Suppose you park your car at a trailhead in a national park an...

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 7 A.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 t 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.

b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a cyclist leaves their home for a 3-hour ride to a park on a Saturday morning at 8:00 a.m. The park is 9 miles away from their home. On a Monday morning, they cycle back home from the park starting at 8 a.m. and take in the same 3 hours. Let PFT be the cyclist's distance from home two hours after 8:00 a.m. on Saturday, and QFT be the distance from home 2 hours after 8:00 a.m. on Monday. If RFT is equal to PFT minus QFT, What are R of 0 and R of 3? We're given 34 possible choices as our answers. For choice A, we have R of 0 is equal to 9, R of 3 is equal to 0. For choice B, we have R of 0 is equal to 0, and R of 3 is equal to 9. For choice C, we have R of 0 is equal to -9, and R of 3 is equal to 9. And for choice D, we have R of 0 is equal to -9, and R of 3 is equal to -9. Now, the question asks us to bind R of 0 and R of 3. Given the formula that RFT is equal to PFT minus QFT. That means 4 are 0. That is going to be equal to P of 0. Minus Q of 0. N or R3. That's gonna be equal to PF 3. Minus Q of 3. So, what we need to do is find the values for the function of E and Q when T is equal to 0 and T is equal to 3. And we can do that using the information that we're given in the problem. So, let's look at our function of PFT. We're told that PFT is the distance from the cyclist's home, 2 hours after 8:00 a.m. on Saturday. So, we need to find P of 0 1st. And since the cyclist is leaving their home at 8:00 a.m. on Saturday morning, That means their distance from home is going to be 0. That means that P of 0 is going to be equal to 0. We're also told that after 3 hours they have arrived at the park, which is 9 miles away. So that means that PF 3. It's going to be equal to 9. We can do something similar for our function QFT. So QFT is the distance from home, tea hours after 8:00 a.m. on Monday. So, if we look at Q of 0, That is going to occur at 8:00 a.m. on Monday, and we're told the problem that on Monday morning, they're cycling back home from the park starting at 8 a.m. So the cyclist is at the park at 8:00 a.m. on Monday morning, which is 9 miles away from their home. That means that Q of 0 is equal to 9. After 3 hours, they have arrived back home on Monday. So that means that queue of 3. is going to be equal to 0. Since they're now at home, the distance from their home is going to be 0. So now we have values or our our functions P and Q at times T equal to 0 and T equal to 3. So we can substitute these values back into our expressions for R. So we're going to look at R of 0 1st. We'll have R of 0. That's gonna be equal to P 0, which is equal to 0, minus Q 0, which is 9. And so simplifying that, we'll have 0 minus 9, which is equal to -9. So R 0. It's going to be equal to -9. Now for our 3, That's going to be equal to P of 3, which is equal to 9, minus Q of 3, which is equal to 0. So simplifying that, we're just gonna get R of 3. is equal to 9. So, those are the two values that are that we needed to calculate. And so now we can compare the values that we calculated to our answer choices, and the correct answer choice turns out to be answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Distance Functions

Function Evaluation

Difference of Functions

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