Question

select the correct answer. a farmers silo is a cone on top of a 100 - foot - tall cylinder. the height of the cone is three - fourths the radius. what is the maximum radius of the silo in feet if there are 6,000 square feet of material? 8 feet 8.5 feet 9 feet 9.5 feet

Explanation:

Step1: Find the surface - area formulas

The surface - area of the silo is the sum of the lateral surface area of the cylinder and the lateral surface area of the cone. The lateral surface area of a cylinder \(A_{cylinder}=2\pi rh\), and the lateral surface area of a cone \(A_{cone}=\pi rl\), where \(l = \sqrt{r^{2}+h_{cone}^{2}}\). Given \(h_{cylinder}=100\) feet, \(h_{cone}=\frac{3}{4}r\). The total surface area \(A = 2\pi r\times100+\pi r\sqrt{r^{2}+(\frac{3}{4}r)^{2}}\). Simplify \(\sqrt{r^{2}+(\frac{3}{4}r)^{2}}=\sqrt{r^{2}(1 + \frac{9}{16})}=\sqrt{\frac{16r^{2}+9r^{2}}{16}}=\sqrt{\frac{25r^{2}}{16}}=\frac{5}{4}r\). So \(A=200\pi r+\frac{5}{4}\pi r^{2}\).

Step2: Set up the equation

We know that \(A = 6000\) square - feet. So \(200\pi r+\frac{5}{4}\pi r^{2}=6000\). Divide the entire equation by \(\pi\): \(200r+\frac{5}{4}r^{2}=\frac{6000}{\pi}\approx\frac{6000}{3.14}\approx1910.83\). Multiply through by 4 to get \(800r + 5r^{2}=7643.32\). Rearrange to the quadratic form \(5r^{2}+800r - 7643.32 = 0\).

Step3: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 5\), \(b = 800\), \(c=-7643.32\)), the quadratic formula is \(r=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). First, calculate the discriminant \(\Delta=b^{2}-4ac=(800)^{2}-4\times5\times(-7643.32)=640000 + 152866.4 = 792866.4\). Then \(r=\frac{-800\pm\sqrt{792866.4}}{10}\). We take the positive root since radius cannot be negative. \(r=\frac{-800+\sqrt{792866.4}}{10}\approx\frac{-800 + 890.43}{10}=\frac{90.43}{10}=9.043\approx9\) feet.

Answer:

9 feet