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Find the volume and surface area of a sphere with radius 3.5 cm and a cone with radius 7 cm and slant height 10 cm.

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4 parts
a

179.59 cm3179.59 \text{ cm}^3

153.94 cm2153.94 \text{ cm}^2

b

366.49 cm3366.49 \text{ cm}^3

373.85 cm2373.85 \text{ cm}^2

c

32 m332 \text{ m}^3

64.8 m264.8 \text{ m}^2

d

64 m364 \text{ m}^3

96 m296 \text{ m}^2

Step-by-step solution

9 steps

Here are the solutions for question 4, as indicated in your image:

4. Find the volume and surface of the following

Part a· 2 steps

sphere radius 3.5 cm

Volume of a sphere: V=43πr3V = \frac{4}{3}\pi r^3
Surface area of a sphere: SA=4πr2SA = 4\pi r^2
Given r=3.5 cmr = 3.5 \text{ cm}.

  1. Calculate the volume.

    V=43×π×(3.5 cm)3V = \frac{4}{3} \times \pi \times (3.5 \text{ cm})^3 V=43×π×42.875 cm3V = \frac{4}{3} \times \pi \times 42.875 \text{ cm}^3 V≈179.59 cm3V \approx 179.59 \text{ cm}^3

  2. Calculate the surface area.

    SA=4×π×(3.5 cm)2SA = 4 \times \pi \times (3.5 \text{ cm})^2 SA=4×π×12.25 cm2SA = 4 \times \pi \times 12.25 \text{ cm}^2 SA≈153.94 cm2SA \approx 153.94 \text{ cm}^2

    The volume is 179.59 cm3\boxed{179.59 \text{ cm}^3} and the surface area is 153.94 cm2\boxed{153.94 \text{ cm}^2}.

Part b

366.49 cm3366.49 \text{ cm}^3

cone radius 7 cm and slant height 10 cm

Volume of a cone: V=13πr2hV = \frac{1}{3}\pi r^2 h
Surface area of a cone: SA=πr(r+l)SA = \pi r (r+l)
Given r=7 cmr = 7 \text{ cm} and l=10 cml = 10 \text{ cm}.

  1. Find the height hh using the Pythagorean theorem

    h=l2−r2h = \sqrt{l^2 - r^2}.

    h=(10 cm)2−(7 cm)2h = \sqrt{(10 \text{ cm})^2 - (7 \text{ cm})^2} h=100 cm2−49 cm2h = \sqrt{100 \text{ cm}^2 - 49 \text{ cm}^2} h=51 cm≈7.14 cmh = \sqrt{51} \text{ cm} \approx 7.14 \text{ cm}

  2. Calculate the volume.

    V=13×π×(7 cm)2×51 cmV = \frac{1}{3} \times \pi \times (7 \text{ cm})^2 \times \sqrt{51} \text{ cm} V=13×π×49 cm2×7.1414 cmV = \frac{1}{3} \times \pi \times 49 \text{ cm}^2 \times 7.1414 \text{ cm} V≈366.49 cm3V \approx 366.49 \text{ cm}^3

  3. Calculate the surface area.

    SA=π×7 cm×(7 cm+10 cm)SA = \pi \times 7 \text{ cm} \times (7 \text{ cm} + 10 \text{ cm}) SA=π×7 cm×17 cmSA = \pi \times 7 \text{ cm} \times 17 \text{ cm} SA=119π cm2SA = 119\pi \text{ cm}^2 SA≈373.85 cm2SA \approx 373.85 \text{ cm}^2

    The volume is 366.49 cm3\boxed{366.49 \text{ cm}^3} and the surface area is 373.85 cm2\boxed{373.85 \text{ cm}^2}.

Part c

32 m332 \text{ m}^3

cuboid of 5 m x 2 m x 3.2 m

Volume of a cuboid: V=lwhV = lwh
Surface area of a cuboid: SA=2(lw+lh+wh)SA = 2(lw + lh + wh)
Given l=5 ml = 5 \text{ m}, w=2 mw = 2 \text{ m}, h=3.2 mh = 3.2 \text{ m}.

  1. Calculate the volume.

    V=5 m×2 m×3.2 mV = 5 \text{ m} \times 2 \text{ m} \times 3.2 \text{ m} V=32 m3V = 32 \text{ m}^3

  2. Calculate the surface area.

    SA=2((5 m×2 m)+(5 m×3.2 m)+(2 m×3.2 m))SA = 2((5 \text{ m} \times 2 \text{ m}) + (5 \text{ m} \times 3.2 \text{ m}) + (2 \text{ m} \times 3.2 \text{ m})) SA=2(10 m2+16 m2+6.4 m2)SA = 2(10 \text{ m}^2 + 16 \text{ m}^2 + 6.4 \text{ m}^2) SA=2(32.4 m2)SA = 2(32.4 \text{ m}^2) SA=64.8 m2SA = 64.8 \text{ m}^2

    The volume is 32 m3\boxed{32 \text{ m}^3} and the surface area is 64.8 m2\boxed{64.8 \text{ m}^2}.

Part d

64 m364 \text{ m}^3

cube of sides 4 m

Volume of a cube: V=s3V = s^3
Surface area of a cube: SA=6s2SA = 6s^2
Given s=4 ms = 4 \text{ m}.

  1. Calculate the volume.

    V=(4 m)3V = (4 \text{ m})^3 V=64 m3V = 64 \text{ m}^3

  2. Calculate the surface area.

    SA=6×(4 m)2SA = 6 \times (4 \text{ m})^2 SA=6×16 m2SA = 6 \times 16 \text{ m}^2 SA=96 m2SA = 96 \text{ m}^2

    The volume is 64 m3\boxed{64 \text{ m}^3} and the surface area is 96 m2\boxed{96 \text{ m}^2}.

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