Chapter 3.2, Problem 89PS

Solution

(a)

To solve the formula $V=\frac{4}{3}\pi r^3$ for r .

  $r=\sqrt[3]{\frac{3V}{4\pi }}$

Given:

Two round balloons are filled with water. One balloon contains twice as much water as the other balloon.

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to solve the given volume formula $V=\frac{4}{3}\pi r^3$ for r , isolate r by performing basic algebraic operations and simplify further using the exponent properties as shown below

  $\begin{array}{c}V=\frac{4}{3}\pi r^3\\ \frac{3}{4}\cdot V=\frac{3}{4}\cdot \frac{4}{3}\pi r^3\\ \frac{3V}{4}=\pi r^3\\ \frac{3V}{4\pi }=r^3\\ r=\sqrt[3]{\frac{3V}{4\pi }}\end{array}$

Therefore, the equation for r is $r=\sqrt[3]{\frac{3V}{4\pi }}$ .

(b)

To show that $S={\left(4\pi \right)}^{\frac{1}{3}}{\left(3V\right)}^{\frac{2}{3}}$ by substituting the value of r from above part into the surface area of a sphere formula $S=4\pi r^2$ .

Given:

Two round balloons are filled with water. One balloon contains twice as much water as the other balloon.

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to show the required result, substitute $r=\sqrt[3]{\frac{3V}{4\pi }}$ into the surface area formula and simplify further using the exponent properties as shown below

  $\begin{array}{c}S=4\pi r^2\\ S=4\pi {\left(\sqrt[3]{\frac{3V}{4\pi }}\right)}^2\\ S=4\pi {\left({\left(\frac{3V}{4\pi }\right)}^{\frac{1}{3}}\right)}^2\\ S=4\pi {\left(\frac{3V}{4\pi }\right)}^{\frac{2}{3}}\\ S=\left(4\pi \right)\cdot \frac{{\left(3V\right)}^{\frac{2}{3}}}{{\left(4\pi \right)}^{\frac{2}{3}}}\\ S={\left(4\pi \right)}^{1-\frac{2}{3}}{\left(3V\right)}^{\frac{2}{3}}\\ S={\left(4\pi \right)}^{\frac{1}{3}}{\left(3V\right)}^{\frac{2}{3}}\end{array}$

Therefore, the surface area of the sphere can be expressed as $S={\left(4\pi \right)}^{\frac{1}{3}}{\left(3V\right)}^{\frac{2}{3}}$ .

(c)

To compare the surface area of the given balloons.

The surface area of the bigger balloon is $2^{\frac{2}{3}}$ times the surface area of the smaller balloon.

Given:

Two round balloons are filled with water. One balloon contains twice as much water as the other balloon.

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

Since one balloon contains twice as much water as the other balloon, the volume of the first balloon must be twice as of the second balloon.

So, the ratio of surface areas of the balloons is

  $\begin{array}{c}\frac{S_1}{S_2}=\frac{{\left(4\pi \right)}^{\frac{1}{3}}{\left(3\left(2V\right)\right)}^{\frac{2}{3}}}{{\left(4\pi \right)}^{\frac{1}{3}}{\left(3V\right)}^{\frac{2}{3}}}\\ =\frac{{\left(6V\right)}^{\frac{2}{3}}}{{\left(3V\right)}^{\frac{2}{3}}}\\ ={\left(\frac{6V}{3V}\right)}^{\frac{2}{3}}\\ ={\left(2\right)}^{\frac{2}{3}}\end{array}$

Therefore, the surface area of the bigger balloon is $2^{\frac{2}{3}}$ times the surface area of the smaller balloon.