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

Step 1: Analyze the Shape

The monument is a composite figure made up of:

1. A rectangular prism (the base), which has dimensions:
   • Height: 33 ft
   • Base dimensions: 11 ft by 11 ft
2. A square pyramid on top, with:
   • Base side length: 11 ft (same as the rectangular prism)
   • Slant height: 13 ft

Step 2: Calculate the Surface Area of the Rectangular Prism

The rectangular prism has three exposed sides (the base is not painted):

• Two vertical sides, each with dimensions 33 ft by 11 ft.
• One vertical side (the front), with dimensions 33 ft by 11 ft.

Surface area of the prism: $\text{Prism Area} = 2 \times (33 \times 11) + 1 \times (33 \times 11) = 3 \times (33 \times 11)$ $\text{Prism Area} = 3 \times 363 = 1089 \, \text{ft}^2$

Step 3: Calculate the Surface Area of the Pyramid

The pyramid has four triangular faces. Each triangle has a base of 11 ft and a slant height of 13 ft. The area of one triangular face is: $\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 11 \times 13 = 71.5 \, \text{ft}^2$ There are four triangular faces, so the total area of the pyramid is: $\text{Pyramid Area} = 4 \times 71.5 = 286 \, \text{ft}^2$

Step 4: Calculate Total Surface Area

The total surface area to be painted is the sum of the areas of the prism and the pyramid: $\text{Total Surface Area} = 1089 + 286 = 1375 \, \text{ft}^2$

Step 5: Calculate the Number of Gallons of Paint Required

One gallon of paint covers 190 ft². To find the number of gallons needed, divide the total surface area by the coverage per gallon: $\text{Gallons Required} = \frac{1375}{190} \approx 7.24$ Since paint can only be bought in whole gallons, you will need to purchase 8 gallons of paint.

Final Answer:

You need to buy 8 gallons of paint to cover the monument.

Would you like more details on any of these steps, or do you have any further questions?

Here are five related questions you might consider:

1. How would the calculation change if the slant height of the pyramid were different?
2. What if the base of the pyramid were rectangular instead of square?
3. How would the cost of the paint be calculated if the price per gallon is known?
4. What formula is used to calculate the surface area of a triangular face?
5. How can you account for paint wasted due to uneven surfaces or texture?

Tip: Always double-check whether all surfaces are considered, especially when certain parts are excluded from painting, as in this case with the base.