Karim is painting a design on a cylindrical barrel. The height of the barrel is 1.2 m. The radius of its base is 0.3 m. What area will the paint have to cover? (Remember to include the bottom and lid of the barrel.)

To find the area that the paint will have to cover, we need to calculate the lateral area, the area of the top and bottom circles, and then add them together.

1. Lateral Area: The lateral area is the surface area of the curved side of the cylinder. It can be calculated by multiplying the height of the cylinder by the circumference of its base. The formula for the lateral area is A = 2πrh, where r is the radius of the base and h is the height of the cylinder. In this case, r = 0.3 m and h = 1.2 m. Therefore, the lateral area is A = 2π(0.3)(1.2) = 2.28π square meters.

2. Top and Bottom Circles: The top and bottom of the barrel are circular, and their areas can be calculated using the formula A = πr^2, where r is the radius of the circular base. In this case, r = 0.3 m. Therefore, the area of the top and bottom circles is A = π(0.3)^2 = 0.09π square meters (for each circle).

3. Total Area: To get the total area, we need to add the lateral area to the areas of the top and bottom circles. Since there are two circles (top and bottom), we need to multiply their area by 2. Therefore, the total area of the paint that will have to cover is:

Total Area = Lateral Area + 2(Top and Bottom Circles)
Total Area = 2.28π + 2(0.09π)
Total Area = 2.28π + 0.18π
Total Area = 2.46π square meters

Therefore, the paint will have to cover an area of approximately 2.46π square meters.

area = 2πr^2 + 2πrh

now just plug in your values for r and h.

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