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: $$ \frac{R}{r} = \frac{L}{l_1} $$ $$ \frac{14}{7} = \frac{L}{l_1} $$ $$ 2 = \frac{L}{l_1} \implies L = 2l_1 $$ 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: $$ L - l_1 = 20 \text{ cm} $$ Step 3: Substitute $L = 2l_1$ into the equation. $$ 2l_1 - l_1 = 20 $$ $$ l_1 = 20 \text{ cm} $$ Step 4: Calculate $L$. $$ L = 2l_1 = 2 \times 20 = 40 \text{ cm} $$ 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. $$ h \approx 18.73499 \text{ cm} $$ Rounding to two decimal places: The height of the lampshade is $\boxed{18.73 \text{ cm}}$. c) The curved surface area of the lampshade. Step 1: Use the formula for the curved surface area (CSA) of a frustum. $$ \text{CSA} = \pi (R + r) l_f $$ Where $R = 14$ cm, $r = 7$ cm, and $l_f = 20$ cm. Step 2: Substitute the values. $$ \text{CSA} = \pi (14 + 7) (20) $$ $$ \text{CSA} = \pi (21) (20) $$ $$ \text{CSA} = 420\pi \text{ cm}^2 $$ Step 3: Calculate the numerical value. $$ \text{CSA} \approx 420 \times 3.14159265 $$ $$ \text{CSA} \approx 1319.4689 \text{ cm}^2 $$ Rounding to two decimal places: The curved surface area of the lampshade is $\boxed{1319.47 \text{ cm}^2}$. 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: $$ H = \sqrt{L^2 - R^2} = \sqrt{40^2 - 14^2} = \sqrt{1600 - 196} = \sqrt{1404} \text{ cm} $$ For the smaller cone: $$ h_1 = \sqrt{l_1^2 - r^2} = \sqrt{20^2 - 7^2} = \sqrt{400 - 49} = \sqrt{351} \text{ cm} $$ Note that $H = \sqrt{4 \times 351} = 2\sqrt{351}$. 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. $$ V = \frac{1}{3}\pi R^2 H - \frac{1}{3}\pi r^2 h_1 $$ Step 3: Substitute the values. $$ V = \frac{1}{3}\pi (14^2) \sqrt{1404} - \frac{1}{3}\pi (7^2) \sqrt{351} $$ $$ V = \frac{1}{3}\pi (196) (2\sqrt{351}) - \frac{1}{3}\pi (49) \sqrt{351} $$ $$ V = \frac{1}{3}\pi \sqrt{351} (196 \times 2 - 49) $$ $$ V = \frac{1}{3}\pi \sqrt{351} (392 - 49) $$ $$ V = \frac{1}{3}\pi \sqrt{351} (343) $$ $$ V = \frac{343\pi \sqrt{351}}{3} \text{ cm}^3 $$ Step 4: Calculate the numerical value. $$ \sqrt{351} \approx 18.7349916 $$ $$ V \approx \frac{343 \times 3.14159265 \times 18.7349916}{3} $$ $$ V \approx \frac{20188.760}{3} $$ $$ V \approx 6729.5866 \text{ cm}^3 $$ 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 $\boxed{6730 \text{ cm}^3}$.