8) Volume of cylinder: \( V=\pi r^{2} \cdot h \), for \( r \)

To solve for the radius \( r \) in the volume formula of a cylinder: \[ V = \pi r^{2} h \] follow these steps: ### Step 1: Start with the Volume Formula \[ V = \pi r^{2} h \] ### Step 2: Isolate \( r^{2} \) To isolate \( r^{2} \), divide both sides of the equation by \( \pi h \): \[ \frac{V}{\pi h} = r^{2} \] \[ r^{2} = \frac{V}{\pi h} \] ### Step 3: Solve for \( r \) Take the square root of both sides to solve for \( r \): \[ r = \sqrt{ \frac{V}{\pi h} } \] ### Final Formula \[ r = \sqrt{ \frac{V}{\pi h} } \] ### Example Calculation Suppose you have a cylinder with a volume \( V = 50 \) cubic units and a height \( h = 10 \) units. To find the radius \( r \): \[ r = \sqrt{ \frac{50}{\pi \times 10} } = \sqrt{ \frac{50}{31.4159} } \approx \sqrt{1.5915} \approx 1.261 \text{ units} \] So, the radius of the cylinder is approximately **1.261 units**. ### Summary To find the radius \( r \) of a cylinder when you know its volume \( V \) and height \( h \), use the formula: \[ r = \sqrt{ \frac{V}{\pi h} } \] This formula allows you to calculate the radius by rearranging the original volume equation to solve for \( r \).