The lampshade is a frustum of a cone with:

Here's the solution to the problem: The lampshade is a frustum of a cone with: Top radius (r) = 7 cm Bottom radius (R) = 14 cm Slant height of the frustum (l_f) = 20 cm Let L be the slant height of the original (larger) cone and l_1 be the slant height of the smaller cone that was cut off. Let H be the height of the original cone and h_1 be the height of the smaller cone. Let h be the height of the lampshade (frustum). a) The slant height of the original cone correct to two decimal places. Step 1: Use similar triangles formed by the radii and slant heights. For the original cone and the smaller cone, we have: (R)/(r) = (L)/(l_1) (14)/(7) = (L)/(l_1) 2 = (L)/(l_1) L = 2l_1 Step 2: Use the given slant height of the frustum. The slant height of the frustum is the difference between the slant height of the original cone and the smaller cone: L - l_1 = 20 cm Step 3: Substitute L = 2l_1 into the equation. 2l_1 - l_1 = 20 l_1 = 20 cm Step 4: Calculate L. L = 2l_1 = 2 × 20 = 40 cm The slant height of the original cone is 40.00 cm. b) The height h, of the lampshade. Step 1: Consider a right-angled triangle formed by the height of the frustum (h), the difference in radii (R-r), and the slant height of the frustum (l_f). The difference in radii is 14 - 7 = 7 cm. The slant height of the frustum is 20 cm. Step 2: Apply the Pythagorean theorem. h^2 + (R-r)^2 = l_f^2 h^2 + (14-7)^2 = 20^2 h^2 + 7^2 = 20^2 h^2 + 49 = 400 h^2 = 400 - 49 h^2 = 351 h = sqrt(351) Step 3: Calculate the numerical value. h ≈ 18.73499 cm Rounding to two decimal places: The height of the lampshade is 18.73 cm. c) The curved surface area of the lampshade. Step 1: Use the formula for the curved surface area (CSA) of a frustum. CSA = (R + r) l_f Where R = 14 cm, r = 7 cm, and l_f = 20 cm. Step 2: Substitute the values. CSA = (14 + 7) (20) CSA = (21) (20) CSA = 420 cm^2 Step 3: Calculate the numerical value. CSA ≈ 420 × 3.14159265 CSA ≈ 1319.4689 cm^2 Rounding to two decimal places: The curved surface area of the lampshade is 1319.47 cm^2. d) The volume of the lampshade correct to 4 significant figures. Step 1: Calculate the height of the original cone (H) and the smaller cone (h_1). From part (a), L = 40 cm and l_1 = 20 cm. For the original cone: H = sqrt(L^2 - R^2) = sqrt(40^2 - 14^2) = sqrt(1600 - 196) = sqrt(1404) cm For the smaller cone: h_1 = sqrt(l_1^2 - r^2) = sqrt(20^2 - 7^2) = sqrt(400 - 49) = sqrt(351) cm Note that H = sqrt(4 × 351) = 2sqrt(351). Step 2: Use the formula for the volume of a frustum, which is the volume of the original cone minus the volume of the smaller cone. V = (1)/(3) R^2 H - (1)/(3) r^2 h_1 Step 3: Substitute the values. V = (1)/(3) (14^2) sqrt(1404) - (1)/(3) (7^2) sqrt(351) V = (1)/(3) (196) (2sqrt(351)) - (1)/(3) (49) sqrt(351) V = (1)/(3) sqrt(351) (196 × 2 - 49) V = (1)/(3) sqrt(351) (392 - 49) V = (1)/(3) sqrt(351) (343) V = 343 sqrt(351)3 cm^3 Step 4: Calculate the numerical value. sqrt(351) ≈ 18.7349916 V ≈ (343 × 3.14159265 × 18.7349916)/(3) V ≈ (20188.760)/(3) V ≈ 6729.5866 cm^3 Step 5: Round to 4 significant figures. The first four significant figures are 6, 7, 2, 9. The fifth digit is 5, so we round up the fourth digit. The volume of the lampshade is 6730 cm^3.