Question

two similar storage tanks are shown in the drawing below. the height of the cone - shaped portion on the larger tank is 5 feet. points c and f are the centers of the bases for both the cylinders and the cones. bc = \frac{5}{3}ef, and bc = 5ft. how much more volume can the larger tank hold than the smaller tank? 795.45 cu ft 533.65 cu ft 451.34 cu ft 771.16 cu ft

Explanation:

Step1: Find the radius of the smaller cone

Given \(BC=\frac{5}{3}EF\) and \(BC = 5\) ft. Then \(5=\frac{5}{3}EF\), so \(EF=3\) ft.

Step2: Find the height of the smaller cone

Since the tanks are similar, the ratio of their corresponding linear dimensions is the same. The ratio of radii \(r_{1}:r_{2}=5:3\). Let the height of the smaller cone be \(h\). For the conical - part (similar solids), if we assume the height of the larger cone is \(H = 5\) ft. Using the similarity ratio \(\frac{H}{h}=\frac{5}{3}\), so \(h = 3\) ft. The height of the larger cylinder \(H_{cylinder}=12 - 5=7\) ft. For the smaller tank, assume the height of the cylinder is also in the ratio \(5:3\). Let the height of the smaller cylinder be \(h_{cylinder}\), then \(\frac{7}{h_{cylinder}}=\frac{5}{3}\), so \(h_{cylinder}=4.2\) ft.

Step3: Calculate the volume of the larger tank

The volume of the larger tank \(V_{1}=V_{cylinder1}+V_{cone1}\).
The formula for the volume of a cylinder \(V_{cylinder}=\pi r^{2}h\) and for a cone \(V_{cone}=\frac{1}{3}\pi r^{2}h\).
\(V_{cylinder1}=\pi\times5^{2}\times7 = 175\pi\)
\(V_{cone1}=\frac{1}{3}\pi\times5^{2}\times5=\frac{125}{3}\pi\)
\(V_{1}=175\pi+\frac{125}{3}\pi=\frac{525\pi + 125\pi}{3}=\frac{650\pi}{3}\approx\frac{650\times3.14}{3}\approx680.33\)

Step4: Calculate the volume of the smaller tank

\(V_{cylinder2}=\pi\times3^{2}\times4.2=37.8\pi\)
\(V_{cone2}=\frac{1}{3}\pi\times3^{2}\times3 = 9\pi\)
\(V_{2}=37.8\pi+9\pi=46.8\pi\approx46.8\times3.14 = 146.95\)

Step5: Calculate the difference in volumes

\(\Delta V=V_{1}-V_{2}\)
\(V_{1}=\frac{650\times3.14}{3}\approx680.33\), \(V_{2}=46.8\times3.14 = 146.95\)
\(\Delta V=680.33-146.95 = 533.38\approx533.65\) (due to rounding differences in intermediate steps)

Answer:

533.65 cu ft