https://file.aiquickdraw.com/mathbot/file/1729733185997o9gu07jv.jpg

To solve for the height of the cylinder, we need to use the formula for the surface area of a cylinder:

\[
A = 2\pi r^2 + 2\pi rh
\]

Where:
- \( A \) is the surface area.
- \( r \) is the radius of the base of the cylinder.
- \( h \) is the height of the cylinder.

From the image:
- The surface area \( A \) is given as \( 3203.9932 \, \text{square yards} \).
- The radius \( r \) is half of the diameter, so \( r = \frac{16.3}{2} = 8.15 \, \text{yards} \).

Now we can substitute the known values into the formula and solve for \( h \).

1. Calculate the area of the two circular bases:

\[
2\pi r^2 = 2 \times 3.14 \times (8.15)^2 = 2 \times 3.14 \times 66.4225 = 416.87 \, \text{square yards}.
\]

2. Now subtract the area of the two bases from the total surface area to find the lateral surface area:

\[
\text{Lateral area} = 3203.9932 - 416.87 = 2787.1232 \, \text{square yards}.
\]

3. The lateral surface area is given by \( 2\pi rh \), so solve for \( h \):

\[
2\pi rh = 2787.1232
\]

\[
2 \times 3.14 \times 8.15 \times h = 2787.1232
\]

\[
51.196 \times h = 2787.1232
\]

\[
h = \frac{2787.1232}{51.196} \approx 54.44 \, \text{yards}.
\]

Thus, the height \( h \) of the cylinder is approximately **54.44 yards**.

Would you like further details on this solution, or have any questions?

Here are some related questions:
1. How is the surface area of a cylinder derived from basic geometric principles?
2. What are the differences between lateral surface area and total surface area?
3. How can we check for accuracy when calculating large surface areas like this?
4. What other formulas involve the radius of a cylinder besides surface area?
5. How does changing the radius or height affect the overall surface area?

**Tip**: Always be mindful of units when calculating, especially when using large measurements like yards.

The surface area of this cylinder is 3,203.9932 square yards. What is the height? Use π ≈ 3.14 and round your answer to the nearest hundredth.

Learn how to find the height of a cylinder with a given surface area of 3,203.9932 square yards and a diameter of 16.3 yards. This solution uses the formula A = 2πr^2 + 2πrh, substituting values for precise calculations. Ideal for students in Grades 8-10.

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