Chapter 3.3, Problem 4MRPS

Solution

(a)

Write an equation that gives the volume V of the pool as a function of the radius x. Use 3.14 for $\pi$ .

  $V=12.56x^2$

Given:

A cylindrical above-ground pool has a height of 5 feet and a radius of x feet.

  

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:

Height of cylindrical pool (h) = 5 feet

Radius of cylindrical pool (r) = x feet

Volume of cylindrical pool $V=\pi r^{2}h$

  $\begin{array}{c}V=\pi r^{2}h\\ $ =3.14{(x)}^{2}4$=3.14\cdot (4)x^2\\ =12.56x^2\end{array}$

(b)

You use a hose to fill the pool with water. Water flows from the hose at a rate of 128 cubic feet per hour. After 8.8 hours the pool is half full. Write an equation that you can use to find the radius x of the pool.

  $x^2=\frac{2252.8}{12.56}$

Given:

A cylindrical above-ground pool has a height of 5 feet and a radius of x feet.

  

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:

Height of cylindrical pool (h) = 5 feet

Radius of cylindrical pool (r) = x feet

Volume of cylindrical pool $V=\pi r^{2}h$

  $\begin{array}{c}V=\pi r^{2}h\\ $ =3.14{(x)}^{2}4$=3.14\cdot (4)x^2\\ =12.56x^2\end{array}$

Water flows from the hose at a rate of 128 cubic feet per hour.

And in 8.8 hours the pool is half full.

That means pool will be filled full in $(8.8)\cdot 2=17.6$ hours

Volume of pool V = (total hours in filling pool) $\cdot$ (rate of water flows)

  $\begin{array}{c}V=(17.6)\cdot 128\\ =2252.8\end{array}$

As $V=12.56x^2$

  $\begin{array}{l}12.56x^2=2252.8\\ x^2=\frac{2252.8}{12.56}\end{array}$

(c)

What is the radius of pool?

Radius of pool = 13.4 feet

Given:

A cylindrical above-ground pool has a height of 5 feet and a radius of x feet.

  

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:

Height of cylindrical pool (h) = 5 feet

Radius of cylindrical pool (r) = x feet

Volume of cylindrical pool $V=\pi r^{2}h$

As solved above $x^2=\frac{2252.8}{12.56}$

  $\begin{array}{c}{x}^2=\frac{2252.8}{12.56}\\ x^2=179.36\\ x=\sqrt{179.36}\\ x=13.4\end{array}$

Radius of cylindrical pool (r) = x feet=13.4 feet

(d)

Find the total number of hours it will take to fill 4/5 of the pool. Your answer should include the 8.8 hours it took to fill the bottom half of the pool in part (b).

15.29hours

Given:

A second hose that outputs 104 cubic feet of water per hour is added after the pool is half full.

  

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:

Height of cylindrical pool (h) = 5 feet

Radius of cylindrical pool (r) = x feet

Volume of cylindrical pool $V=\pi r^{2}h$

  $asV=2252.8$

Water flows from the first hose at a rate of 128 cubic feet of water per hour.

And in 8.8 hours the pool is half full.

Water flows from the second hose at a rate of 104 cubic feet of water per hour.

(Duration of first hose open)(per hour fill) + (Duration of second hose opens)(per hour fill) = $\frac{4}{5}(2252.8)$

  $\begin{array}{c}(8.8)128+x\cdot (104)=\frac{4}{5}(2252.8)\\ 1126.4+104x=\frac{9011.2}{5}\\ 1126.4+104x=1802.24\\ 104x=1802.24-1126.4\\ 104x=675.84\\ x=\frac{675.84}{104}\\ x=6.49\end{array}$

So the total number of hours it will take to fill 4/5 of the pool $=8.8+6.49=15.29$