Question

8-85. solve each equation below for the indicated variable. hint (all): review appendix a on solving equations and the beginning of chapter 6. a. solve for ( l ): ( p = 2l + 2w ) hint (a): b. solve for ( pi ): ( c = 2pi r ) hint (b): use division property of equality to isolate ( pi ). c. solve for ( r ): ( v = \frac{4}{3}pi r^3 ) hint (c):

Explanation:

Part a: Solve for \( l \) in \( P = 2l + 2w \)

Step 1: Subtract \( 2w \) from both sides

To isolate the term with \( l \), we subtract \( 2w \) from both sides of the equation.
\( P - 2w = 2l + 2w - 2w \)
Simplifying the right side, \( 2w - 2w = 0 \), so we get:
\( P - 2w = 2l \)

Step 2: Divide both sides by 2

To solve for \( l \), we divide both sides of the equation by 2.
\( \frac{P - 2w}{2} = \frac{2l}{2} \)
Simplifying the right side, \( \frac{2l}{2} = l \), so we get:
\( l = \frac{P - 2w}{2} \)

Part b: Solve for \( \pi \) in \( C = 2\pi r \)

Step 1: Divide both sides by \( 2r \)

To isolate \( \pi \), we divide both sides of the equation by \( 2r \) (assuming \( r
eq 0 \)).
\( \frac{C}{2r} = \frac{2\pi r}{2r} \)

Step 2: Simplify the right side

Simplifying the right side, \( \frac{2\pi r}{2r} = \pi \) (since \( 2r \) cancels out), so we get:
\( \pi = \frac{C}{2r} \)

Part c: Solve for \( r \) in \( V = \frac{4}{3}\pi r^3 \)

Step 1: Multiply both sides by \( \frac{3}{4} \)

To eliminate the fraction, we multiply both sides of the equation by \( \frac{3}{4} \).
\( \frac{3}{4}V = \frac{3}{4} \cdot \frac{4}{3}\pi r^3 \)

Step 2: Simplify the right side

Simplifying the right side, \( \frac{3}{4} \cdot \frac{4}{3} = 1 \), so we get:
\( \frac{3V}{4} = \pi r^3 \)

Step 3: Divide both sides by \( \pi \)

To isolate \( r^3 \), we divide both sides by \( \pi \) (assuming \( \pi
eq 0 \)).
\( \frac{3V}{4\pi} = r^3 \)

Step 4: Take the cube root of both sides

To solve for \( r \), we take the cube root of both sides.
\( r = \sqrt[3]{\frac{3V}{4\pi}} \)

Answer:

s:
a. \( \boldsymbol{l = \frac{P - 2w}{2}} \)
b. \( \boldsymbol{\pi = \frac{C}{2r}} \)
c. \( \boldsymbol{r = \sqrt[3]{\frac{3V}{4\pi}}} \)