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A pyramidal monument with a square base of side 10 meters and a height of 15 meters is being renovated. The renovation cost is $40 per square meter. If the monument is sold at a 25% profit, what is the selling price of the

To solve this problem, we need to calculate the surface area of the pyramidal monument, determine the renovation cost, and then find the selling price after including a 25% profit. Here are the steps:

1. **Calculate the Surface Area of the Pyramid:**
   - The pyramid has a square base with side length $a = 10$ meters.
   - The height of the pyramid $h = 15$ meters.
   - The slant height $l$ of the pyramid can be found using the Pythagorean theorem in the right triangle formed by the height of the pyramid, half of the base side, and the slant height:
     \[
     l = \sqrt{\left(\frac{a}{2}\right)^2 + h^2} = \sqrt{\left(\frac{10}{2}\right)^2 + 15^2} = \sqrt{5^2 + 15^2} = \sqrt{25 + 225} = \sqrt{250} = 5\sqrt{10}
     \]

2. **Surface Area Calculation:**
   - The area of the square base $A_{\text{base}} = a^2 = 10^2 = 100$ square meters.
   - The area of one triangular face $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \times 5\sqrt{10} = 25\sqrt{10}$ square meters.
   - The total area of the four triangular faces $A_{\text{lateral}} = 4 \times 25\sqrt{10} = 100\sqrt{10}$ square meters.
   - The total surface area of the pyramid $A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} = 100 + 100\sqrt{10}$ square meters.

3. **Renovation Cost:**
   - The renovation cost per square meter is $40.
   - Total renovation cost $C = 40 \times A_{\text{total}} = 40 \times (100 + 100\sqrt{10}) = 4000 + 4000\sqrt{10}$.

4. **Selling Price Calculation:**
   - To find the selling price with a 25% profit, calculate the total cost including profit:
     \[
     \text{Selling Price} = C + 0.25 \times C = 1.25 \times C = 1.25 \times (4000 + 4000\sqrt{10}) = 5000 + 5000\sqrt{10}
     \]

Therefore, the selling price of the renovated pyramidal monument is $5000 + 5000\sqrt{10}$ dollars.

Would you like further details or have any questions?

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Here are some related questions that might interest you:

1. How do you find the volume of a pyramidal structure?
2. What is the Pythagorean theorem and how is it used in this context?
3. How do you calculate the area of a triangle given its base and height?
4. What are some applications of surface area calculations in real life?
5. How do you determine profit percentages in financial calculations?
6. Can you explain the difference between slant height and perpendicular height in pyramids?
7. How do material costs impact the overall expense in construction projects?
8. What other geometric shapes have you encountered that are used in monuments?

**Tip:** Always break down complex problems into smaller steps to make them easier to solve.

Learn how to calculate the renovation cost and selling price of a pyramidal monument with a square base and given dimensions. Includes step-by-step calculations for surface area and profit margins.

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