Question

Andrea must paint an antique sign composed of three spheres, each of which has a diameter of 12 inches. Each can of paint covers 288 square inches. How many cans of paint must Andrea buy to do the job? A. 4 B. 5 C. 6 D. 7

Short answer

B. 5

Step-by-step solution

Calculate the radius of one sphere

The diameter of each sphere is 12 inches. The radius is half of the diameter. Therefore, the radius is \[ r = \frac{12}{2} = 6 \text{ inches} \].

Calculate the surface area of one sphere

The formula for the surface area of a sphere is \[ 4 \pi r^2 \]. Using the radius from Step 1 (\[ r = 6 \text{ inches} \]):\[ 4 \pi (6)^2 = 4 \pi (36) = 144\pi \text{ square inches} \].

Calculate the total surface area of three spheres

Multiply the surface area of one sphere by 3:\[ 3 \times 144\pi = 432\pi \text{ square inches} \].

Calculate the total surface area covered by one can of paint

Each can of paint covers 288 square inches. First, determine how many cans are needed by dividing the total surface area by the area one can covers.\[ \frac{432\pi}{288} \approx 4.71 \].

Round up the number of cans required

As 4.71 cans is not an integer, round up to the next whole number. Thus, Andrea needs 5 cans.

Key concepts

radius calculation

To get the radius of one of the spheres, you need to know its diameter. The diameter is the length of a line that goes from one side of the circle, through its center, to the other side. For our spheres, the diameter is 12 inches. The radius, which is half of the diameter, can be calculated using the formula: \( r = \frac{d}{2} \). So for a diameter of 12 inches:
\( r = \frac{12}{2} = 6 \) inches.
This step is important because you need the radius to determine other measurements, like surface area.

surface area formula

The surface area of a sphere is the total area that covers the outside part of the sphere. The formula to find this is: \( 4\pi r^2 \). Using the radius we found earlier, 6 inches, we can substitute it into the formula:
\( 4\pi (6)^2 = 4\pi (36) = 144\pi \) square inches.
This formula helps you understand how much paint is needed to cover one sphere. The result tells you that the surface area of one of our spheres is \( 144\pi \) square inches.

total surface area

To find how much area needs to be painted for all three spheres, you multiply the surface area of one sphere by the number of spheres. In our case, we have three spheres, each with a surface area of \( 144\pi \):
\( 3 \times 144\pi = 432\pi \) square inches.
This calculation shows the total surface area Andrea needs to cover with paint. Total surface area is important when you want to understand the scope of a project like this.

paint coverage calculation

Finally, we need to determine how many cans of paint are necessary. Each can covers 288 square inches. Divide the total surface area by the area one can covers:
\( \frac{432\pi}{288} \approx 4.71 \).
Since you can’t buy a fraction of a can, you round up to the next whole number. Rounding up from 4.71 gives us 5. So, Andrea should buy 5 cans of paint to cover the three spheres adequately.