https://file.aiquickdraw.com/mathbot/file/17294638168453z19rv0t.jpg

We are given that the surface area of the shape, consisting of a cylinder and a cone, is \( 285\pi \ \text{cm}^2 \). The objective is to find the length of the cylinder. Here's how we can break this down:

### 1. Surface Area Components

The total surface area consists of:
- The curved surface area of the cone.
- The curved surface area of the cylinder.
- The area of the circular base (which is common to both the cone and the cylinder and hence counted once).

#### a. Surface Area of the Cone:
The formula for the curved surface area of a cone is:
\[
A_{\text{cone}} = \pi r l
\]
Where:
- \( r = 5 \ \text{cm} \) (the radius),
- \( l = 18 \ \text{cm} \) (the slant height).

So,
\[
A_{\text{cone}} = \pi \times 5 \times 18 = 90\pi \ \text{cm}^2
\]

#### b. Surface Area of the Cylinder:
The formula for the curved surface area of a cylinder is:
\[
A_{\text{cylinder}} = 2\pi r h
\]
Where:
- \( r = 5 \ \text{cm} \) (the radius),
- \( h \) is the height or length of the cylinder (which we need to find).

#### c. Base Area:
The area of the circular base is:
\[
A_{\text{base}} = \pi r^2 = \pi \times 5^2 = 25\pi \ \text{cm}^2
\]

### 2. Total Surface Area Equation:
The total surface area is the sum of the cone's curved surface area, the cylinder's curved surface area, and the base area:
\[
A_{\text{total}} = A_{\text{cone}} + A_{\text{cylinder}} + A_{\text{base}} 
\]
\[
285\pi = 90\pi + 2\pi \times 5 \times h + 25\pi
\]

Simplifying:
\[
285\pi = 115\pi + 10\pi h
\]
\[
285\pi - 115\pi = 10\pi h
\]
\[
170\pi = 10\pi h
\]
\[
h = \frac{170\pi}{10\pi} = 17 \ \text{cm}
\]

### Conclusion:
The length of the cylinder is \( \boxed{17.0} \ \text{cm} \).

Let me know if you'd like more details or have further questions. Here are some related questions to explore:

1. What would the total surface area be if the radius of the cylinder doubled?
2. How would changing the slant height of the cone affect the total surface area?
3. What is the volume of the cylinder in this shape?
4. How does the formula for surface area change if the shape is made of a hemisphere and a cylinder?
5. What is the ratio of the surface area of the cone to the surface area of the cylinder?

**Tip**: Always ensure all areas contributing to the total surface area are accounted for, especially shared components like the base in combined solids.

The shape below is made from a cylinder joined to the base of a cone. The surface area of this shape is 285π cm². Calculate the length of the cylinder. If your answer is a decimal, give it to 1 d.p.

Find the length of the cylinder in a composite shape made of a cone and a cylinder, with a total surface area of 285π cm². This problem involves key geometric concepts and surface area formulas for cones, cylinders, and circles. Step-by-step solution included, suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation