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Math Help - Calculating the Time of a Trip

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    Calculating the Time of a Trip

    Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved road that parallels the ocean. Sally can jog 8 miles per hour along the paved road but only 3 miles an hour in the sand on the beach. Because of a river directly between the two houses, it is necessary to jog in the sand to the road, continue on the road, and then jog directly back in the sand to get from one house to the other. The time time T to get from one house to the other as a function of the angle theta t is given by the following:

    T(t) = 1 + (2/[3 sin(t)]) - (1/[4 tan(t)]), where theta t lies between 0 degrees and 90 degrees.

    Calculate the time T for theta t = 30 degrees. How long is Sally on the paved road?


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    Quote Originally Posted by magentarita View Post
    Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved road that parallels the ocean. Sally can jog 8 miles per hour along the paved road but only 3 miles an hour in the sand on the beach. Because of a river directly between the two houses, it is necessary to jog in the sand to the road, continue on the road, and then jog directly back in the sand to get from one house to the other. The time time T to get from one house to the other as a function of the angle theta t is given by the following:

    T(t) = 1 + (2/[3 sin(t)]) - (1/[4 tan(t)]), where theta t lies between 0 degrees and 90 degrees.

    Calculate the time T for theta t = 30 degrees. How long is Sally on the paved road?

    You need to draw a good diagram (unless one has been provided). The question doesn't make it clear how theta is defined (which is why I suspect you've been provided with a diagram on which theta is defined).

    Nevertheless, to get the given formula the angle theta must be the angle between the horizintal line joining the two houses and the line joining the first house to the point on the road that Sally starts jogging from. It's also the acute angle between road and the line joining the point that Sally leaves the road from to the second house (so the path Sally follows is symmetric).


    To answer the first question, you substitute the given value of theta into the formula. You're expected to know that \sin 30^o = \frac{1}{2} and \tan 30^o = \frac{1}{\sqrt{3}} .....


    From the diagram it should be clear that the total distance on the beach that Sally jogs is \frac{1}{\sin 30^o} + \frac{1}{\sin 30^o} = \frac{2}{ \sin 30^o}. Therefore the time Sally spends jogging on the beach is \frac{\text{distance}}{\text{speed}} = \frac{2}{3 \sin 30^o}.

    Therefore the time Sally spends jogging on the road is


    Total time minus time on beach = T - \frac{2}{3 \sin 30^o} = \left( 1 + \frac{2}{3 \sin 30^o} - \frac{1}{4 \tan 30^o} \right) - \frac{2}{3 \sin 30^o} = 1 - \frac{1}{4 \tan 30^o} = \, ....
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