Question

the composite figure shown has a surface area of 844 square centimeters. type the correct answer in the box. use numerals instead of words. what is the height of the rectangular prism? the height of the prism is centimeters.

Explanation:

Step1: Analyze the composite figure

The composite figure consists of a rectangular prism and a pyramid. First, we need to find the surface area components. Let's assume the height of the prism is \( h \) (which is \( x \) in the diagram, but we'll use \( h \) for clarity). The base of the prism (and the base of the pyramid) has dimensions 10 cm and 18 cm. The slant height of the pyramid is 12 cm.

Step2: Calculate surface area of the prism (excluding the top face, since it's attached to the pyramid)

The surface area of the prism (excluding the top face) is: \( 2(10h + 18h) + 10\times18 \) (two lateral faces with dimensions 10 and \( h \), two lateral faces with dimensions 18 and \( h \), and the bottom face 10×18). Simplify: \( 2(28h) + 180 = 56h + 180 \).

Step3: Calculate surface area of the pyramid (lateral faces only, since the base is attached to the prism)

The pyramid has a rectangular base, so the lateral surface area is the sum of the areas of the four triangular faces. The two triangles with base 10 cm: area of each is \( \frac{1}{2}\times10\times12 \), so two of them: \( 2\times\frac{1}{2}\times10\times12 = 120 \). The two triangles with base 18 cm: area of each is \( \frac{1}{2}\times18\times12 \), so two of them: \( 2\times\frac{1}{2}\times18\times12 = 216 \). Total lateral surface area of pyramid: \( 120 + 216 = 336 \).

Step4: Total surface area of composite figure

Total surface area = surface area of prism (excluding top) + surface area of pyramid (lateral) = \( (56h + 180) + 336 = 56h + 516 \).

Step5: Solve for \( h \)

