Questions: A 35-mm camera allows you to control the diameter d of the aperture by changing the f-stop, which varies inversely with the diameter according to the formula f = 48/d. a. If f = 2, find the diameter. b. If the diameter is tripled, by what factor will the f-stop change? c. The radius r is one-half of the diameter. Describe the proportional relationship between the radius and the diameter. d. Area A is proportional to the radius squared according to the equation A = πr^2. What is the equation that relates area and diameter? e. What is the equation that relates area and f-stop? a. If f = 2, then the diameter is mm . (Type an integer or a simplified fraction.) b. If the diameter is tripled, the f-stop will change by a factor of (Type an integer or a simplified fraction.)

Solution Steps

To solve the given problems, we will use the provided formula \( f = \frac{48}{d} \).

a. To find the diameter when \( f = 2 \), we will rearrange the formula to solve for \( d \).

b. To determine the factor by which the \( f \)-stop changes when the diameter is tripled, we will substitute \( 3d \) into the formula and compare the new \( f \)-stop with the original.

c. The relationship between the radius \( r \) and the diameter \( d \) is direct, as \( r = \frac{d}{2} \).

Given the formula for the \( f \)-stop, \( f = \frac{48}{d} \), we can find the diameter when \( f = 2 \):

\[ d = \frac{48}{f} = \frac{48}{2} = 24 \]

If the diameter is tripled, the new diameter becomes \( d' = 3d = 3 \times 24 = 72 \). We can find the new \( f \)-stop using the same formula:

\[ f' = \frac{48}{d'} = \frac{48}{72} = \frac{2}{3} \]

To find the factor by which the \( f \)-stop changes, we compare the original \( f \)-stop with the new one:

\[ \text{Change Factor} = \frac{f}{f'} = \frac{2}{\frac{2}{3}} = 3 \]

The radius \( r \) is defined as half of the diameter:

\[ r = \frac{d}{2} \]

This shows a direct proportional relationship between the radius and the diameter.

Final Answer