If two identical spherical balls with radii of 2.2 cm fit exactly inside a cylinder, what is the surface area of the closed cylinder?

Round your answer to one decimal place.

To solve the problem of finding the surface area of a closed cylinder that exactly fits two identical spherical balls with a radius of 2.2 cm, we need to follow these steps:

### Step 1: Understanding the Geometry
- Each sphere has a radius \( r = 2.2 \) cm.
- Since the two spheres fit exactly inside the cylinder, the height \( h \) of the cylinder must be twice the diameter of one sphere.

### Step 2: Calculate the Dimensions of the Cylinder
- The diameter of one sphere is \( 2r = 2 \times 2.2 = 4.4 \) cm.
- Therefore, the height \( h \) of the cylinder is \( 2 \times 4.4 = 8.8 \) cm.
- The radius \( R \) of the base of the cylinder is equal to the radius of the sphere, \( R = 2.2 \) cm.

### Step 3: Calculate the Surface Area of the Closed Cylinder
The surface area \( A \) of a closed cylinder is given by the formula:

\[
A = 2\pi R^2 + 2\pi R h
\]

Where:
- \( 2\pi R^2 \) is the area of the two circular bases.
- \( 2\pi R h \) is the lateral surface area.

Substitute the values of \( R \) and \( h \):

\[
A = 2\pi (2.2)^2 + 2\pi (2.2)(8.8)
\]

\[
A = 2\pi (4.84) + 2\pi (19.36)
\]

\[
A = 9.68\pi + 38.72\pi
\]

\[
A = 48.4\pi
\]

\[
A \approx 48.4 \times 3.1416 = 152.1 \text{ square centimeters}
\]

### Final Answer
The surface area of the closed cylinder is **152.1 cm²** (rounded to one decimal place).

Would you like more details, or do you have any other questions?

### Related Questions
1. What is the volume of the cylinder in this scenario?
2. How would the surface area change if the radius of the spheres were doubled?
3. Can you find the surface area of the cylinder if only one sphere fits inside it?
4. What is the lateral surface area of the cylinder alone?
5. How do you derive the formula for the surface area of a closed cylinder?
6. If the cylinder was open at the top, what would its surface area be?
7. What is the ratio of the surface area of the cylinder to the surface area of the spheres?
8. How would the surface area change if the number of spheres inside the cylinder increased to three?

### Tip
Always double-check that the units you are using are consistent throughout the problem to avoid mistakes in your calculations.

Discover how to calculate the surface area of a cylinder that perfectly fits two identical spherical balls with a radius of 2.2 cm inside. This problem involves geometry, surface area calculation, and volume relationships. 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