Question

A cylinder has a surface area of 1,714 square inches and a lateral area of 653 square inches. What is the radius of the base of the cylinder?Use ​𝜋≈3.14 and round your answer to the nearest whole number.inches

Answer 1

To find the radius of the base of the cylinder, we can use the formulas for the lateral area and surface area of a cylinder.

1. **Lateral Area of a Cylinder**:
The formula for the lateral area \( A_L \) of a cylinder is given by:
\[
A_L = 2\pi rh
\]
where \( r \) is the radius and \( h \) is the height.

2. **Surface Area of a Cylinder**:
The formula for the total surface area \( A_S \) of a cylinder is:
\[
A_S = 2\pi r(h + r)
\]

### Given Data:
- Lateral area \( A_L = 653 \) square inches
- Surface area \( A_S = 1714 \) square inches

### Step 1: Solve for Height using Lateral Area

From the lateral area formula, we can rearrange to find \( h \):
\[
653 = 2 \pi r h
\]

Using \( \pi \approx 3.14 \):
\[
653 = 2 \times 3.14 \times r \times h
\]
\[
653 = 6.28rh
\]
\[
rh = \frac{653}{6.28} \approx 104.0 \quad \text{(1)}
\]

### Step 2: Substitute into Surface Area Formula

Now, we substitute \( h \) from equation (1) into the surface area formula:
\[
1714 = 2\pi r(h + r)
\]
Substituting \( h \):
\[
1714 = 2\pi \left(r \left(\frac{104.0}{r}\right) + r\right)
\]
This simplifies to:
\[
1714 = 2\pi \left(104.0 + r\right)
\]
\[
1714 = 2 \times 3.14 \times (104.0 + r)
\]
\[
1714 = 6.28(104.0 + r)
\]
Dividing both sides by 6.28:
\[
\frac{1714}{6.28} = 104.0 + r
\]
Calculating the left side:
\[
273.62 \approx 104 + r
\]
So:
\[
r \approx 273.62 - 104.0 = 169.62 \quad \text{(2)}
\]

### Final Step: Round to the Nearest Whole Number

We find the radius \( r \):
\[
r \approx 170 \text{ inches}
\]

Thus, the radius of the base of the cylinder is approximately **170 inches**.