Q60. - Calculus

Question

Annie is paddling her kayak through the San Juan Islands

and is a quarter of a mile away from where she wants to

cross a channel. She sees a ferry in the channel approaching

fast from her left, about 2 miles away. The ferry travels

at about 20 mph, while Annie can do about 3 mph if she

jams.

(a) To set up a model for this problem, suppose Annie

is travelling on the x-axis and is approaching the origin

from the right. Suppose also that the ferry is travelling

on the y-axis and is approaching the origin

from above. Then x = x(t) represents Annie’s position

at time t and y = y(t) represents the ferry’s position.

Given what you know about this problem, what

are x(t) and y(t)?

(b) Construct an equation in terms of x = x(t) and y =

y(t) that describes the distance between Annie and

the ferry at time t.

(c) Use implicit differentiation to determine how fast the

distance between Annie and the ferry is decreasing

when she first sees the ferry.

(d) If Annie decides to jam across the channel, will the

ferry hit her?

Step-by-Step Solution

Answer

(a) $x\left(t\right)$ represents the speed of Annie at any time

$y\left(t\right)$ represents the speed of ferry at any time

(b) $x\left(t\right)=3t+0.25$

$y\left(t\right)=20t+2$

(c) The distance between Annie and ferry is decreasing at the rate of $\frac{20}{3}$ mph.

(d) The ferry will not hit Annie. 

Part (a) Step 1. Given Information

Speed of Annie $3$ mph

Speed of ferry $20$ mph

Part (a) Step 2. Explanation

Since $x\left(t\right) \& y\left(t\right)$represents Annie's and ferry's position respectively.

Then $x\text{'}\left(t\right) \& y\text{'}\left(t\right)$ represents their respective speed.

Part (b) Step 1. Given Information

Speed of Annie $3$mph

Speed of ferry $20$mph

Part (b) Step 2. Calculation

Since $x\text{'}\left(t\right)=3$then integrating both sides with respect to $t$we get

$x\left(t\right)=3t+c$, integration constant

At $t=0, x\left(t\right)=0.25$

Therefore, $c=0.25$

Then $x\left(t\right)=3t+0.25$

Now, for $y\text{'}\left(t\right)=20$, proceeding same as above we get

$y\left(t\right)=20t+k,$integration constant

At $t=0, y\left(t\right)=2$

Then $k=2$

Therefore $y\left(t\right)=20t+2$

Part (c) Step 1. Given Information

Speed of Annie $3$mph

Speed of ferry $20$mph

Part (c) Step 2. Calculation

Here $x\left(t\right)=2t+0.25 \& y\left(t\right)=20t+2$

Now, $x\text{'}\left(t\right)=3 \& y\text{'}\left(t\right)=20$ 

Therefore, $\frac{dy}{dx}=\frac{20}{3}$

The distance between Annie and ferry is decreasing at the rate of $\frac{20}{3}$mph.

 

Part (d) Step 1. Given Information

Speed of Annie $3$mph

Speed of Ferry $20$mph

Part (d) Step 2. Calculation

Since $x\left(t\right)=3t+0.25$

Then time is taken to reach the origin by Annie is

$0=3t+0.25$

$or,t=\frac{0.25}{3}$ sec

Now,$y=20t-2$

The time taken to reach the origin by ferry is 

$0=20t-2$

or, $t=\frac{1}{10}$sec

Therefore the ferry will not hit Annie.