Question

15 in physics class, esther learned that force due to gravity can be determined by using the formula ( f = \frac{gm_1m_2}{r^2} ). solve for ( r ) in terms of ( f, g, m_1, ) and ( m_2 ).
16 shoe sizes and foot length are related by the formula ( s = 3f - 24 ), where ( s ) represents the shoe size and ( f ) represents the length of the foot, in inches.
a solve the formula for ( f ).
b to the nearest tenth of an inch, how long is the foot of a person who wears a size ( 10\frac{1}{2} ) shoe?
17 the slant height, ( ell ), of the conical water tank shown in the accompanying diagram is ( ell = sqrt3{\frac{8v}{pi}} ). solve for ( v ), in terms of ( ell ) and ( pi ).
18 the volume of earth can be calculated by using formula ( v = \frac{4}{3}pi r^3 ). solve for ( r ) in terms of ( v ).

Explanation:

Question 15

Step1: Start with the formula \( F = \frac{Gm_1m_2}{r^2} \)

We need to isolate \( r \). First, multiply both sides by \( r^2 \):
\( F r^2 = Gm_1m_2 \)

Step2: Divide both sides by \( F \)

\( r^2 = \frac{Gm_1m_2}{F} \)

Step3: Take the square root of both sides

Since \( r \) represents a distance, we take the positive square root:
\( r = \sqrt{\frac{Gm_1m_2}{F}} \) (or equivalently \( r = \frac{\sqrt{Gm_1m_2F}}{F} \) after rationalizing, but the first form is also correct)

Step1: Start with the formula \( S = 3F - 24 \)

We want to solve for \( F \). First, add 24 to both sides:
\( S + 24 = 3F \)

Step2: Divide both sides by 3

\( F = \frac{S + 24}{3} \) (or \( F=\frac{1}{3}S + 8 \))

Step1: Identify the value of \( S \)

The shoe size \( S = 10\frac{1}{2}=\frac{21}{2} = 10.5 \)

Step2: Substitute \( S = 10.5 \) into the formula for \( F \) from part (a)

Using \( F=\frac{S + 24}{3} \), substitute \( S = 10.5 \):
\( F=\frac{10.5 + 24}{3} \)

Step3: Calculate the numerator

\( 10.5+24 = 34.5 \)

Step4: Divide by 3

\( F=\frac{34.5}{3}=11.5 \)

Answer:

\( r = \sqrt{\frac{Gm_1m_2}{F}} \)

Question 16a