22. Find the amount of paper needed for the label on the can. Use 3.14 for $$ \pi $$ and round to the nearest tenth. A $$ 21.2 \mathrm{in}^{2} $$ B $$ 25.5 \mathrm{in}^{2} $$ C $$ 29.1 \mathrm{in}^{2} $$ D $$ 42.4 \mathrm{in}^{2} $$ 23. A cylinder has a surface area of 5652 square millimeters. If the diameter of the cylinder is 30 millimeters, what is the height? Use 3.14 for $$ \pi $$. F 10 mm G 45 mm H 59 mm J 105 mm

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UpStudy AI · Accepted Answer

To solve these problems, we will use the formulas for the surface area of a cylinder and the necessary calculations step by step.

### Problem 22: Find the amount of paper needed for the label on the can.

The surface area $$ A $$ of a cylinder (which is the label area in this case) is given by the formula:

$$
A = 2\pi r(h + r)
$$

Where:
- $$ r $$ is the radius of the base of the cylinder.
- $$ h $$ is the height of the cylinder.

However, we need the dimensions of the cylinder to calculate the surface area. Since the problem does not provide these, we cannot solve it directly.

### Problem 23: Find the height of the cylinder.

Given:
- Surface area $$ A = 5652 \, \text{mm}^2 $$
- Diameter $$ d = 30 \, \text{mm} $$

First, we find the radius $$ r $$:

$$
r = \frac{d}{2} = \frac{30}{2} = 15 \, \text{mm}
$$

Now, we can use the surface area formula for a cylinder:

$$
A = 2\pi r(h + r)
$$

Substituting the known values:

$$
5652 = 2 \cdot 3.14 \cdot 15(h + 15)
$$

Now, we simplify and solve for $$ h $$:

1. Calculate $$ 2 \cdot 3.14 \cdot 15 $$:

$$
2 \cdot 3.14 \cdot 15 = 94.2
$$

2. Substitute back into the equation:

$$
5652 = 94.2(h + 15)
$$

3. Divide both sides by 94.2:

$$
h + 15 = \frac{5652}{94.2}
$$

4. Calculate $$ \frac{5652}{94.2} $$:

$$
h + 15 \approx 60
$$

5. Subtract 15 from both sides to find $$ h $$:

$$
h \approx 60 - 15 = 45 \, \text{mm}
$$

Thus, the height of the cylinder is approximately $$ 45 \, \text{mm} $$.

### Final Answers:
- For Problem 22, we cannot determine the answer without additional information.
- For Problem 23, the height of the cylinder is $$ 45 \, \text{mm} $$, corresponding to option G.

Let me know if you need further assistance!
For Problem 22, there's not enough information to find the amount of paper needed for the label. For Problem 23, the height of the cylinder is 45 mm, which is option G.

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