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A rectangular prism has a length of 12 cm, width of 5 cm, and height of 8 cm. a) Calculate its total surface area. b) Calculate its volume. c) If all dimensions are doubled, what is the new total surf

Mathematics

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3 parts
a

392 cm2\text{392 cm}^2

b

480 cm3\text{480 cm}^3

c

3840 cm3\text{3840 cm}^3

8

Step-by-step solution

10 steps

Here's the solution for Question 7:

Part a· 4 steps

  1. Identify the given dimensions.

    Length (LL) = 12 cm, Width (WW) = 5 cm, Height (HH) = 8 cm.

  2. Substitute the dimensions into the Total Surface Area (TSA) formula.

    TSA=2(LW+WH+LH)\text{TSA} = 2(LW + WH + LH) TSA=2((12 cm×5 cm)+(5 cm×8 cm)+(12 cm×8 cm))\text{TSA} = 2((12 \text{ cm} \times 5 \text{ cm}) + (5 \text{ cm} \times 8 \text{ cm}) + (12 \text{ cm} \times 8 \text{ cm}))

  3. Calculate the products and sum them.

    TSA=2(60 cm2+40 cm2+96 cm2)\text{TSA} = 2(60 \text{ cm}^2 + 40 \text{ cm}^2 + 96 \text{ cm}^2) TSA=2(196 cm2)\text{TSA} = 2(196 \text{ cm}^2)

  4. Calculate the total surface area.

    TSA=392 cm2\text{TSA} = 392 \text{ cm}^2

    The Total Surface Area is 392 cm2\boxed{\text{392 cm}^2}.

Part b

480 cm3\text{480 cm}^3

  1. Identify the given dimensions.

    Length (LL) = 12 cm, Width (WW) = 5 cm, Height (HH) = 8 cm.

  2. Substitute the dimensions into the volume (VV) formula.

    V=L×W×HV = L \times W \times H V=12 cm×5 cm×8 cmV = 12 \text{ cm} \times 5 \text{ cm} \times 8 \text{ cm}

  3. Calculate the volume.

    V=60 cm2×8 cmV = 60 \text{ cm}^2 \times 8 \text{ cm} V=480 cm3V = 480 \text{ cm}^3

    The volume of the rectangular prism is 480 cm3\boxed{\text{480 cm}^3}.

Part c

3840 cm3\text{3840 cm}^3

  1. Determine the new dimensions by doubling the original dimensions.

    New Length (L′L') = 2×12 cm=24 cm2 \times 12 \text{ cm} = 24 \text{ cm}
    New Width (W′W') = 2×5 cm=10 cm2 \times 5 \text{ cm} = 10 \text{ cm}
    New Height (H′H') = 2×8 cm=16 cm2 \times 8 \text{ cm} = 16 \text{ cm}

  2. Calculate the new volume (V′V') using the new dimensions.

    V′=L′×W′×H′V' = L' \times W' \times H' V′=24 cm×10 cm×16 cmV' = 24 \text{ cm} \times 10 \text{ cm} \times 16 \text{ cm} V′=240 cm2×16 cmV' = 240 \text{ cm}^2 \times 16 \text{ cm} V′=3840 cm3V' = 3840 \text{ cm}^3

    The new volume is 3840 cm3\boxed{\text{3840 cm}^3}.

  3. Calculate the ratio of the new volume to the original volume.

    Ratio=New VolumeOriginal Volume\text{Ratio} = \frac{\text{New Volume}}{\text{Original Volume}} Ratio=3840 cm3480 cm3\text{Ratio} = \frac{3840 \text{ cm}^3}{480 \text{ cm}^3} Ratio=8\text{Ratio} = 8

    The ratio of new volume to original volume is 8\boxed{8}.

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