"Calculating the Time of a Trip"

ChristaJoy

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Sep 23, 2012
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Hi, I've been trying to solve this problem over and over again but I can't seem to get anywhere with it, and my answer seems unreasonable. Here it is:

Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. Sally can jog 8 miles per hour on the paved path, but only 3 miles per hour on the sand. Because a river flows directly between the two houses, it's necessary to jog in the sand to the road, continue on the path, and then jog directly back in the sand to get from one house to the other. The time T to get from one house to the other as a function of the angle t is:

T(t)=1+(2/(3sin(t)))-(1/(4tan(t))) and 0<t<(pi/2)

a) Calculate the time T for tan(t)=1/4

b)Describe the path taken.

c) Explain why t must be larger than 14degrees.

I have no idea how to find b or c, and when I was trying to find a, I kept getting 2.75 hours but that just doesn't seem right. Any help would be appreciated :) Thanks!
 
The time T to get from one house to the other as a function of the angle t is:

T(t) = 1 + 2/(3sin(t)) - 1/(4tan(t))

0 < t < pi/2

Note: I deleted two sets of unnecessary grouping symbols -- for better readability

Was this function definition for T given to you?

I also get T = 2.75, using the facts that angle t is acute AND tan(t) = 1/4, to first find t.


If, instead, it was you who determined the above expression for T(t), then I would need a description or diagram defining angle t before I could verify your work. Phrases like "jog directly back in the sand" are too ambiguous. (Well, I suppose that I might be able to figure out where angle t is located by trial-and-error, going through each possibility until I was confident, but I don't have the motivation for that.)
 
Was this function definition for T given to you?

I also get T = 2.75, using the facts that angle t is acute AND tan(t) = 1/4, to first find t.


If, instead, it was you who determined the above expression for T(t), then I would need a description or diagram defining angle t before I could verify your work. Phrases like "jog directly back in the sand" are too ambiguous. (Well, I suppose that I might be able to figure out where angle t is located by trial-and-error, going through each possibility until I was confident, but I don't have the motivation for that.)


Yes, this definition of T was given to me. I'll upload a diagram to make it more clear. ThanksTrig Problem.jpg
 
Ah, I see that t is for theta.

Okay -- 2.75 hrs seems reasonable enough to me, as the jogging route could be over 15 miles (looking at the diagram). If a slow jogger moves twice as fast as the average walker, that speed will be about 6mph. (It takes 2.6 hrs to move 16 miles at 6mph, so 2.75 hrs seems reasonable, to me.)

Part (b) is kinda vague. After all, the diagram shows the path. How about describing the path in terms of the distance of each segment, along with some directional statements? It seems that you have freedom to describe it in terms of your own choosing.

Part (c) requires you to visualize what happens to those diagonal segments of the path when theta gets smaller. In particular, what happens to the endpoint locations, when theta is reduced?
 
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