Question

select the correct answer. for an art project, a cone is covered with paper without any gaps or overlaps. the height of the cone is 28 inches and its diameter is 14 inches. what is the surface area of the covering to the nearest square inch? a. 635 in² b. 1,376 in² c. 1,993 in² d. 789 in²

Explanation:

Step1: Find the radius of the cone

The diameter is 14 inches, so the radius \( r=\frac{14}{2} = 7 \) inches.

Step2: Find the slant height \( l \) of the cone

Using the Pythagorean theorem for the cone (\( l=\sqrt{r^{2}+h^{2}} \)), where \( h = 28 \) inches and \( r = 7 \) inches.
\( l=\sqrt{7^{2}+28^{2}}=\sqrt{49 + 784}=\sqrt{833}\approx28.86 \) inches.

Step3: Calculate the surface area of the cone (lateral surface area + base area, but since it's covered without gaps/overlaps, we need to check if it's a closed cone or just lateral. Wait, when covering a cone with paper without gaps/overlaps for an art project, usually we consider the total surface area? Wait, no, if it's a cone (like a party hat), sometimes it's lateral, but let's check the formula. Wait, the total surface area of a cone is \( \pi rl+\pi r^{2} \), lateral is \( \pi rl \). Wait, let's re - check the problem: "a cone is covered with paper without any gaps or overlaps". So we need to cover the entire cone, so total surface area? Wait, no, maybe it's a closed cone? Wait, no, a cone has a base (a circle) and a lateral surface. Wait, let's calculate both. Wait, first, let's see the options. Let's compute lateral surface area \( \pi rl \) and total surface area \( \pi r(l + r) \).

First, lateral surface area: \( \pi\times7\times28.86\approx3.14\times7\times28.86\approx3.14\times202.02\approx634.34\approx635 \) (if we consider lateral). Wait, but let's check total surface area: \( \pi r(l + r)=3.14\times7\times(28.86 + 7)=3.14\times7\times35.86\approx3.14\times251.02\approx788.2\approx789 \). Wait, now we are confused. Wait, the problem says "a cone is covered with paper without any gaps or overlaps". If the cone is a solid (like a model), we need to cover the base and the lateral surface. But let's re - examine the slant height calculation. Wait, \( h = 28 \), \( r = 7 \), so \( l=\sqrt{7^{2}+28^{2}}=\sqrt{49 + 784}=\sqrt{833}\approx28.86 \). Now, total surface area of a cone is \( S=\pi r(r + l) \). Let's compute that:

\( r = 7 \), \( l\approx28.86 \)

\( S=\pi\times7\times(7 + 28.86)=7\pi\times35.86\approx7\times3.14\times35.86\approx21.98\times35.86\approx788.2\approx789 \) square inches. Wait, but let's check the lateral surface area: \( \pi rl=3.14\times7\times28.86\approx634.3\approx635 \). Now, why the difference? Let's think about the cone: if we are covering a cone (like a hollow cone, open at the base), then we only need lateral surface area. But if it's a solid cone (with a base), we need total. The problem says "a cone is covered with paper without any gaps or overlaps". If the cone is, for example, a paper cone (like a party hat), we don't cover the base (since it's open). But maybe in the art project, it's a closed cone? Wait, let's check the answer options. Option A is 635, D is 789. Let's recalculate the slant height more accurately. \( l=\sqrt{7^{2}+28^{2}}=\sqrt{49 + 784}=\sqrt{833}\approx28.8617 \)

Lateral surface area: \( \pi rl=\pi\times7\times28.8617\approx3.1416\times7\times28.8617\approx3.1416\times202.0319\approx634.7\approx635 \)

Total surface area: \( \pi rl+\pi r^{2}=\pi r(l + r)=3.1416\times7\times(28.8617 + 7)=3.1416\times7\times35.8617\approx3.1416\times251.0319\approx788.7\approx789 \)

Now, the problem says "a cone is covered with paper without any gaps or overlaps". If the cone is being covered (like wrapping a cone - shaped object), we need to cover the entire surface, including the base? Wait, but a cone's base is a circle. Wait, maybe the problem is about the lateral surface area. Wait, let's check the answer options. Let's see the calculations again.

Wait, maybe I made a mistake in assuming total surface area. Let's check the formula for the surface area of a cone when covering it (like a net). The net of a cone is a sector and a circle (the base). But if we are covering the cone without gaps or overlaps, we need to include both? Wait, but let's check the numbers.

Wait, the height is 28, diameter 14, so radius 7.

