A cylinder has a base radius of 2 and a height of 9 . To the nearest whole number , by how much does the lateral area exceed the sum of the areas of the two bases ?

Options

A. 101
B. 96
C. 88
D. 81

Explanation

To solve the problem step by step, we will calculate the lateral area of the cylinder and the sum of the areas of the two bases, and then find the difference between them.

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

Given:
- Radius \( r = 2 \)
- Height \( h = 9 \)

Substituting the values into the formula:
\[ A_L = 2 \pi (2)(9) = 36 \pi \]

### Step 2: Calculate the Area of One Base
The area \( A_B \) of one base of the cylinder (which is a circle) is given by:
\[ A_B = \pi r^2 \]

Substituting the radius:
\[ A_B = \pi (2^2) = 4 \pi \]

### Step 3: Calculate the Sum of the Areas of Two Bases
Since there are two bases, the total area \( A_{total\_bases} \) of the two bases is:
\[ A_{total\_bases} = 2 \times A_B = 2 \times 4 \pi = 8 \pi \]

### Step 4: Calculate the Difference Between the Lateral Area and the Sum of the Areas of the Bases
Now, we need to find out by how much the lateral area exceeds the sum of the areas of the two bases:
\[ \text{Difference} = A_L - A_{total\_bases} \]
Substituting the values:
\[ \text{Difference} = 36 \pi - 8 \pi = 28 \pi \]

### Step 5: Convert the Difference to a Numerical Value
To find the numerical value, we can approximate \( \pi \) as 3.14:
\[ 28 \pi \approx 28 \times 3.14 \approx 87.92 \]
Rounding this to the nearest whole number gives:
\[ \text{Difference} \approx 88 \]

### Final Answer
The lateral area exceeds the sum of the areas of the two bases by approximately **88**.

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