/(2)(0)=0\ mile\\f(2)=(3)/(2)(2)=3\ miles`

`g(t)=3-(3)/(2)t\\g(0)=3-(3)/(2)(0)=3\ miles\\g(2)=3-(3)/(2)(2)=2-2=0\ mile\\`

b)

`h(t)=f(t)-g(t)=(3)/(2)t -(3-(3)/(2)t)=(3)/(2)t -3+(3)/(2)t=3t-3\\h(t)=3t-3\\h(0)=3(0)-3=0-3=-3\\h(2)=3(2)-3=6-3=3`

c)

According to Intermediate Value Theorem, there exist a point b where f(b) = g(b). i.e. f(b) - g(b) = 0

`h(b)=f(b)-g(b)=0\\h(0)=-3,h(2)=3`

This means there exist a point b within the interval [-3, 3] where f(b) - g(b) = 0