Acylinder has a surface area of about 221.1 square meters and a height of 7.8 meters. What is the approxomate radius of the cylinder?
A. 6.4 meters
B. 4.5 meters
C. 2.1 meters
D. 3.2 meters

Question

Acylinder has a surface area of about 221.1 square meters and a height of 7.8 meters. What is the approxomate radius of the cylinder? A. 6.4 meters B. 4.5 meters C. 2.1 meters D. 3.2 meters

UpStudy AI · Accepted Answer

We start with the formula for the total surface area of a cylinder:

$$
S = 2\pi r^2 + 2\pi r h
$$

Given:
- $$ S \approx 221.1 \, \text{m}^2 $$
- $$ h = 7.8 \, \text{m} $$

Substitute these values into the formula:

$$
221.1 = 2\pi r^2 + 2\pi \cdot 7.8 \cdot r
$$

Factor out $$ 2\pi $$:

$$
221.1 = 2\pi (r^2 + 7.8r)
$$

Divide both sides by $$ 2\pi $$:

$$
r^2 + 7.8r = \frac{221.1}{2\pi}
$$

Approximate $$ 2\pi $$ as about $$ 6.2832 $$:

$$
r^2 + 7.8r \approx \frac{221.1}{6.2832} \approx 35.17
$$

Now, the equation becomes:

$$
r^2 + 7.8r - 35.17 \approx 0
$$

This is a quadratic equation in $$ r $$ with coefficients:
- $$ a = 1 $$
- $$ b = 7.8 $$
- $$ c = -35.17 $$

Use the quadratic formula:

$$
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Plug in the values:

$$
r = \frac{-7.8 \pm \sqrt{7.8^2 - 4(1)(-35.17)}}{2}
$$

Compute the discriminant:

$$
7.8^2 = 60.84
$$

$$
-4(1)(-35.17) = 140.68
$$

$$
\text{Discriminant} = 60.84 + 140.68 = 201.52
$$

Now, take the square root:

$$
\sqrt{201.52} \approx 14.2
$$

Thus:

$$
r = \frac{-7.8 \pm 14.2}{2}
$$

Since the radius must be positive, we take the positive solution:

$$
r = \frac{-7.8 + 14.2}{2} = \frac{6.4}{2} = 3.2
$$

The approximate radius of the cylinder is $$ 3.2 $$ meters.
The approximate radius of the cylinder is 3.2 meters.

Acylinder has a surface area of about 221.1 square meters and a height of 7.8 meters. What is the approxomate radius of the cylinder?
A. 6.4 meters
B. 4.5 meters
C. 2.1 meters
D. 3.2 meters

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