# 69 - 71 Shortest and most economical path of motorboat

**Problem 69**

A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of \$2.00 per hour for the time it is actually used, and the cost of land transportation is \$0.06 per mile, how much must he pay for the trip?

**Solution:**

$s = \sqrt{12^2 + x^2}$

$s = \sqrt{144 + x^2}$

Note: time = distance/speed

Total cost of travel:

$C = \dfrac{s}{20}(2) + (20 - x)0.06$

$C = \frac{1}{10}\sqrt{144 + x^2} + 1.2 - 0.06x$

$\dfrac{dC}{dx} = \dfrac{1}{10} \left( \dfrac{2x}{2\sqrt{144 + x^2}} \right) - 0.06 = 0$

$\dfrac{x}{10\sqrt{144 + x^2}} = 0.06$

$x = 0.6\sqrt{144 + x^2}$

$x^2 = 0.36(144 + x^2)$

$0.64x^2 = 51.84$

$x = 9 \text{ miles}$

$C = \frac{1}{10}\sqrt{144 + 9^2} + 1.2 - 0.06(9)$

$C = \$ 2.16$ *answer*

**Problem 70**

A man in a motorboat at A (Figure 42) receives a message at noon calling him to B. A bus making 40 miles per hour leaves C, bound for B, at 1:00 PM. If AC = 40 miles, what must be the speed of the boat to enable the man to catch the bus.

**Solution:**

$(rt)^2 = 40^2 + 40^2(t - 1)^2$

$r^2t^2 = 1600 + 1600t^2 - 3200t + 1600$

$r^2 = \dfrac{1600(t^2 - 2t + 2}{t^2}$

$r = \dfrac{40}{t}\sqrt{t^2 - 2t + 2}$

$\dfrac{dr}{dt} = \dfrac{40}{t}\dfrac{2t - 2}{2\sqrt{t^2 - 2t + 2}} + \dfrac{-40}{t^2}\sqrt{t^2 - 2t + 2} = 0$

$\dfrac{t - 1}{\sqrt{t^2 - 2t + 2}} = \dfrac{1}{t}\sqrt{t^2 - 2t + 2}$

$t^2 - t = t^2 - 2t + 2$

$t = 2 \text{ hours}$

$r = \dfrac{40}{2}\sqrt{2^2 - 2(2) + 2}$

$r = 20 \sqrt{2} \text{ miles/hour}$

$r = 28.28 \text{ miles/hour}$ *answer*

**Problem 71**

In Problem 70, if the speed of the boat is 30 miles per hour, what is the greatest distance offshore from which the bus can be caught?

**Solution:**

$y = \sqrt{900t^2 - 1600(t - 1)^2}$

$\dfrac{dy}{dt} = \dfrac{1800t - 3200(t - 1)}{2\sqrt{900t^2 - 1600(t - 1)^2}} = 0$

$9t - 16(t - 1) = 0$

$7t = 16$

$t = 16/7$

$y = \sqrt{900(16/7)^2 - 1600(16/7 - 1)^2}$

$y = 45.35 \text{ miles}$ *answer*

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