Question

You travel 3 miles downstream in a kayak at a speed of r miles per hour. You turn around and travel 4 miles upstream at a speed of 0.5r miles per hour. Finally, you turn around and return to your original starting point at a speed of r+1 miles per hour.
a)Write a model in simplified form for the total time t (in hours) of the trip.
b)How long does your trip take when r is 4 miles per hour?

Answer 1

Write a model in simplified form for the total time t (in hours) of the trip:

$Total\ time\ taken = \frac{3}{r} + \frac{4}{0.5r} + \frac{1}{r+1}$

It takes 2.95 hours when r is 4 miles per hour

Solution:

Given that,

You travel 3 miles downstream in a kayak at a speed of r miles per hour

Downstream:

Speed = r miles per hour

Distance = 3 miles

Time taken is given as:

$Downstream\ time = \frac{distance}{speed}\\\\Downstream\ time = \frac{3}{r} ------- eqn\ 1$

You turn around and travel 4 miles upstream at a speed of 0.5r miles per hour

Upstream:

Speed = 0.5r miles per hour

Distance = 4 miles

Therefore,

$Upstream\ time = \frac{4}{0.5r} -------- eqn 2$

Finally, you turn around and return to your original starting point at a speed of  r + 1 miles per hour

Therefore,

$time = \frac{1}{r+1} ----------- eqn\ 3$

a)Write a model in simplified form for the total time t (in hours) of the trip

The total time taken is:

eqn 1 + eqn 2 + eqn 3

$Total\ time\ taken = \frac{3}{r} + \frac{4}{0.5r} + \frac{1}{r+1}$

b)How long does your trip take when r is 4 miles per hour?

Substitute r = 4

$Total\ time\ taken = \frac{3}{4} + \frac{4}{0.5 \times 4} + \frac{1}{4+1}\\\\Total\ time\ taken = 0.75 + 2 + 0.2\\\\Total\ time\ taken = 2.95$

Thus it takes 2.95 hours when r is 4 miles per hour