Page 150 - How To Solve Word Problems In Calculus

Solution

P

x s
θ 2θ
A B
1 O 1

Let P be the point where the boat reaches the shore. Since
triangle APB is inscribed in a semicircle, it is a right triangle
with hypotenuse 2.
x
= cos θ
2
x = 2 cos θ

A geometry theorem says that a central angle is measured by
its intercepted arc, while an inscribed angle is measured by

half its intercepted arc. Since PB is the intercepted arc for both
POB and PAB, it follows that POB is twice PAB or 2θ. The
arc length s = r(2θ) = 2θ.
If r is the constant rate of a moving object and d is the
d
distance traveled, the time of travel t = . It follows that the
r
time to make the journey from A to B as a function of θ is

t = t row + t jog
x s
t = +
3 6
2 cos θ 2θ
t(θ) = +
3 6
2 θ π
= cos θ + 0 ≤ θ ≤
3 3 2

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