Chapter 3.3, Problem 46PS

Solution

(a)

Elvis’s running speed is about 6.4 meters per second, Write a function r(x) for the time he spends running from point A to point D. Elvis’s swimming speed is 0.9 meter per second. Write a function s(x) for the time he spends swimming from point D to point B.

  $r(x)=\frac{20-x}{6.4}$

  $s(x)=\frac{\sqrt{{12}^2+x^2}}{0.9}$

Given:

A mathematician at a lake throws a tennis ball from point A along the water’s edge to point B in the water, as shown. His dog, Elvis, runs along the beach from point A to point D and then swims to fetch the ball at point B.

  

Concept Used:

• The composition function $f\circ g$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

Elvis’s running speed is about 6.4 meters per second

Function for the time he spends running from point A to point D $=r(x)$

Distance A to D = $20-x$

  $\begin{array}{c}r(x)=\frac{dis\tan ceAtoD}{runningspeed}\\ =\frac{20-x}{6.4}\end{array}$

Function for the time he spends swimming from point D to point B $=s(x)$

Elvis’s swimming speed is 0.9 meter per second

Distance D to B = $\sqrt{{12}^2+x^2}$

  $\begin{array}{c}s(x)=\frac{dis\tan ceDtoB}{swim\min gspeed}\\ =\frac{\sqrt{{12}^2+x^2}}{0.9}\end{array}$

(b)

Write a function t(x) that represents the total time Elvis spends travelling from point A to point D to point B.

  $t(x)=\frac{20-x}{6.4}+\frac{\sqrt{{12}^2+x^2}}{0.9}$

Given:

A mathematician at a lake throws a tennis ball from point A along the water’s edge to point B in the water, as shown. His dog, Elvis, runs along the beach from point A to point D and then swims to fetch the ball at point B.

  

Concept Used:

• The composition function $f\circ g$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

Function that represents the total time Elvis spends travelling from point A to point D to point B = $t\left(x\right)$

  $r(x)=\frac{20-x}{6.4}$

  $s(x)=\frac{\sqrt{{12}^2+x^2}}{0.9}$

  $\begin{array}{c}t(x)=r(x)+s(x)\\ =\frac{20-x}{6.4}+\frac{\sqrt{{12}^2+x^2}}{0.9}\end{array}$

Function for the total time he spends swimming from A to point D to point B $t\left(x\right)$

  $=\frac{20-x}{6.4}+\frac{\sqrt{{12}^2+x^2}}{0.9}$

(c)

Use a graphing calculator to graph t(x). Find the value of x that minimizes t(x). Explain the meaning of this value.

x=1.704m

Given:

A mathematician at a lake throws a tennis ball from point A along the water’s edge to point B in the water, as shown. His dog, Elvis, runs along the beach from point A to point D and then swims to fetch the ball at point B.

  

Concept Used:

• The composition function $f\circ g$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

Function that represents the total time Elvis spends travelling from point A to point D to point B = $t\left(x\right)$

  $t(x)=\frac{20-x}{6.4}+\frac{\sqrt{{12}^2+x^2}}{0.9}$

The graph of $t(x)=\frac{20-x}{6.4}+\frac{\sqrt{{12}^2+x^2}}{0.9}$ is -

  

At x=1.704 the value of t(x) will be minimum because if x = 1.704 Elvis have to cover minimum distance to reach B from A to D to B.