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The lampshade is a frustum of a cone with:

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3 parts
b

18.73 cm18.73 \text{ cm}

c

1319.47 cm21319.47 \text{ cm}^2

d

6730 cm36730 \text{ cm}^3

Step-by-step solution

15 steps

Here's the solution to the problem:

The lampshade is a frustum of a cone with:
Top radius (rr) = 7 cm
Bottom radius (RR) = 14 cm
Slant height of the frustum (lfl_f) = 20 cm

Let LL be the slant height of the original (larger) cone and l1l_1 be the slant height of the smaller cone that was cut off.
Let HH be the height of the original cone and h1h_1 be the height of the smaller cone.
Let hh be the height of the lampshade (frustum).

Part a· 4 steps

The slant height of the original cone correct to two decimal places.

  1. Use similar triangles formed by the radii and slant heights.

    For the original cone and the smaller cone, we have:

    Rr=Ll1\frac{R}{r} = \frac{L}{l_1} 147=Ll1\frac{14}{7} = \frac{L}{l_1} 2=Ll1  ⟹  L=2l12 = \frac{L}{l_1} \implies L = 2l_1

  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−l1=20 cmL - l_1 = 20 \text{ cm}

  3. Substitute L=2l1L = 2l_1 into the equation.

    2l1−l1=202l_1 - l_1 = 20 l1=20 cml_1 = 20 \text{ cm}

  4. Calculate LL.

    L=2l1=2×20=40 cmL = 2l_1 = 2 \times 20 = 40 \text{ cm}

    The slant height of the original cone is 40.00 cm.

Part b

18.73 cm18.73 \text{ cm}

The height h, of the lampshade.

  1. Consider a right-angled triangle formed by the height of the frustum (hh), the difference in radii (R−rR-r), and the slant height of the frustum (lfl_f).

    The difference in radii is 14−7=714 - 7 = 7 cm.
    The slant height of the frustum is 20 cm.

  2. Apply the Pythagorean theorem.

    h2+(R−r)2=lf2h^2 + (R-r)^2 = l_f^2 h2+(14−7)2=202h^2 + (14-7)^2 = 20^2 h2+72=202h^2 + 7^2 = 20^2 h2+49=400h^2 + 49 = 400 h2=400−49h^2 = 400 - 49 h2=351h^2 = 351 h=351h = \sqrt{351}

  3. Calculate the numerical value.

    h≈18.73499 cmh \approx 18.73499 \text{ cm}

    Rounding to two decimal places:
    The height of the lampshade is 18.73 cm\boxed{18.73 \text{ cm}}.

Part c

1319.47 cm21319.47 \text{ cm}^2

The curved surface area of the lampshade.

  1. Use the formula for the curved surface area (CSA) of a frustum.

    CSA=π(R+r)lf\text{CSA} = \pi (R + r) l_f

    Where R=14R = 14 cm, r=7r = 7 cm, and lf=20l_f = 20 cm.

  2. Substitute the values.

    CSA=π(14+7)(20)\text{CSA} = \pi (14 + 7) (20) CSA=π(21)(20)\text{CSA} = \pi (21) (20) CSA=420π cm2\text{CSA} = 420\pi \text{ cm}^2

  3. Calculate the numerical value.

    CSA≈420×3.14159265\text{CSA} \approx 420 \times 3.14159265 CSA≈1319.4689 cm2\text{CSA} \approx 1319.4689 \text{ cm}^2

    Rounding to two decimal places:
    The curved surface area of the lampshade is 1319.47 cm2\boxed{1319.47 \text{ cm}^2}.

Part d

6730 cm36730 \text{ cm}^3

The volume of the lampshade correct to 4 significant figures.

  1. Calculate the height of the original cone (HH) and the smaller cone (h1h_1).

    From part (a), L=40L = 40 cm and l1=20l_1 = 20 cm.
    For the original cone:

    H=L2−R2=402−142=1600−196=1404 cmH = \sqrt{L^2 - R^2} = \sqrt{40^2 - 14^2} = \sqrt{1600 - 196} = \sqrt{1404} \text{ cm}

    For the smaller cone:

    h1=l12−r2=202−72=400−49=351 cmh_1 = \sqrt{l_1^2 - r^2} = \sqrt{20^2 - 7^2} = \sqrt{400 - 49} = \sqrt{351} \text{ cm}

    Note that H=4×351=2351H = \sqrt{4 \times 351} = 2\sqrt{351}.

  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=13πR2H−13πr2h1V = \frac{1}{3}\pi R^2 H - \frac{1}{3}\pi r^2 h_1

  3. Substitute the values.

    V=13π(142)1404−13π(72)351V = \frac{1}{3}\pi (14^2) \sqrt{1404} - \frac{1}{3}\pi (7^2) \sqrt{351} V=13π(196)(2351)−13π(49)351V = \frac{1}{3}\pi (196) (2\sqrt{351}) - \frac{1}{3}\pi (49) \sqrt{351} V=13π351(196×2−49)V = \frac{1}{3}\pi \sqrt{351} (196 \times 2 - 49) V=13π351(392−49)V = \frac{1}{3}\pi \sqrt{351} (392 - 49) V=13π351(343)V = \frac{1}{3}\pi \sqrt{351} (343) V=343π3513 cm3V = \frac{343\pi \sqrt{351}}{3} \text{ cm}^3

  4. Calculate the numerical value.

    351≈18.7349916\sqrt{351} \approx 18.7349916 V≈343×3.14159265×18.73499163V \approx \frac{343 \times 3.14159265 \times 18.7349916}{3} V≈20188.7603V \approx \frac{20188.760}{3} V≈6729.5866 cm3V \approx 6729.5866 \text{ cm}^3

  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 cm3\boxed{6730 \text{ cm}^3}.

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