Question

this container is composed of a right circular cylinder and a right circular cone.
which is closest to the surface area of the container?
a 490 ft²
b 754 ft²
c 1,243 ft²
d 1,696 ft²

Explanation:

Step1: Find the radius of the cylinder and cone

The diameter of the cylinder is 24 ft, so the radius \( r=\frac{24}{2} = 12\) ft.

Step2: Calculate the lateral (curved) surface area of the cylinder

The formula for the lateral surface area of a cylinder is \( 2\pi rh \). Here, \( r = 12\) ft and \( h = 10\) ft. So, \( 2\pi\times12\times10=240\pi\) square feet.

Step3: Calculate the lateral (curved) surface area of the cone

The formula for the lateral surface area of a cone is \( \pi rl \), where \( l \) is the slant height. Here, \( r = 12\) ft and \( l = 13\) ft. So, \( \pi\times12\times13 = 156\pi\) square feet.

Step4: Calculate the area of the base of the cylinder

The base of the cylinder is a circle, and the formula for the area of a circle is \( \pi r^{2}\). So, \( \pi\times12^{2}=144\pi\) square feet.

Step5: Sum up the surface areas

The total surface area of the container is the sum of the lateral surface area of the cylinder, the lateral surface area of the cone, and the area of the base of the cylinder (since the top of the cylinder is attached to the cone, we don't include the top circle of the cylinder, and the base of the cone is attached to the cylinder, so we don't include it either). So, total surface area \(=240\pi + 156\pi+144\pi=(240 + 156+144)\pi=540\pi\).

Now, \( \pi\approx3.14\), so \( 540\times3.14 = 1695.6\approx1696\) square feet. But wait, wait, did we do that right? Wait, no, wait. Wait, the container: the cylinder has a base (the bottom), a lateral surface, and then the cone on top. Wait, the cone's base is attached to the cylinder, so we don't include the base of the cone. The cylinder: we include the lateral surface and the bottom base, and the cone's lateral surface. Wait, let's re - check.

Wait, the cylinder: lateral surface area \( 2\pi rh=2\times\pi\times12\times10 = 240\pi\). The bottom base: \( \pi r^{2}=\pi\times12^{2}=144\pi\). The cone: lateral surface area \( \pi rl=\pi\times12\times13 = 156\pi\). Now, sum them: \( 240\pi+144\pi + 156\pi=(240 + 144+156)\pi=540\pi\approx540\times3.14 = 1695.6\approx1696\) \( ft^{2}\). But wait, the options have D as 1,696 \( ft^{2}\). But wait, maybe I made a mistake. Wait, let's check again. Wait, is the top of the cylinder open? Yes, because it's attached to the cone. So the cylinder's surface area is lateral + bottom. The cone's surface area is lateral (since its base is attached to the cylinder, so we don't include the base of the cone). So that's correct. So the total surface area is \( 2\pi rh+\pi r^{2}+\pi rl\). Plugging in the numbers: \( 2\times3.14\times12\times10+3.14\times12^{2}+3.14\times12\times13\)

Calculate each term:

- \( 2\times3.14\times12\times10 = 753.6\)
- \( 3.14\times12^{2}=3.14\times144 = 452.16\)
- \( 3.14\times12\times13=3.14\times156 = 489.84\)

Now sum them: \( 753.6+452.16 + 489.84=753.6+(452.16 + 489.84)=753.6 + 942=1695.6\approx1696\) \( ft^{2}\). So the closest is 1696, which is option D. Wait, but let me check again. Wait, maybe I misinterpreted the surface area. Wait, maybe the problem is that the container is like a cylinder with a cone on top, so the surface area is: the lateral surface of the cylinder, the lateral surface of the cone, and the bottom base of the cylinder. Yes, that's what we did. So the calculation leads to approximately 1696, which is option D. Wait, but the options are A 490, B 754, C 1243, D 1696. So our calculation gives approximately 1696, so the answer is D. Wait, but let me check the steps again.

Wait, radius \( r = 12\) (correct, since diameter 24). Lateral surface area of cylinder: \( 2\pi rh=2\times\pi\times12\times10 = 240\pi\) (correct). Lateral surface area of cone: \( \pi rl=\pi\times12\times13 = 156\pi\) (correct). Area of the bottom base of cylinder: \( \pi r^{2}=\pi\times12^{2}=144\pi\) (correct). Sum: \( 240\pi+156\pi + 144\pi=540\pi\). \( 540\times3.14 = 1695.6\approx1696\). So that's correct. So the answer is D.

Answer:

D \( 1,696\space ft^{2}\)