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To solve this problem, we need to calculate the surface area of the monument and determine how many gallons of paint are needed based on the coverage per gallon.

The monument has two parts: 
1. A rectangular prism at the bottom.
2. A square pyramid on top.

### Step 1: Calculate the Surface Area of the Rectangular Prism
The prism has:
- A base of $12 \, \text{ft} \times 12 \, \text{ft}$,
- A height of $33 \, \text{ft}$.

Only the four vertical sides need painting (the base is not painted).

The area of each vertical side is:
\[
12 \, \text{ft} \times 33 \, \text{ft} = 396 \, \text{ft}^2
\]

Since there are four sides, the total area for the prism is:
\[
4 \times 396 \, \text{ft}^2 = 1584 \, \text{ft}^2
\]

### Step 2: Calculate the Surface Area of the Square Pyramid
The pyramid has:
- A base edge of $12 \, \text{ft}$ (matching the prism),
- A slant height of $13 \, \text{ft}$.

The surface area of the four triangular faces (which need painting) is calculated as follows:

The area of one triangular face is:
\[
\frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 12 \, \text{ft} \times 13 \, \text{ft} = 78 \, \text{ft}^2
\]

With four triangular faces, the total area for the pyramid is:
\[
4 \times 78 \, \text{ft}^2 = 312 \, \text{ft}^2
\]

### Step 3: Calculate the Total Surface Area
The total paintable surface area of the monument is:
\[
1584 \, \text{ft}^2 + 312 \, \text{ft}^2 = 1896 \, \text{ft}^2
\]

### Step 4: Calculate the Number of Gallons Needed
Each gallon covers $172 \, \text{ft}^2$. So, the number of gallons needed is:
\[
\frac{1896 \, \text{ft}^2}{172 \, \text{ft}^2/\text{gallon}} \approx 11.02 \, \text{gallons}
\]

Since paint can only be bought by the gallon, round up to the next whole number:
\[
\text{Gallons needed} = 12
\]

### Final Answer
The number of gallons required is **12**.

Would you like further explanation or have any questions?

### Related Questions
1. How would the answer change if the paint coverage was different?
2. What would be the surface area if the slant height of the pyramid was shorter?
3. How is the total cost affected if each gallon of paint costs $20?
4. What if the monument had a different base shape? How would you calculate the surface area?
5. How could we determine the paint needed if the base also required painting?

#### Tip:
Always verify if any part of the surface area is excluded from painting, such as the base in this example.

A monument outside city hall has dimensions as shown in the figure below. If one gallon of paint can cover 172 ft², how many gallons of paint must be bought in order to paint the monument? Assume that the base of the monument cannot be painted and that paint can be bought only by the gallon.

Learn how to calculate the gallons of paint needed to cover a monument with a rectangular prism and pyramid structure, excluding the base. This solution demonstrates surface area calculations for each shape, taking into account the 172 ft² coverage per gallon. Ideal for high school geometry students.

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