This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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11. Find the height of the cylinder. Step 1: Calculate the radius of the cylinder. The diameter d = 10 cm. The radius r is half of the diameter. r = (d)/(2) = 10 cm2 = 5 cm Step 2: Use the volume formula to find the height. The volume of a cylinder is given by V = r^2 h. We are given V = 800 cm^3 and we found r = 5 cm. 800 cm^3 &= (5 cm)^2 h \\ 800 cm^3 &= (25 cm^2) h \\ h &= 800 cm^325 cm^2 \\ h &= (32)/() cm The height of the cylinder is approximately: h ≈ 10.19 cm The exact height is (32)/() cm. 12. The height of the diagonal in a square is 12 ft. What is the area and perimeter of the square? Step 1: Find the side length of the square. For a square with side length s, the diagonal L is given by L = ssqrt(2). We are given L = 12 ft. 12 ft &= ssqrt(2) \\ s &= 12 ftsqrt(2) \\ s &= 12sqrt(2)2 ft \\ s &= 6sqrt(2) ft Step 2: Calculate the area of the square. The area of a square is A = s^2. A &= (6sqrt(2) ft)^2 \\ A &= (6^2)(sqrt(2)^2) ft^2 \\ A &= (36)(2) ft^2 \\ A &= 72 ft^2 Step 3: Calculate the perimeter of the square. The perimeter of a square is P = 4s. P &= 4(6sqrt(2) ft) \\ P &= 24sqrt(2) ft The area of the square is 72 ft^2 and the perimeter is 24sqrt(2) ft. 13. What is the surface area of a rectangular prism with the dimensions 15 cm x 8 cm x 9 cm? Step 1: Identify the dimensions of the rectangular prism. Length l = 15 cm Width w = 8 cm Height h = 9 cm Step 2: Calculate the surface area. The surface area SA of a rectangular prism is given by the formula SA = 2(lw + lh + wh). SA &= 2((15 cm)(8 cm) + (15 cm)(9 cm) + (8 cm)(9 cm)) \\ SA &= 2(120 cm^2 + 135 cm^2 + 72 cm^2) \\ SA &= 2(327 cm^2) \\ SA &= 654 cm^2 The surface area of the rectangular prism is 654 cm^2. What's next?