Chapter 9.7, Problem 30HP

Solution

(a)

The formula $V=\pi r^{2}h$ for h .

  $h=\frac{V}{\pi r^2}$

Given information:

The formula: $V=\pi r^{2}h$ .

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to solve the formula for h , first divide each sides by pr², to simplify further as shown below:

  $\begin{array}{l}V=\pi r^{2}h\\ \frac{V}{\pi r^2}=\frac{\pi r^{2}h}{\pi r^2}\\ h=\frac{V}{\pi r^2}\end{array}$

Thus, the formula is $h=\frac{V}{\pi r^2}$ .

(b)

The equation $V=\pi r^{2}h$ for r .

  $r=\pm \sqrt{\frac{V}{\pi h}}$

Given information:

The equation: $V=\pi r^{2}h$ .

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to solve the formula for r , first divide both sides by ph, and then applying square root both sides to simplify further as shown below:

  $\begin{array}{l}V=\pi r^{2}h\\ \frac{V}{\pi h}=\frac{\pi r^{2}h}{\pi h}\\ r^2=\frac{V}{\pi h}\\ \sqrt{r^2}=\pm \sqrt{\frac{V}{\pi h}}\\ r=\pm \sqrt{\frac{V}{\pi h}}\end{array}$

Thus, the formula is $r=\pm \sqrt{\frac{V}{\pi h}}$ .