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Question

An illustration shows a three dimensional composite figure formed by a right triangular prism placed on top of a rectangular prism. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 12 feet and a height of 8 feet. The width of the rectangular prism is 8 feet. The edges of the hidden side face of the right triangular prism align perfectly with the edges of the hidden top face of the rectangular prism. The right triangular prism is aligned to the left of the rectangular prism. The height of the right triangular prism is not shown. The total height of the left side of the figure is 20 feet. The right triangular prism has a width of 8 feet. The hypotenuse side of the right triangular prism has a length of 15 feet.

What is the surface area of the figure?

ft.2

What is the surface area of the figure?

ft.2

GPT-4o mini · Answer

To find the surface area of the composite figure, we'll calculate the surface area of both the rectangular prism and the right triangular prism separately, and then combine them, taking care not to double count the area where they meet.

Step 1: Surface Area of the Rectangular Prism

The rectangular prism has:

Length (L) = 12 ft
Width (W) = 8 ft
Height (H) = 8 ft

The surface area (SA) of a rectangular prism is calculated as:

$S A = 2 (L W + L H + W H)$

Calculating each term:

$L W = 12 \times 8 = 96$ ft²
$L H = 12 \times 8 = 96$ ft²
$W H = 8 \times 8 = 64$ ft²

Now plug in these values:

$S A_{r e c t a n g u l a r p r i s m} = 2 (96 + 96 + 64) = 2 (256) = 512 f t ²$

Step 2: Surface Area of the Right Triangular Prism

The right triangular prism has:

Base width (b) = 8 ft
Height (h) (triangular prism) = 20 ft - 8 ft = 12 ft (the triangle's upward height since total height is 20 ft)
Hypotenuse (side) $c$ = 15 ft

Finding the Area of the Triangle:

The area of the triangle base can be calculated using:

$A r e a_{t r i a n g l e} = \frac{1}{2} \times b a s e \times h e i g h t = \frac{1}{2} \times 8 \times 12 = 48 f t ²$

Surface Area of the Triangular Prism:

The surface area of a triangular prism is given by:

$S A = (B a s e A r e a) + (P e r i m e t e r \times H e i g h t)$

First, calculate the perimeter of the triangular base. The sides of the triangle are:

Base = 8 ft
Height side (from right angle) = 12 ft (found using Pythagorean Theorem since $b^{2} + h^{2} = c^{2}$ , $8^{2} + h^{2} = 15^{2}$ )
Hypotenuse = 15 ft

So, the perimeter $P$ is:

$P = 8 + 12 + 15 = 35 f t$

Now, calculate the surface area using the height (12 ft):

$S A_{t r i a n g u l a r p r i s m} = (A r e a_{t r i a n g l e}) + (P e r i m e t e r \times H e i g h t) = 48 + (35 \times 12) = 48 + 420 = 468 f t ²$

Step 3: Combine Surface Areas

Next, we need to observe that the rectangular prism's top surface is shared with the triangular prism’s base. Therefore, we should subtract the area of the base of the triangular prism from the rectangular prism's surface area.

$S A_{c o m p o s i t e f i g u r e} = S A_{r e c t a n g u l a r p r i s m} + S A_{t r i a n g u l a r p r i s m} - B a s e A r e a_{t r i a n g l e}$

Calculate:

$S A_{c o m p o s i t e f i g u r e} = 512 + 468 - 48 = 932 f t ²$

Thus, the surface area of the composite figure is

$932 f t ²$