Slant height \( l=\sqrt{7^{2}+28^{2}}=\sqrt{49 + 784}=\sqrt{833}\approx28.86 \)

Lateral surface area \( LSA=\pi rl = \pi\times7\times28.86\approx3.14\times7\times28.86\approx3.14\times202.02 = 634.34\approx635 \)

Total surface area \( TSA=\pi rl+\pi r^{2}=\pi r(l + r)=3.14\times7\times(28.86 + 7)=3.14\times7\times35.86=3.14\times251.02 = 788.2\approx789 \)

Now, let's see the answer options. Option A is 635, D is 789. Let's think about the context: "an art project, a cone is covered with paper without any gaps or overlaps". If it's a cone - shaped object (like a model), maybe we need to cover the lateral surface (the curved part) and the base? Wait, no, if you have a cone, the base is a flat circle. If you are covering it with paper, maybe you are making a cone (so the lateral surface and the base? But when you make a cone from paper, you cut a sector (lateral) and a circle (base) and attach them. But the problem says "covered with paper without any gaps or overlaps", so maybe it's the total surface area. But let's check the answer options. Wait, maybe I made a mistake in the slant height. Wait, 7 - 28 - l triangle. 7 squared is 49, 28 squared is 784, sum is 833, square root of 833 is approximately 28.86.

Wait, let's check the answer options again. Option D is 789, which is close to the total surface area. Option A is close to lateral. Let's see the problem statement again: "a cone is covered with paper without any gaps or overlaps". If the cone is a solid (like a 3D object), we need to cover all its surface, including the base. So total surface area. So let's recalculate total surface area:

\( r = 7 \), \( h = 28 \), \( l=\sqrt{7^{2}+28^{2}}=\sqrt{49 + 784}=\sqrt{833}\approx28.86 \)

\( TSA=\pi r^{2}+\pi rl=\pi r(r + l)=3.14\times7\times(7 + 28.86)=3.14\times7\times35.86 = 3.14\times251.02=788.2\approx789 \)

So the answer should be D. 789 \( \text{in}^2 \)

Wait, but wait, maybe the problem is about lateral surface area. Let's check the options again. If we calculate lateral surface area: \( \pi rl=3.14\times7\times28.86\approx634\approx635 \), which is option A. Now we are confused. Let's check the formula for the surface area of a cone. The lateral (or curved) surface area of a cone is \( \pi rl \), where \( l \) is the slant height. The total surface area is \( \pi rl+\pi r^{2} \).

Now, let's check the problem statement: "a cone is covered with paper without any gaps or overlaps". If the cone is, for example, a party hat, we don't cover the base (since it's open at the bottom). So we only need the lateral surface area. But if it's a closed cone (like a solid), we need the total. Let's see the numbers. The height is 28, radius 7. The slant height is about 28.86, which is almost equal to the height. So the lateral surface area is \( \pi\times7\times28.86\approx635 \), and total is about 789.

Wait, maybe the problem is considering the lateral surface area. Let's check the answer options. Option A is 635, which is close to the lateral surface area. Let's recalculate the lateral surface area with more precise \( \pi \) (using \( \pi\approx3.1416 \)):

\( l=\sqrt{7^{2}+28^{2}}=\sqrt{49 + 784}=\sqrt{833}\approx28.8617 \)

\( LSA=\pi rl=3.1416\times7\times28.8617\approx3.1416\times202.0319\approx634.7\approx635 \)

So if we consider the lateral surface area (covering the curved part, not the base), then the answer is A. But why the confusion? Let's think about the cone: when you cover a cone with paper, if it's a hollow cone (open at the base), you only need the lateral surface. If it's a solid cone, you need both. The problem says "a cone is covered with paper without any gaps or overlaps". It doesn't specify if it's open or closed. But in most art projects, if you are making a cone - shaped cover, you might not cover the base (since it's attached to something). Wait, but let's check the answer options. The options are A:635, D:789.

Wait, maybe I made a mistake in the slant height. Wait, 7 - 28 - l. Wait, 28 is the height, 7 is the radius. So by Pythagoras, \( l=\sqrt{h^{2}+r^{2}}=\sqrt{28^{2}+7^{2}}=\sqrt{784 + 49}=\sqrt{833}\approx28.86 \). That's correct.

Wait, let's check the total surface area again: \( \pi r^{2}+\pi rl=\pi r(r + l)=3.14\times7\times(7 + 28.86)=3.14\times7\times35.86 = 3.14\times251.02 = 788.2\approx789 \). So if we include the base, it's 789. If we don't, it's 635.

Now, let's see the problem statement again: "a cone is covered with paper without any gaps or overlaps". If the cone is a 3D object, the paper has to cover all its outer surface, including the base. So total surface area. So the answer should be D. 789 \( \text{in}^2 \)

Answer:

D. \( 789\ \text{in}^2 \)