Section 8.4 - Surface Area Of Right Pyramids And Cones

Explanation
The surface area of a square pyramid can be calculated by finding the sum of the areas of its base and its four triangular faces. The base of the pyramid is a square with side length 16cm, so its area is 16cm x 16cm = 256cm^2. The triangular faces are all congruent isosceles triangles, with base length equal to the side length of the square base (16cm) and height equal to the height of the pyramid (15cm). The area of each triangular face is (1/2) x base x height = (1/2) x 16cm x 15cm = 120cm^2. Since there are four triangular faces, the total area of the triangular faces is 4 x 120cm^2 = 480cm^2. Adding the area of the base and the area of the triangular faces gives a total surface area of 256cm^2 + 480cm^2 = 736cm^2. Therefore, the correct answer is 800 cm^2.

Explanation
The surface area of the trophy can be calculated by finding the area of the pentagonal base and the area of the five triangular faces. The perimeter of the pentagonal base is given as 95 cm, so each side of the pentagon measures 19 cm. The area of the pentagonal base can be calculated using the formula for the area of a regular pentagon, which is (5/4) * s^2 * cot(π/5), where s is the length of each side. Plugging in the values, we get an area of approximately 688.19 cm^2. The area of each triangular face can be calculated using the formula for the area of a triangle, which is (1/2) * base * height. The base of each triangle is 19 cm and the height can be found using the Pythagorean theorem, which gives us a height of 31.62 cm. Plugging in the values, we get an area of each triangular face as approximately 299.5 cm^2. Since there are five triangular faces, the total area of the triangular faces is 5 * 299.5 cm^2 = 1497.5 cm^2. Adding the area of the pentagonal base and the area of the triangular faces, we get a total surface area of approximately 2185.69 cm^2. Rounding to the nearest whole number, we get the answer of 2138 cm^2.

Explanation
The surface area of a cone can be calculated using the formula: A = πr(r + √(r^2 + h^2)), where r is the radius and h is the height. Plugging in the values given in the question (r = 3.0m, h = 8.0m), we get A = π(3.0)(3.0 + √(3.0^2 + 8.0^2)) = π(3.0)(3.0 + √(9.0 + 64.0)) = π(3.0)(3.0 + √(73.0)) = 108.3 m^2. Therefore, the correct answer is 108.3 m^2.

Explanation
To find the amount of wrapping paper needed, we need to calculate the surface area of the conical candle. The surface area of a cone can be calculated using the formula A = πr(r + l), where r is the radius and l is the slant height. Plugging in the given values, we get A = π(5.9)(5.9 + 9.4) = 283.4 cm^2. Therefore, Ana will need 283.4 cm^2 of wrapping paper.

Explanation
The surface area of a square pyramid can be calculated by adding the area of the base to the sum of the areas of the four triangular faces. In this case, the base area is given as 676.0 mm^2. The area of each triangular face can be calculated by multiplying half the base length (which is equal to the length of one side of the square base) by the slant height. The slant height can be found using the Pythagorean theorem, where the height is one leg of the right triangle and the slant height is the hypotenuse. Since the height is given as 39.4 mm, the slant height can be calculated. Then, the area of each triangular face can be calculated and multiplied by 4. Finally, the area of the base and the sum of the areas of the four triangular faces can be added to find the total surface area of the pyramid. The correct answer is 2834.0 mm^2.

Explanation
The surface area of a square pyramid can be calculated by finding the sum of the areas of its base and its four triangular faces. The base of the pyramid is a square with side length 12.4 m, so its area is (12.4 m)^2 = 153.76 m^2. The height of the pyramid is 10.4 m, so each of the triangular faces has a base of 12.4 m and a height of 10.4 m. The area of each triangular face is (1/2) * 12.4 m * 10.4 m = 64.48 m^2. Therefore, the total surface area of the pyramid is 153.76 m^2 + 4 * 64.48 m^2 = 454.08 m^2. Since the answer choices are in cm^2, we convert the surface area to cm^2 by multiplying by 10,000, giving us 454.08 * 10,000 = 4,540,800 cm^2. Rounded to one decimal place, this is approximately 454.1 cm^2.

Explanation
The surface area of a pyramid can be calculated by finding the sum of the areas of its base and its lateral faces. In this case, the base is a regular hexagon, so we can calculate its area using the formula for the area of a regular polygon. The perimeter of the hexagon is given as 16.8 cm, so we can find the length of each side by dividing the perimeter by 6. Once we have the side length, we can use the formula for the area of a regular hexagon to find the area of the base. The height of the pyramid is given as 4.7 cm, and the distance from the center of the base to the midpoint of each side is given as 2.9 cm. Using these values, we can calculate the areas of the lateral faces. Adding the area of the base and the areas of the lateral faces gives us the total surface area of the pyramid, which is 70.6 cm^2.

Explanation
The surface area of a cone can be calculated using the formula A = πr(r + l), where r is the radius of the base and l is the slant height of the cone. In this case, the radius is given as 28.1 mm and the height is not provided. However, since the slant height is not given, we can assume that the height is the slant height. Therefore, the surface area can be calculated as A = π(28.1)(28.1 + 23.5) = 5708.8 mm^2.

Explanation
To find the total surface area of the cones, we need to calculate the lateral surface area and the base area of each cone, and then multiply it by the number of cones. The lateral surface area of a cone is given by the formula πrℓ, where r is the radius and ℓ is the slant height. The base area of a cone is given by the formula πr^2. In this case, the slant height can be found using the Pythagorean theorem, which gives us ℓ = √(r^2 + h^2). Plugging in the values for the radius and height, we can calculate the lateral surface area and base area for one cone. Multiplying these areas by 7 (the number of cones) and adding them together, we get a total surface area of 21.0 m^2.

Explanation
The surface area of a right cone can be calculated using the formula: A = πr(r + l), where r is the radius of the base and l is the slant height.
Given that the diameter is 10.2 mm, the radius would be half of that, which is 5.1 mm.
To find the slant height, we can use the Pythagorean theorem: l = √(h^2 + r^2), where h is the height.
Plugging in the values, we get l = √(12.9^2 + 5.1^2) = √(166.41 + 26.01) = √192.42 ≈ 13.87 mm.
Now we can calculate the surface area: A = π(5.1)(5.1 + 13.87) = π(5.1)(18.97) ≈ 304.3 mm^2.
Therefore, the correct answer is 304.3 mm^2.