Question

calculate the slant height of a cone whose radius is 3 cm and surface area is 102 square cm.

Explanation:

Step1: Recall the formula for the surface area of a cone

The total surface area of a cone is given by \( SA=\pi r(r + l) \), where \( r \) is the radius and \( l \) is the slant height. We know \( r = 3\space cm \) and \( SA=102\space cm^{2} \), and we use \( \pi=\frac{22}{7} \) (or we can use \( \pi\approx3.14 \), but let's use \( \frac{22}{7} \) for exactness). So we substitute the known values into the formula: \( 102=\frac{22}{7}\times3\times(3 + l) \)

Step2: Simplify the equation

First, simplify the left - hand side and the right - hand side. \( 102=\frac{66}{7}\times(3 + l) \)

Multiply both sides by \( \frac{7}{66} \) to get: \( 102\times\frac{7}{66}=3 + l \)

Simplify \( 102\times\frac{7}{66}=\frac{102\div6\times7}{66\div6}=\frac{17\times7}{11}=\frac{119}{11}\approx10.818 \)

So the equation becomes \( \frac{119}{11}=3 + l \)

Step3: Solve for \( l \)

Subtract 3 from both sides. \( l=\frac{119}{11}-3=\frac{119 - 33}{11}=\frac{86}{11}\approx7.818 \) (If we use \( \pi\approx3.14 \))

Let's redo the calculation with \( \pi\approx3.14 \)

The formula for the total surface area of a cone is \( SA=\pi r^{2}+\pi rl=\pi r(r + l) \)

Substitute \( SA = 102 \), \( r = 3 \), \( \pi\approx3.14 \)

\( 102=3.14\times3\times(3 + l) \)

\( 102 = 9.42\times(3 + l) \)

Divide both sides by 9.42: \( 3 + l=\frac{102}{9.42}\approx10.828 \)

Subtract 3 from both sides: \( l=\frac{102}{9.42}-3\approx10.828 - 3=7.828\approx7.83 \) (or if we consider only the lateral (curved) surface area, the formula for lateral surface area is \( LSA=\pi rl \). Maybe the problem is about lateral surface area? Let's check. If it's lateral surface area, \( LSA = 102=\pi rl \), \( r = 3 \), then \( l=\frac{102}{\pi r}=\frac{102}{3\pi}=\frac{34}{\pi}\approx10.82 \). But the original problem says "surface area", which is ambiguous, but usually total surface area includes the base. But maybe it's a typo and it's lateral surface area. Let's assume it's lateral surface area (curved surface area) for a more reasonable answer.

Let's re - define: If the curved surface area (lateral surface area) \( CSA=\pi rl \), \( CSA = 102 \), \( r = 3 \)

Then \( l=\frac{CSA}{\pi r}=\frac{102}{3\pi}=\frac{34}{\pi}\approx\frac{34}{3.14}\approx10.83 \)

Wait, maybe the problem has a mistake in the surface area value or maybe we made a wrong assumption. Let's check with total surface area again.

If \( SA=\pi r^{2}+\pi rl \), \( r = 3 \), \( SA = 102 \)

\( \pi r^{2}=3.14\times9 = 28.26 \)

Then \( \pi rl=SA-\pi r^{2}=102 - 28.26 = 73.74 \)

Then \( l=\frac{73.74}{\pi r}=\frac{73.74}{3.14\times3}=\frac{73.74}{9.42}=7.83 \)

So depending on whether it's total surface area or lateral surface area, but let's go with the total surface area formula.

From \( SA=\pi r(r + l) \)

\( 102 = 3.14\times3\times(3 + l) \)

\( 102=9.42\times(3 + l) \)

\( 3 + l=\frac{102}{9.42}\approx10.83 \)

\( l=10.83 - 3 = 7.83\space cm \) (approx)

If we use \( \pi=\frac{22}{7} \)

\( 102=\frac{22}{7}\times3\times(3 + l) \)

\( 102\times7=22\times3\times(3 + l) \)

\( 714 = 66\times(3 + l) \)

\( 3 + l=\frac{714}{66}=\frac{119}{11}\approx10.818 \)

\( l=\frac{119}{11}-3=\frac{119 - 33}{11}=\frac{86}{11}\approx7.818\approx7.82\space cm \)

Answer:

The slant height of the cone is approximately \( \boldsymbol{7.82\space cm} \) (or \( \frac{86}{11}\space cm\approx7.82\space cm \))