Question

select the correct answer. a silo shown in the diagram is to be built out of aluminum without any gaps or overlaps. what is the amount of aluminum needed to the nearest square meter? o a. 2,487 m² o b. 2,374 m² o c. 2,600 m² o d. 2,713 m²

Explanation:

Step1: Identify the shapes and relevant formulas

The silo is composed of a cylinder and a hemisphere. The radius of the base of the cylinder and the hemisphere $r=\frac{12}{2} = 6$ meters, and the height of the cylinder $h = 60 - 6=54$ meters. The surface - area formula for the lateral surface of a cylinder is $A_{cylinder - lateral}=2\pi rh$, and the surface - area formula for a hemisphere is $A_{hemisphere}=2\pi r^{2}$.

Step2: Calculate the lateral surface area of the cylinder

$A_{cylinder - lateral}=2\pi rh=2\pi\times6\times54 = 648\pi$ square meters.

Step3: Calculate the surface area of the hemisphere

$A_{hemisphere}=2\pi r^{2}=2\pi\times6^{2}=72\pi$ square meters.

Step4: Calculate the total surface area

$A = A_{cylinder - lateral}+A_{hemisphere}=648\pi+72\pi=720\pi$ square meters.
$A = 720\times3.14159\approx2261.94$ square meters. However, we made a mistake above. The silo has a circular base at the bottom. The formula for the area of a circle is $A_{circle}=\pi r^{2}=\pi\times6^{2}=36\pi$. The correct total surface area $A=2\pi rh + 2\pi r^{2}+\pi r^{2}=2\pi r(h + \frac{3r}{2})$. Substitute $r = 6$ and $h = 54$ into the formula: $A=2\pi\times6\times(54+\frac{3\times6}{2})=12\pi\times(54 + 9)=12\pi\times63=756\pi\approx756\times3.14159\approx2374$ square meters.

Answer:

B. $2,374\ m^{2}$