Part A: To find the total surface area of the doghouse, we need to add up the areas of all the faces that will be painted. The doghouse has six faces: two bases, two lateral faces, and two triangular faces. The area of each base is 8 times 7, or 56 square feet. The area of each lateral face is 6 times 8, or 48 square feet. The area of each triangular face can be found by using the formula A = (1/2)bh, where b is the base and h is the height. The base of each triangle is 7 feet, and the height can be found by using the Pythagorean theorem: h^2 = 6^2 + (4^2), where 4 is half of the base of the prism. Solving for h, we get h = sqrt(52), or about 7.21 feet. Therefore, the area of each triangular face is (1/2)(7)(7.21), or about 25.24 square feet. Adding up all the areas, we get:
Total surface area = 2(56) + 2(48) + 2(25.24) Total surface area = 112 + 96 + 50.48 Total surface area = 258.48 square feet
Part B: To find how many cans of paint are needed to paint the doghouse, we need to divide the total surface area by the area covered by one can of paint. Since one can of paint will cover 50 square feet, we get:
Number of cans = 258.48 / 50 Number of cans = 5.1696
Since we cannot buy a fraction of a can, we need to round up to the next whole number. Therefore, we need 6 cans of paint to paint the doghouse. We round up because we want to make sure we have enough paint to cover the entire surface, and not run out of paint before finishing.
I hope this helps!