Let $A$ be the lateral surface area of the cone below, and let $B$ be the lateral surface area of the cylinder below. Find $\frac{A}{B}.$

[asy]
unitsize(0.5 cm);

pair A, B, C, trans;

A = (0,6);
B = (-3,0);
C = (3,0);

draw(B--A--C);
draw(yscale(0.2)*arc((0,0),3,0,180), dashed);
draw(yscale(0.2)*arc((0,0),3,180,360));
draw((0,0)--C, dashed);
draw((0,0)--A, dashed);

dot((0,0));
label("$3$", C/2, N, red, UnFill);
label("$5$", A/2, W, red, UnFill);

trans = (10,0);

draw(shift(trans)*yscale(0.2)*arc((0,0),3,0,180), dashed);
draw(shift(trans)*yscale(0.2)*arc((0,0),3,180,360));
draw(shift(trans)*(B--(-3,6)));
draw(shift(trans)*(C--(3,6)));
draw(shift(trans + A)*yscale(0.2)*Circle((0,0),3));
draw(shift(trans)*((0,0)--C), dashed);
label("$3$", C/2 + trans, N, red, UnFill);
label("$5$", (3,3) + trans, E, red);
dot(trans);
[/asy]


To solve for $\frac{A}{B}$, we need to determine the lateral surface areas of both the cone and the cylinder, then take their ratio.

### Step 1: Find the Lateral Surface Area of the Cone, $ A $
The formula for the lateral surface area of a cone is:
\[
A = \pi r \ell
\]
where:
- $ r $ is the radius of the base of the cone,
- $ \ell $ is the slant height of the cone.

From the diagram, we see that:
- $ r = 3 $,
- $ \ell = 5 $.

Substituting these values:
\[
A = \pi \cdot 3 \cdot 5 = 15\pi
\]

### Step 2: Find the Lateral Surface Area of the Cylinder, $ B $
The formula for the lateral surface area of a cylinder is:
\[
B = 2\pi r h
\]
where:
- $ r $ is the radius of the base of the cylinder,
- $ h $ is the height of the cylinder.

From the diagram, we have:
- $ r = 3 $,
- $ h = 5 $.

Substituting these values:
\[
B = 2 \pi \cdot 3 \cdot 5 = 30\pi
\]

### Step 3: Find the Ratio $\frac{A}{B}$
Now, we calculate the ratio of the lateral surface areas:
\[
\frac{A}{B} = \frac{15\pi}{30\pi} = \frac{15}{30} = \frac{1}{2}
\]

### Conclusion
The ratio $\frac{A}{B}$ is:
\[
\boxed{\frac{1}{2}}
\]

Would you like more details on the cone and cylinder surface area formulas, or have any questions? 

Here are some related questions:
1. How does changing the radius affect the surface areas of the cone and cylinder?
2. What happens to the ratio $\frac{A}{B}$ if the height of the cylinder changes?
3. How does the ratio $\frac{A}{B}$ compare for different slant heights of the cone?
4. How would the surface area change if we doubled the radius but kept the height and slant height constant?
5. What is the effect of changing only the height on the lateral area of the cylinder?

**Tip:** For cones and cylinders, the lateral surface area depends heavily on the radius, which scales linearly in both cases.

Let $A$ be the lateral surface area of the cone below, and let $B$ be the lateral surface area of the cylinder below. Find $\frac{A}{B}.$

Learn how to find the ratio of the lateral surface area of a cone to that of a cylinder with a shared radius and height dimensions. This problem explores key geometry concepts including lateral surface area calculation, with solutions suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation