The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate, then the ratio of the change of their areas, is

Options

A. `1 : pi`
B. `pi : 1`
C. `2 : pi`
D. `1 : 2`

Answer

To find the ratio of the change of the areas of a square and a circle, given that the side of the square is equal to the diameter of the circle and that the side and radius change at the same rate, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let the side of the square be \( a \).
- The diameter of the circle is equal to the side of the square, so \( d = a \).
- The radius of the circle is \( r = \frac{d}{2} = \frac{a}{2} \).

2. **Area Formulas:**
- The area of the square \( A_1 \) is given by:
\[ A_1 = a^2 \]
- The area of the circle \( A_2 \) is given by:
\[ A_2 = \pi r^2 = \pi \left(\frac{a}{2}\right)^2 = \frac{\pi a^2}{4} \]

3. **Differentiate Areas with Respect to Time:**
- Differentiate \( A_1 \):
\[ \frac{dA_1}{dt} = \frac{d}{dt}(a^2) = 2a \frac{da}{dt} \]
- Differentiate \( A_2 \):
\[ \frac{dA_2}{dt} = \frac{d}{dt}\left(\frac{\pi a^2}{4}\right) = \frac{\pi}{4} \cdot 2a \frac{da}{dt} = \frac{\pi a}{2} \frac{da}{dt} \]

4. **Express the Rates of Change:**
- We know that the side \( a \) and the radius \( r \) change at the same rate, which means:
\[ \frac{da}{dt} = \frac{dr}{dt} \]

5. **Calculate the Ratio of the Changes in Areas:**
- Now, we can find the ratio of the changes in areas:
\[ \text{Ratio} = \frac{\frac{dA_1}{dt}}{\frac{dA_2}{dt}} = \frac{2a \frac{da}{dt}}{\frac{\pi a}{2} \frac{da}{dt}} \]
- Simplifying this gives:
\[ \text{Ratio} = \frac{2a}{\frac{\pi a}{2}} = \frac{2a \cdot 2}{\pi a} = \frac{4}{\pi} \]

6. **Final Result:**
- Thus, the ratio of the change of their areas is:
\[ \frac{dA_1/dt}{dA_2/dt} = \frac{4}{\pi} \]