Chapter 2.6, Problem 23AYU

23. Time Required to Go from an Island to a Town An island is 2 miles from the nearest point P on a straight shoreline. A town is 12 miles down the shore from P. See the illustration.

(a) If a person can row a boat at an average speed of 3 miles per hour and the same person can walk 5 miles per hour, build a model that expresses the time T that it takes to go from the island to town as a function of the distance X from P to where the person lands the boat.

(b) What is the domain of Τ?

(c) How long will it take to travel from the island to town if the person lands the boat 4 miles from P?

(d) How long will it take if the person lands the boat 8 miles from Ρ?

To determine

To find:

a. To model that expresses the time $T$ that it takes to go from the island to the town as a function of distance $x$ from $p$.

Answer to Problem 23AYU

Solution:

a. $T\left(x\right)=\frac{\sqrt{x^2+4}}{3}+\frac{12-x}{5}$

Explanation of Solution

Given:

1. The shortest distance from the island to shore is 2 miles (Point $P$).

2. The town is 12 miles from the point $P$.

3. The person can row boat at 3 miles per hour.

4. The person can walk at 5 miles per hour.

Calculation:

The can be expressed as the figure below

Let $x$ be the distance from point $p$ where person lands in the shore.

Let $d_1$ be the distance the person row the boat.

Let $d_2$ be the distance the person walks.

Let $v_1$ be the average speed of the person when he rows the boat.

Let $v_2$ be the average speed of the person when he walks.

a. Let $T$ be the time taken to reach the town form the island.

$T=\frac{d_1}{v_1}+\frac{d_2}{v_2}$

Where,

$d_1=\sqrt{x^2+2^2}$

$d_2=12-x$

$v_1=3$

$v_2=5$

$T\left(x\right)=\frac{\sqrt{x^2+4}}{3}+\frac{12-x}{5}$

To determine

To find:

b. To find the domain of $T$.

Answer to Problem 23AYU

Solution:

b. $\text{Domain of }T\left(x\right)= \left\{x|0\le x\le 12\right\}$

Explanation of Solution

Given:

1. The shortest distance from the island to shore is 2 miles (Point $P$).

2. The town is 12 miles from the point $P$.

3. The person can row boat at 3 miles per hour.

4. The person can walk at 5 miles per hour.

Calculation:

The can be expressed as the figure below

Let $x$ be the distance from point $p$ where person lands in the shore.

Let $d_1$ be the distance the person row the boat.

Let $d_2$ be the distance the person walks.

Let $v_1$ be the average speed of the person when he rows the boat.

Let $v_2$ be the average speed of the person when he walks.

b. $\text{Domain of }T\left(x\right)= \left\{x|0\le x\le 12\right\}$

To determine

To find:

c. To find time $T$, if the person lands the boat 4 miles from $p$.

Answer to Problem 23AYU

Solution:

c. $=3.09\text{ Hours}$

Explanation of Solution

Given:

1. The shortest distance from the island to shore is 2 miles (Point $P$).

2. The town is 12 miles from the point $P$.

3. The person can row boat at 3 miles per hour.

4. The person can walk at 5 miles per hour.

Calculation:

The can be expressed as the figure below

Let $x$ be the distance from point $p$ where person lands in the shore.

Let $d_1$ be the distance the person row the boat.

Let $d_2$ be the distance the person walks.

Let $v_1$ be the average speed of the person when he rows the boat.

Let $v_2$ be the average speed of the person when he walks.

c. To find time $T$, if the person lands the boat 4 miles from $p$ .

$T\left(4\right)=\frac{\sqrt{4^2+4}}{3}+\frac{12-4}{5}$

$=3.09\text{ Hours}$

To determine

To find:

d. To find time $T$, if the person lands the boat 8 miles from $p$.

Answer to Problem 23AYU

Solution:

d. $=3.55\text{ Hours}$

Explanation of Solution

Given:

1. The shortest distance from the island to shore is 2 miles (Point $P$).

2. The town is 12 miles from the point $P$.

3. The person can row boat at 3 miles per hour.

4. The person can walk at 5 miles per hour.

Calculation:

The can be expressed as the figure below

Let $x$ be the distance from point $p$ where person lands in the shore.

Let $d_1$ be the distance the person row the boat.

Let $d_2$ be the distance the person walks.

Let $v_1$ be the average speed of the person when he rows the boat.

Let $v_2$ be the average speed of the person when he walks.

d. To find time $T$, if the person lands the boat 8 miles from $p$ .

$T\left(8\right)=\frac{\sqrt{8^2+4}}{3}+\frac{12-8}{5}$

$=3.55\text{ Hours}$.