Math / Geometry

QuestionThis object is composed of a right rectangular prism attached to the side of a larger rig Determine the surface area of the object.

Studdy Solution

STEP 1

1. The object is composed of two right rectangular prisms.
2. The larger prism has dimensions $13 \, \text{cm} \times 13 \, \text{cm} \times 20 \, \text{cm}$ .
3. The smaller prism has dimensions $11 \, \text{cm} \times 6 \, \text{cm} \times 6 \, \text{cm}$ .
4. The smaller prism is attached to the side of the larger prism.
5. The shared face between the prisms is not counted twice in the surface area.

STEP 2

1. Calculate the surface area of the larger prism.
2. Calculate the surface area of the smaller prism.
3. Determine the area of the shared face.
4. Subtract the shared face area from the total surface area of both prisms.

STEP 3

Calculate the surface area of the larger prism using the formula:
$SA_{\text{large}} = 2(lw + lh + wh)$
where $l = 13 \, \text{cm}$ , $w = 13 \, \text{cm}$ , $h = 20 \, \text{cm}$ .
$SA_{\text{large}} = 2(13 \times 13 + 13 \times 20 + 13 \times 20)$ $= 2(169 + 260 + 260)$ $= 2(689)$ $= 1378 \, \text{cm}^2$

STEP 4

Calculate the surface area of the smaller prism using the formula:
$SA_{\text{small}} = 2(lw + lh + wh)$
where $l = 11 \, \text{cm}$ , $w = 6 \, \text{cm}$ , $h = 6 \, \text{cm}$ .
$SA_{\text{small}} = 2(11 \times 6 + 11 \times 6 + 6 \times 6)$ $= 2(66 + 66 + 36)$ $= 2(168)$ $= 336 \, \text{cm}^2$

STEP 5

Determine the area of the shared face. The shared face is the face of the smaller prism that is attached to the larger prism:
$\text{Shared Face Area} = 11 \, \text{cm} \times 6 \, \text{cm} = 66 \, \text{cm}^2$

STEP 6

Subtract the shared face area from the total surface area of both prisms:
$\text{Total Surface Area} = SA_{\text{large}} + SA_{\text{small}} - \text{Shared Face Area}$ $= 1378 \, \text{cm}^2 + 336 \, \text{cm}^2 - 66 \, \text{cm}^2$ $= 1714 \, \text{cm}^2 - 66 \, \text{cm}^2$ $= 1648 \, \text{cm}^2$
The surface area of the object is:
$\boxed{1648 \, \text{cm}^2}$

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