Question

1. john, a farmer, wants to paint his silo. the silo has a cylinder base and a half - spherical roof as shown in the diagram.

a. calculate the surface area of the silo to the nearest tenth of a square metre. hint: use half of a sphere, and use a cylinder but without the circular bases.
b. john wants to paint the silo. each can of paint covers approximately 40 m² of area. estimate how many cans of paint he will need to paint the silo.

Explanation:

Step1: Calculate surface - area of half - sphere

The formula for the surface area of a sphere is $A_{sphere}=4\pi r^{2}$. For a half - sphere, $A_{half - sphere}=2\pi r^{2}$. Given the diameter of the base of the cylinder (which is the same as the diameter of the half - sphere) is $15$ m, so the radius $r = 7.5$ m. Then $A_{half - sphere}=2\pi(7.5)^{2}=2\pi\times56.25 = 112.5\pi$ $m^{2}$.

Step2: Calculate lateral surface area of cylinder

The formula for the lateral surface area of a cylinder is $A_{cylinder}=2\pi rh$. Here, $r = 7.5$ m and $h = 40$ m. So $A_{cylinder}=2\pi\times7.5\times40=600\pi$ $m^{2}$.

Step3: Calculate total surface area of silo

The total surface area of the silo $A = A_{half - sphere}+A_{cylinder}=112.5\pi+600\pi=(112.5 + 600)\pi=712.5\pi$ $m^{2}$.
$A\approx712.5\times3.14159\approx2238.9$ $m^{2}$.

Step4: Calculate number of paint cans

If each can of paint covers approximately $40$ $m^{2}$, then the number of cans $n=\frac{A}{40}=\frac{2238.9}{40}\approx56.0$ cans. Since we can't buy a fraction of a can, we need to round up to $57$ cans.

Answer:

a. $2238.9$ $m^{2}$
b. $57$ cans