We are tasked with determining how many gallons of paint are needed to cover the surface of the monument, excluding its base, and with the knowledge that one gallon covers 233 square feet.

Step 1: Calculate the surface area of the monument.

The monument consists of two main parts:

1. The cylindrical part (the vertical section).
2. The conical top.

Part 1: Surface Area of the Cylinder (without the base)

The formula for the lateral surface area of a cylinder is: $A_{\text{cylinder}} = 2\pi r h$ where:

• $r$ is the radius of the base of the cylinder (half of the width, so $r = \frac{14}{2} = 7$ ft),
• $h$ is the height of the cylinder (44 ft).

Thus, the surface area of the cylindrical part is: $A_{\text{cylinder}} = 2 \pi (7)(44) = 616 \pi \approx 1935.97 \, \text{ft}^2.$

Part 2: Surface Area of the Cone

The formula for the lateral surface area of a cone is: $A_{\text{cone}} = \pi r l$ where:

• $r = 7 \, \text{ft}$ (same as the radius of the base),
• $l$ is the slant height of the cone, which we can find using the Pythagorean theorem: $l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 10^2} = \sqrt{49 + 100} = \sqrt{149} \approx 12.21 \, \text{ft}.$ Thus, the lateral surface area of the cone is: $A_{\text{cone}} = \pi (7)(12.21) = 85.47 \pi \approx 268.48 \, \text{ft}^2.$

Step 2: Total Surface Area to be Painted

Now, sum the areas of the cylinder and the cone: $A_{\text{total}} = A_{\text{cylinder}} + A_{\text{cone}} = 1935.97 + 268.48 = 2204.45 \, \text{ft}^2.$

Step 3: Calculate the Number of Gallons of Paint Required

One gallon of paint covers 233 ft², so the number of gallons needed is: $\text{Gallons} = \frac{A_{\text{total}}}{233} = \frac{2204.45}{233} \approx 9.46.$ Since we can only buy whole gallons, we'll round up to 10 gallons.

Thus, 10 gallons of paint are required to paint the monument.

Would you like further details or clarification on any part of the process?

Here are some related questions for deeper understanding:

1. How is the formula for the lateral surface area of a cylinder derived?
2. Can you explain how the slant height of the cone was calculated?
3. What would happen if the monument had two bases to be painted?
4. How would the required gallons change if the paint coverage was less efficient?
5. What is the surface area of a similar monument with different dimensions?

Tip: When rounding in practical applications, it's important to always round up when dealing with quantities like gallons to ensure full coverage.