8) Calculate the height of a cone whose volume is \( 37680 \mathrm{~cm}^{3} \), radius 30 cm and slant length is 50 cm.

1. The volume of a cone is given by \[ V = \frac{1}{3}\pi r^2 h. \] We are given \( V = 37680 \, \text{cm}^3 \) and \( r = 30 \, \text{cm} \). Plugging these into the formula, we have \[ 37680 = \frac{1}{3}\pi (30)^2 h. \] 2. Compute \( (30)^2 \) which is \[ 30^2 = 900. \] So the equation becomes \[ 37680 = \frac{1}{3}\pi \cdot 900 \cdot h. \] 3. Multiply \( \frac{1}{3} \) and \( 900 \) to get \( 300 \): \[ 37680 = 300\pi h. \] 4. Solve for \( h \) by dividing both sides by \( 300\pi \): \[ h = \frac{37680}{300\pi}. \] 5. Simplify the fraction \( \frac{37680}{300} \): \[ \frac{37680}{300} = 125.6. \] Thus, \[ h = \frac{125.6}{\pi}. \] 6. When you calculate numerically using \( \pi \approx 3.14 \): \[ h \approx \frac{125.6}{3.14} \approx 40 \, \text{cm}. \] 7. We also have the slant height \( l = 50 \, \text{cm} \) of the cone. The relationship between the height, radius, and slant height is given by the Pythagorean theorem: \[ l^2 = r^2 + h^2. \] Plugging in \( r = 30 \, \text{cm} \) and \( l = 50 \, \text{cm} \), we get \[ 50^2 = 30^2 + h^2. \] This simplifies to \[ 2500 = 900 + h^2. \] 8. Solve for \( h^2 \): \[ h^2 = 2500 - 900 = 1600. \] Taking the square root of both sides gives \[ h = \sqrt{1600} = 40 \, \text{cm}. \] Thus, the height of the cone is \[ h = 40 \, \text{cm}. \]