To solve the problem of how the area of a circle changes as its radius increases, we need to follow a few steps. Let's break it down clearly. ### Step 1: Understand the Problem We have a circle with an initial radius of 5 cm, and this radius is increasing at a rate of 0.2 m/s. We need to find out how much the area of the circle increases as the radius changes. ### Step 2: Formula for the Area of a Circle The area \( A \) of a circle is given by the formula: \[ A = \pi r^2 \] where \( r \) is the radius of the circle. ### Step 3: Differentiate the Area with Respect to Time To find how the area changes with time as the radius changes, we need to differentiate the area with respect to time \( t \). Using the chain rule, we have: \[ \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} \] First, we find \( \frac{dA}{dr} \): \[ \frac{dA}{dr} = 2\pi r \] Now, substituting this into our equation gives: \[ \frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt} \] ### Step 4: Substitute Known Values We know: - The initial radius \( r = 5 \) cm (which we need to convert to meters for consistency with the rate of change of radius, since \( 0.2 \) m/s is in meters). - The rate of change of the radius \( \frac{dr}{dt} = 0.2 \) m/s. First, convert the radius from centimeters to meters: \[ r = 5 \text{ cm} = 0.05 \text{ m} \] Now, substitute \( r \) and \( \frac{dr}{dt} \) into the differentiated area formula: \[ \frac{dA}{dt} = 2\pi (0.05) \cdot (0.2) \] ### Step 5: Calculate the Increase in Area Now, perform the calculation: \[ \frac{dA}{dt} = 2\pi (0.05) (0.2) = 2\pi (0.01) = 0.02\pi \] ### Step 6: Compare with Options The calculated increase in area is \( 0.02\pi \) square meters. However, the options provided are in terms of \( \pi \) without a coefficient. To express \( 0.02\pi \) in terms of the options given: - \( 0.02\pi = \frac{2}{100}\pi = \frac{1}{50}\pi \) None of the options A, B, C, or D match \( 0.02\pi \) directly. However, if we consider the context of the problem, it seems there might be a misunderstanding in the options provided or the interpretation of the question. ### Step 7: Analyze the Options - **Option A: \( 2\pi \)** - This is too large compared to our calculated value. - **Option B: \( 5\pi \)** - This is also too large. - **Option C: \( \pi \)** - This is still larger than our result. - **Option D: \( 4\pi \)** - This is also too large. ### Conclusion The correct answer based on our calculations is \( 0.02\pi \), which does not match any of the provided options. Therefore, it seems there may be an error in the options given or a misunderstanding in the question's context. ### Revision Summary - The area of a circle is calculated using \( A = \pi r^2 \). - The rate of change of area with respect to time is found using differentiation. - The increase in area when the radius increases can be calculated using \( \frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt} \). - Always ensure units are consistent when performing calculations (convert cm to m in this case). If you encounter similar problems, remember to check your calculations and ensure the options provided align with your results.