Question

1. find the surface area:

(figure of a three - dimensional shape with dimensions 6 ft, 8 ft, 7.5 ft, 10 ft)
(color this answer black)

Explanation:

Step1: Identify the shape (triangular prism)

The figure is a triangular prism. The formula for the surface area of a triangular prism is \( SA = 2B + Ph \), where \( B \) is the area of the triangular base, \( P \) is the perimeter of the triangular base, and \( h \) is the length of the prism (the distance between the two triangular bases).

Step2: Calculate the area of the triangular base (\( B \))

The triangular base has a base of \( 6 \) ft and a height of \( 8 \) ft (since it's a right triangle, area \( B=\frac{1}{2}\times base\times height \)).
\( B=\frac{1}{2}\times6\times8 = 24 \) square feet.

Step3: Calculate the perimeter of the triangular base (\( P \))

The sides of the triangular base are \( 6 \) ft, \( 8 \) ft, and we can find the hypotenuse using Pythagoras: \( \sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10 \) ft. So perimeter \( P=6 + 8+10 = 24 \) ft.

Step4: Identify the length of the prism (\( h \))

The length of the prism (the distance between the two triangular bases) is \( 7.5 \) ft.

Step5: Calculate the surface area using \( SA = 2B+Ph \)

Substitute \( B = 24 \), \( P = 24 \), and \( h = 7.5 \) into the formula:
\( SA=2\times24+24\times7.5 \)
\( SA = 48+180 \)
\( SA = 228 \) square feet. Wait, no, wait. Wait, maybe I misidentified the base. Wait, looking at the diagram again, the triangular base: maybe the legs are 6 and 8, hypotenuse 10, and the length of the prism is 7.5? Wait, no, the other dimension is 10 ft? Wait, maybe I made a mistake. Wait, let's re - examine the diagram. The triangular base: the right triangle has legs 6 ft and 8 ft, hypotenuse 10 ft. The length of the prism (the distance along the 7.5 ft? Wait, no, the prism's length is 10 ft? Wait, no, the diagram has 10 ft, 7.5 ft. Wait, maybe the triangular base area is \( \frac{1}{2}\times6\times8 = 24 \), perimeter \( 6 + 8+10 = 24 \), and the length of the prism (the lateral edge) is 7.5? No, wait, maybe the length of the prism is 10 ft? Wait, I think I messed up the length. Let's re - do:

Wait, the triangular base: base 6, height 8, area \( \frac{1}{2}\times6\times8 = 24 \). The perimeter of the triangle: 6 + 8+10 = 24. The length of the prism (the distance between the two triangles) is 7.5? No, the diagram has 7.5 ft as the height of the prism? Wait, no, let's look at the labels again. The triangle has sides 6, 8, 10 (since 6 - 8 - 10 is a right triangle). The prism has a length (the distance along the direction perpendicular to the triangle) of 7.5 ft? Wait, no, the other dimension is 10 ft? Wait, maybe the formula is different. Wait, another way: the surface area of a triangular prism is the sum of the areas of the two triangular bases and the three rectangular faces.

The two triangular bases: area of each is \( \frac{1}{2}\times6\times8 = 24 \), so two of them is \( 2\times24 = 48 \).

The three rectangular faces:

1. The rectangle with dimensions 6 ft and 7.5 ft: area \( 6\times7.5 = 45 \)
2. The rectangle with dimensions 8 ft and 7.5 ft: area \( 8\times7.5 = 60 \)
3. The rectangle with dimensions 10 ft and 7.5 ft: area \( 10\times7.5 = 75 \)

Now sum the areas of the rectangles: \( 45+60 + 75=180 \)

Then add the area of the two triangles: \( 180 + 48=228 \)? Wait, but that seems wrong. Wait, no, maybe the length of the prism is 10 ft? Wait, let's check the diagram again. The diagram has 10 ft, 7.5 ft, 6 ft, 8 ft. Wait, maybe the triangular base is with legs 6 and 8, hypotenuse 10, and the length of the prism is 7.5 ft. Then the lateral faces are:

- 6×7.5, 8×7.5, 10×7.5. And the two triangles: 2×(0.5×6×8). So total surface area:

\( 2\times(0.5\times6\times8)+6\times7.5 + 8\times7.5+10\times7.5 \)

\( = 48+(45 + 60+75) \)

\( = 48 + 180 \)

\( = 228 \) square feet. Wait, but let's check again.

Wait, maybe I misread the length. If the length of the prism is 10 ft, then:

Triangular base area: \( 0.5\times6\times8 = 24 \), two triangles: 48.

Lateral faces: 6×10, 8×10, 10×10? No, that doesn't make sense. Wait, the diagram has 7.5 ft and 10 ft. Wait, the correct approach:

The triangular prism has a triangular base with base \( b = 6 \) ft, height \( h_{triangle}=8 \) ft, hypotenuse \( c = 10 \) ft. The length of the prism (the distance between the two triangular bases) is \( l = 7.5 \) ft.

Surface area formula for triangular prism: \( SA=2\times(\frac{1}{2}\times b\times h_{triangle})+b\times l + h_{triangle}\times l + c\times l \)

Substitute the values:

\( SA = 2\times(\frac{1}{2}\times6\times8)+6\times7.5 + 8\times7.5+10\times7.5 \)

\( SA=48+(45 + 60 + 75) \)

\( SA = 48+180 \)

\( SA = 228 \) square feet.

Wait, but let's confirm the formula. The surface area of a triangular prism is the sum of the areas of the two triangular bases and the three rectangular lateral faces. The two triangular bases have area \( 2\times(\frac{1}{2}\times base\times height) \), and the three rectangular faces have areas equal to the product of each side of the triangle and the length of the prism.

Yes, so with base 6, height 8, hypotenuse 10, and length of prism 7.5, the calculation is correct.

Answer:

The surface area is \(\boldsymbol{228}\) square feet.