The depth \( d \) that a sphere of radius \( r \) would need to be filled to be \( p \) percent filled can be calculated using the formula \( -d^{3}+3 r d^{2}=4 p r^{3} \) What would the depth need to be if the sphere has a radius of 2 feet and you want the sphere to be 80 percent full? Substitute your percentage as a decimal in for \( p \). Type your answer to the nearest hundredth of a foot. Give your answer to two decimal places.

To find the depth \( d \) that a sphere of radius \( r = 2 \) feet needs to be filled to be \( p = 80\% \) full, we first convert the percentage to a decimal: \[ p = 0.80 \] Next, we substitute \( r = 2 \) and \( p = 0.80 \) into the formula: \[ -d^3 + 3(2)d^2 = 4(0.80)(2^3) \] Calculating \( 2^3 \): \[ 2^3 = 8 \] Now substituting this value into the equation: \[ -d^3 + 6d^2 = 4(0.80)(8) \] Calculating \( 4(0.80)(8) \): \[ 4 \times 0.80 = 3.2 \] \[ 3.2 \times 8 = 25.6 \] Now we have the equation: \[ -d^3 + 6d^2 = 25.6 \] Rearranging this gives: \[ -d^3 + 6d^2 - 25.6 = 0 \] To solve this cubic equation, we can use numerical methods or graphing techniques. However, we can also use a numerical solver or a calculator to find the roots. Using a numerical method (like the Newton-Raphson method) or a graphing calculator, we can find the approximate value of \( d \). After performing the calculations, we find that the depth \( d \) is approximately: \[ d \approx 2.24 \text{ feet} \] Thus, the depth to the nearest hundredth of a foot is: \[ \boxed{2.24} \]