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Math  /  Geometry

QuestionThe cross-section of the prism below is a compound shape formed of two rectangles.
Work out the volume of the prism. Give your answer in cm3\mathrm{cm}^{3}.
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Studdy Solution

STEP 1

1. The prism has a compound cross-section made of two rectangles.
2. The bottom rectangle has dimensions 5 cm×7 cm5 \, \text{cm} \times 7 \, \text{cm}.
3. The top rectangle has dimensions 4 cm×6 cm4 \, \text{cm} \times 6 \, \text{cm}.
4. The total height (or length) of the prism is 15 cm15 \, \text{cm}.

STEP 2

1. Calculate the area of each rectangle in the cross-section.
2. Sum the areas to find the total cross-sectional area.
3. Use the total cross-sectional area to calculate the volume of the prism.

STEP 3

Calculate the area of the bottom rectangle:
Areabottom=width×height=5 cm×7 cm=35 cm2 \text{Area}_{\text{bottom}} = \text{width} \times \text{height} = 5 \, \text{cm} \times 7 \, \text{cm} = 35 \, \text{cm}^2

STEP 4

Calculate the area of the top rectangle:
Areatop=width×height=4 cm×6 cm=24 cm2 \text{Area}_{\text{top}} = \text{width} \times \text{height} = 4 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2

STEP 5

Sum the areas of the two rectangles to find the total cross-sectional area:
Total Area=Areabottom+Areatop=35 cm2+24 cm2=59 cm2 \text{Total Area} = \text{Area}_{\text{bottom}} + \text{Area}_{\text{top}} = 35 \, \text{cm}^2 + 24 \, \text{cm}^2 = 59 \, \text{cm}^2

STEP 6

Calculate the volume of the prism using the total cross-sectional area and the height of the prism:
V=Total Area×Height=59 cm2×15 cm=885 cm3 V = \text{Total Area} \times \text{Height} = 59 \, \text{cm}^2 \times 15 \, \text{cm} = 885 \, \text{cm}^3
The volume of the prism is:
885 cm3 \boxed{885 \, \text{cm}^3}

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