https://file.aiquickdraw.com/mathbot/file/1727370561521j88tw7a5.jpg

Let's address the problem step by step based on the image provided.

### Part (a):
You need to show that the surface area of the sphere is \(36\pi y^2\).

From the formula given for the surface area of a sphere:
\[
\text{Surface area of a sphere} = 4\pi r^2
\]
The radius of the sphere is shown as \(3y\) in the image.

So, substituting \(r = 3y\) into the formula:
\[
\text{Surface area of sphere} = 4\pi (3y)^2 = 4\pi \times 9y^2 = 36\pi y^2
\]
Thus, the surface area of the sphere is indeed \(36\pi y^2\), which confirms part (a).

### Part (b):
The curved surface area of the cone is equal to the surface area of the sphere. You are asked to find the expression for \(l\) (the slant height of the cone) in terms of \(y\).

The curved surface area of a cone is given by:
\[
\text{Curved surface area of a cone} = \pi r l
\]
From the image, the base radius of the cone is also \(3y\). Therefore, substitute \(r = 3y\) into the formula:
\[
\text{Curved surface area of cone} = \pi (3y) l = 3\pi y l
\]
We are told that the curved surface area of the cone is equal to the surface area of the sphere, which is \(36\pi y^2\). So we set them equal:
\[
3\pi y l = 36\pi y^2
\]
Now, divide both sides by \(3\pi y\) to solve for \(l\):
\[
l = \frac{36\pi y^2}{3\pi y} = \frac{36y}{3} = 12y
\]
Thus, \(l = 12y\).

### Final Answers:
- (a) The surface area of the sphere is \(36\pi y^2\).
- (b) The slant height \(l\) of the cone is \(12y\).

Would you like more details or further explanations?

Here are 5 related questions to explore the topic further:
1. What is the formula for the total surface area of a cone, including the base?
2. How does the slant height \(l\) affect the volume of the cone?
3. If the radius of the sphere were doubled, how would the surface area change?
4. Can the surface area of a cone ever be greater than that of a sphere with the same radius?
5. How does the volume of a sphere compare to that of a cone with the same radius and height?

**Tip**: When comparing different shapes, equating their areas or volumes can help in finding relationships between dimensions like slant height, radius, or height.

a) Show that the surface area of the sphere below is 36πy². b) The curved surface area of the cone shown is equal to the surface area of the sphere. Write an expression for l in terms of y, in its simplest form.

Learn how to find the surface area of a sphere and calculate the slant height of a cone when the curved surface area equals the surface area of the sphere. This problem involves key geometry concepts and surface area formulas for spheres and cones. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation