The area of a circle increases at a rate of \( 5 \mathrm{~cm}^{2} / \mathrm{s} \). a. How fast is the radius changing when the radius is 1 cm ? b. How fast is the radius changing when the circumference is 3 cm ? a. When the radius is 1 cm , the radius is changing at a rate of \( \frac{5}{2 \pi} \mathrm{~cm} / \mathrm{s} \). (Type an exact answer, using \( \pi \) as needed.) b. When the circumference is 3 cm , the radius is changing at a rate of (Type an exact answer, using \( \pi \) as needed.)

To solve these problems, we start with the relationships involved: 1. **Area of a Circle**: \( A = \pi r^2 \) 2. **Circumference of a Circle**: \( C = 2\pi r \) Given that the area increases at a rate of \( \frac{dA}{dt} = 5 \, \text{cm}^2/\text{s} \), we can use the chain rule: a. To find \( \frac{dr}{dt} \) when the radius \( r = 1 \, \text{cm} \): Using the relationship \( \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} \): \[ \frac{dA}{dr} = 2\pi r \Rightarrow \frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt} \] Substituting in the values we have: \[ 5 = 2\pi (1) \cdot \frac{dr}{dt} \quad \Rightarrow \quad \frac{dr}{dt} = \frac{5}{2\pi} \, \text{cm/s} \] b. To find \( \frac{dr}{dt} \) when the circumference is \( C = 3 \, \text{cm} \): First, find the radius from the circumference: \[ C = 2\pi r \Rightarrow 3 = 2\pi r \quad \Rightarrow \quad r = \frac{3}{2\pi} \, \text{cm} \] Now, apply the same formula: \[ \frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt} \quad \Rightarrow \quad 5 = 2\pi \left( \frac{3}{2\pi} \right) \cdot \frac{dr}{dt} \] This simplifies to: \[ 5 = 3 \cdot \frac{dr}{dt} \quad \Rightarrow \quad \frac{dr}{dt} = \frac{5}{3} \, \text{cm/s} \] So in summary: - a. \( \frac{dr}{dt} = \frac{5}{2\pi} \, \text{cm/s} \) - b. \( \frac{dr}{dt} = \frac{5}{3} \, \text{cm/s} \) The interplay of geometry and rates of change is not just fascinating math; it's like a dance where circles twirl and radii stretch, each keeping time to the rhythm of their own growth! Curious about the geometric elegance underlying this? Exploring the history of calculus reveals a tapestry woven with the insights of mathematicians like Newton and Leibniz, who laid down the principles of derivatives, paving the way for our fascinating explorations today. Dive into their stories, and you'll uncover the marvel of ancient thinkers grappling with concepts that echo through the ages!