Question

select the correct answer. the first attempt of the basic design of a new toy is shown. diagram: rectangular prism (length 24 in, width 8 in, height 10 in) with a cylinder (diameter 6 in, height 16 in) on top what is the surface area of the new toy to the nearest square inch?
a. 1,326 in²
b. 1,410 in²
c. 1,382 in²
d. 1,354 in²

Explanation:

Step1: Calculate surface area of rectangular prism

The formula for the surface area of a rectangular prism is \( SA_{prism} = 2(lw + lh + wh) \), where \( l = 24 \) in, \( w = 8 \) in, \( h = 10 \) in.
\[ \begin{align*} SA_{prism}&= 2(24\times8 + 24\times10 + 8\times10)\\ &= 2(192 + 240 + 80)\\ &= 2(512)\\ &= 1024 \text{ in}^2 \end{align*} \]

Step2: Calculate lateral surface area of cylinder

The formula for the lateral surface area of a cylinder is \( SA_{cylinder - lateral} = 2\pi rh \), where \( r = \frac{6}{2}= 3 \) in, \( h = 16 \) in.
\[ SA_{cylinder - lateral}= 2\times\pi\times3\times16 = 96\pi \approx 301.59 \text{ in}^2 \]

Step3: Adjust for the overlapping area

The overlapping area is the area of the circular base of the cylinder, which is \( \pi r^2=\pi\times3^2 = 9\pi\approx28.27 \text{ in}^2 \). But since the cylinder is on top of the prism, we subtract twice this area? No, actually, the total surface area of the toy is the surface area of the prism plus the lateral surface area of the cylinder (because the top of the prism has a circle covered by the cylinder, so we subtract the area of the circle from the prism's surface area and add the lateral surface area of the cylinder, but wait, the prism's surface area includes the top face. So when we put the cylinder on it, we remove the area of the circle from the prism's top face and add the lateral surface area of the cylinder. Wait, actually, the correct way is: Surface area of toy = Surface area of prism - area of the circular base (since it's covered) + lateral surface area of cylinder. Wait, no, the prism's surface area is \( 2(lw + lh + wh) \), which includes the top face (lw). When we place the cylinder on the top face, we cover a circular area ( \( \pi r^2 \)) on the top face, so we need to subtract that circular area from the prism's surface area and add the lateral surface area of the cylinder (since the cylinder's top and bottom: the bottom is covered, the top is exposed? Wait, the diagram shows a cylinder on top of the prism. So the cylinder has a bottom base (attached to the prism) and a top base (exposed). The prism has a top face with a circle covered. So:

Surface area of toy = Surface area of prism - area of the circular base (covered) + lateral surface area of cylinder + area of the top circular base of cylinder.

Wait, but the area of the top circular base of the cylinder is \( \pi r^2 \), and the area we subtracted from the prism is \( \pi r^2 \), so actually, it's Surface area of prism + lateral surface area of cylinder. Wait, let's think again.

Original prism surface area: all six faces. When we add the cylinder, we cover a circle on the top face of the prism (so we lose that circle's area from the prism's surface area) but we gain the lateral surface area of the cylinder and the top circle of the cylinder. So:

Surface area of toy = (Surface area of prism - area of circle) + (lateral surface area of cylinder + area of top circle of cylinder) = Surface area of prism + lateral surface area of cylinder. Because the -area of circle and +area of top circle cancel out. Oh right! Because the bottom circle of the cylinder is attached to the prism (so we don't count it), and the top circle of the cylinder is exposed (so we count it), and the prism's top face had a circle covered (so we subtract that circle from the prism's surface area). So:

Prism SA: 1024 (from step1)

Cylinder lateral SA: 96π ≈ 301.59

Now, the prism's top face area is 24*8=192. We covered a circle of area π*3²=9π≈28.27, so the exposed area of the prism's top face is 192 - 28.27. Then the cylinder contributes its lateral surface area (96π≈301.59) and its top circle (9π≈28.27). So total surface area:

(SA of prism - area of circle) + (lateral SA of cylinder + area of top circle of cylinder) = SA of prism + lateral SA of cylinder. Because (192 - 28.27) + (301.59 + 28.27) = 192 + 301.59 = (SA of prism - 192 + 192) + 301.59? Wait, no, the SA of the prism is 2(lw + lh + wh) = 2(192 + 240 + 80) = 2(512)=1024. That includes the top face (192) and bottom face (192), front/back (24*10*2=480), left/right (8*10*2=160). So when we add the cylinder, we take the prism's SA, subtract the area of the circle (28.27) from the top face, and add the lateral SA of the cylinder (301.59) and the top circle of the cylinder (28.27). So:

1024 - 28.27 + 301.59 + 28.27 = 1024 + 301.59 = 1325.59 ≈ 1326? Wait, but let's check the calculations again.

Wait, maybe I made a mistake in the prism's dimensions. Wait, the prism has length 24, width 8, height 10. So:

lw = 24*8=192, lh=24*10=240, wh=8*10=80. Then 2(lw + lh + wh)=2(192+240+80)=2(512)=1024. Correct.

Cylinder: diameter 6, so radius 3. Height 16. Lateral surface area: 2πrh=2*π*3*16=96π≈301.5928947.

Now, the overlapping area: the circle where the cylinder meets the prism. So the surface area of the toy is the surface area of the prism plus the lateral surface area of the cylinder, because the area of the circle on the prism's top face is covered (so we subtract it) but the cylinder's top face is exposed (so we add it), and the cylinder's bottom face is covered (so we don't add it). Wait, no: the prism's surface area includes the top face (192). When we place the cylinder on it, we cover a circle (area πr²=9π≈28.27) on the top face, so we need to subtract that from the prism's surface area. Then, the cylinder has a lateral surface area (96π≈301.59) and a top surface area (9π≈28.27). So total surface area:

Prism SA: 1024

Minus covered circle: -9π

Plus cylinder lateral SA: +96π

Plus cylinder top SA: +9π

So total: 1024 + 96π ≈ 1024 + 301.59 ≈ 1325.59 ≈ 1326 in².

Wait, but let's check the answer options. Option A is 1326 in². So that should be the answer.

Answer:

A. \( 1,326 \text{ in}^2 \)