Jane has to point a cylindrical column that is 14 feet high and that has a circular base with a radius of 3 feet. If one bucket of paint will cover `10pi` square feet, how many whole buckets does jane need to buy in order to paint the column, including the top and bottom?

Answer

To solve the problem step by step, we will calculate the total surface area of the cylindrical column and then determine how many buckets of paint Jane needs to buy.

### Step 1: Understand the dimensions of the cylinder
- Height (h) = 14 feet
- Radius (r) = 3 feet

### Step 2: Calculate the total surface area of the cylinder
The total surface area (A) of a cylinder is given by the formula:
\[ A = 2\pi r^2 + 2\pi rh \]
Where:
- \( 2\pi r^2 \) is the area of the top and bottom circles (2 bases)
- \( 2\pi rh \) is the curved surface area

### Step 3: Substitute the values into the formula
Substituting \( r = 3 \) feet and \( h = 14 \) feet into the formula:
\[ A = 2\pi (3^2) + 2\pi (3)(14) \]

### Step 4: Calculate the area of the top and bottom circles
First, calculate \( 3^2 \):
\[ 3^2 = 9 \]
Now substitute this back into the formula:
\[ A = 2\pi (9) + 2\pi (3)(14) \]
\[ A = 18\pi + 2\pi (42) \]
\[ A = 18\pi + 84\pi \]

### Step 5: Combine the areas
Now combine the two areas:
\[ A = (18 + 84)\pi \]
\[ A = 102\pi \]

### Step 6: Determine the coverage of one bucket of paint
It is given that one bucket of paint covers \( 10\pi \) square feet.

### Step 7: Calculate the number of buckets needed
To find the number of buckets needed, divide the total area by the coverage of one bucket:
\[ \text{Number of buckets} = \frac{102\pi}{10\pi} = \frac{102}{10} = 10.2 \]

### Step 8: Round up to the nearest whole bucket
Since Jane needs whole buckets, we round 10.2 up to the next whole number:
\[ \text{Whole buckets needed} = 11 \]

### Final Answer
Jane needs to buy **11 whole buckets** of paint to cover the cylindrical column.

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