We know the total surface area is 844. So:
\( 56h + 516 = 844 \)
Subtract 516 from both sides:
\( 56h = 844 - 516 = 328 \) Wait, that can't be right. Wait, maybe I made a mistake in the prism's surface area. Wait, the prism's dimensions: length 18, width 10, height \( h \). The surface area of the prism (including all faces) is \( 2(10\times18 + 10\times h + 18\times h) = 2(180 + 28h) = 360 + 56h \). But when attached to the pyramid, we subtract the area of the top face (10×18) from the prism's surface area, and add the lateral surface area of the pyramid. So correct total surface area: (prism surface area - top face) + pyramid lateral surface area = \( (360 + 56h - 180) + 336 = (180 + 56h) + 336 = 56h + 516 \). Wait, but 844 - 516 = 328, 328 ÷ 56 ≈ 5.857, which is not an integer. Wait, maybe the pyramid's base is a square? No, the diagram shows 10 and 18. Wait, maybe I messed up the pyramid's lateral faces. Wait, the height of the pyramid (the perpendicular height) is not 12, the slant height is 12? Wait, the diagram has a right angle, so maybe the slant height is 12 for the triangular faces. Wait, another approach: maybe the base of the prism is 10 and 18, and the height of the prism is \( h \), and the pyramid has a base of 10 and 18, slant height 12. Wait, let's recalculate the prism's surface area correctly. The prism has length 18, width 10, height \( h \). The faces: two 10×18 (top and bottom), two 10×\( h \) (front and back), two 18×\( h \) (left and right). But when combined with the pyramid, the top face of the prism (10×18) is covered, so we exclude that. So prism's surface area: bottom face (10×18) + two 10×\( h \) + two 18×\( h \) = \( 180 + 20h + 36h = 180 + 56h \). Pyramid's lateral surface area: for a rectangular pyramid, lateral surface area is \( \frac{1}{2}\times \text{perimeter of base} \times \text{slant height} \). Perimeter of base: \( 2(10 + 18) = 56 \). Slant height is 12. So lateral surface area: \( \frac{1}{2}\times56\times12 = 336 \). So total surface area: \( 180 + 56h + 336 = 56h + 516 \). Set equal to 844: \( 56h = 844 - 516 = 328 \). Wait, 328 divided by 56: 56×5=280, 328-280=48, 48/56=6/7≈0.857. That's not a nice number. Maybe the slant height is for the triangular faces with base 10 or 18? Wait, maybe the pyramid has a square base? No, the diagram shows 10 and 18. Wait, maybe I misread the diagram. Wait, the problem says "the height of the prism is" and we need to find it. Wait, maybe the composite figure's surface area is calculated as: prism's surface area (all faces) plus pyramid's lateral surface area minus 2×base area (since the base of the pyramid and the top of the prism are both covered, so we subtract twice? No, no: when you attach two solids, you subtract the area of the overlapping face once. So prism's surface area (all faces) + pyramid's lateral surface area - area of the overlapping face (since it's internal now). Prism's surface area: \( 2(10×18 + 10×h + 18×h) = 360 + 56h \). Pyramid's lateral surface area: 336 (as before). Overlapping area: 10×18=180. So total surface area: \( 360 + 56h + 336 - 180 = 516 + 56h \). Same as before. So 56h + 516 = 844 → 56h=328 → h=328/56=41/7≈5.857. But that's not an integer. Wait, maybe the slant height is not 12 for all faces? Wait, maybe the pyramid has a square base with side 10? No, the diagram shows 10 and 18. Wait, maybe I made a mistake in the problem. Wait, the user's diagram: the prism has length 18, width 10, and the pyramid has a base with length 18, width 10, and the slant height (the height of the triangular face) is 12. Wait, maybe the height of the prism is 4? Let's check: 56×4 + 516=224+516=740≠844. 56×6 +516=336+516=852, which is close to 844. 852-844=8, so 56h=844-516=328, h=328/56=41/7≈5.857. But maybe the problem has a typo, or I misread the numbers. Wait, maybe the surface area is 852, then h=6. But the given surface area is 844. Wait, maybe the pyramid's lateral surface area is calculated differently. Wait, another way: the composite figure's surface area includes the bottom of the prism, the four sides of the prism, the four triangular sides of the pyramid. So bottom: 10×18=180. Prism's sides: two 10×h and two 18×h: 20h + 36h=56h. Pyramid's sides: two triangles with base 10, height 12: 2×(0.5×10×12)=120. Two triangles with base 18, height 12: 2×(0.5×18×12)=216. Total: 180 + 56h + 120 + 216=56h + 516. So 56h=844-516=328→h=328/56=41/7≈5.857. But maybe the question is asking for the height of the prism, and the answer is 41/7 or approximately 5.86, but that's not an integer. Wait, maybe I made a mistake in the slant height. Wait, maybe the 12 cm is the height of the pyramid (perpendicular height), not the slant height. Let's try that. If the pyramid's height is 12, then the slant height for the face with base 10: slant height \( l_1 = \sqrt{12^2 + (18/2)^2} = \sqrt{144 + 81} = \sqrt{225}=15 \). Slant height for the face with base 18: \( l_2 = \sqrt{12^2 + (10/2)^2} = \sqrt{144 + 25} = \sqrt{169}=13 \). Then lateral surface area of pyramid: 2×(0.5×10×15) + 2×(0.5×18×13)=150 + 234=384. Then total surface area: 180 + 56h + 384=56h + 564. Set to 844: 56h=280→h=5. Ah, that makes sense! So I misread the 12 cm as slant height, but it's the perpendicular height of the pyramid. So that's the mistake.

Step1 (corrected): Pyramid's slant height calculation

If the pyramid's height (perpendicular to base) is 12, then for the triangular face with base 10 (the width of the base), the slant height \( l_1 \) is the height of the triangle, which is \( \sqrt{12^2 + (18/2)^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \) (since the base of the triangle is 10, wait no: the base of the pyramid is 18 (length) and 10 (width). So the two triangular faces with base 10 (width) have a slant height calculated from the pyramid's height and half the length (18/2=9). So slant height \( l_1 = \sqrt{12^2 + 9^2} = 15 \). The two triangular faces with base 18 (length) have a slant height calculated from the pyramid's height and half the width (10/2=5). So slant height \( l_2 = \sqrt{12^2 + 5^2} = 13 \).

Step2 (corrected): Lateral surface area of pyramid

Lateral surface area = 2×(0.5×10×15) + 2×(0.5×18×13) = 150 + 234 = 384.

Step3 (corrected): Total surface area

Total surface area = bottom of prism (10×18) + lateral surface area of prism (2×10×h + 2×18×h) + lateral surface area of pyramid = 180 + 56h + 384 = 56h + 564.

Step4 (corrected): Solve for h

Set total surface area to 844:
\( 56h + 564 = 844 \)
Subtract 564:
\( 56h = 844 - 564 = 280 \)
Divide by 56:
\( h = \frac{280}{56} = 5 \)

Ah, that works! So I misread the 12 cm as the slant height, but it's the perpendicular height of the pyramid. So the height of the prism is 5 cm.

Answer:

5