d. \(110\pi \space \text { units} ^ 2\)

UpStudy Step-by-Step Solution:

To find the surface area of the cylinder, we need to consider both the lateral surface area and the area of the two bases.

Given:

- Radius \(r = 5\)

- Height \(h = 6\)

The formula for the total surface area \(A\) of a cylinder is:

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

Let's calculate each part step by step:

1. Calculate the area of the two bases:

\(\text { Area of one base} = \pi r^ 2\)

\(\text { Area of two bases} = 2\pi r^ 2 = 2\pi ( 5^ 2) = 2\pi ( 25) = 50\pi \)

2. Calculate the lateral surface area:

\(\text { Lateral surface area} = 2\pi r h\)

\(\text { Lateral surface area} = 2\pi ( 5) ( 6) = 60\pi \)

3. Calculate the total surface area:

\(\text { Total surface area} = 50\pi + 60\pi = 110\pi \)

So, the total surface area of the cylinder is:

\(110\pi \space \text { units} ^ 2\)

Supplemental Knowledge:

When dealing with the surface area of a cylinder, it's important to account for both the lateral surface area and the areas of the two circular bases. A cylinder can be thought of as a combination of these two components. The lateral surface area is calculated using the height of the cylinder and the circumference of its base, while the total area of the bases involves the radius of the cylinder.

1. Area of the two bases:

Each base is a circle with the formula \(\pi r^ 2\). Since there are two bases, we multiply this by 2.

\(2\pi r^ 2\)

2. Lateral surface area:

This is calculated using the circumference of the base (\(2\pi r\)) multiplied by the height \(h\).

\(2\pi rh\)

Combining these gives the total surface area formula for a cylinder:

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

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