Problem

A construction company is planning their projects for the year $2025.$ The company is considering the construction of a certain number of $1,000m^2$ buildings. In $2022,$ the company has built $5$ similar buildings but with smaller areas, and the material costs associated with those $5$ buildings are as shown in the following table:

$\backslash$table$[[$Building #$,$Area $(m^2),$Cost $($JD$)],[1,680,85,443],[2,727,90,011],[3,760,92,488],[4,790,95,082],[5,825,97,499]]$

The material cost index values are displayed in the following table:

$\backslash$table$[[$Year$,$Cost index$],[2021,336],[2022,346],[2023,348],[2024,351],[2025,353]]$

The company started to build $1,000m^2$ buildings in $2023,$ and the project management team observed that $21,690$ labor hours were needed to construct the first $1,000m^2$ building; however, the number of labor hours per building decreased thereafter at a $93\%$ learning slope. Use the $8^{th}$ building as the basis for estimating the average labor hours per building and assume the $2025$ average labor wage is JD $3\mathrm{.}5$ per hour. The company's fixed cost is JD $250,000$ per year and the demand on the buildings $(D)$ is governed by the following relationship: $p=228,000-1,400D,$ where $p$ is the selling price per building.

$($a$)$ What is the range of profitable demand?

$($b$)$ How many buildings should the company construct in $2025$ in order to maximize its profit?

$($c$)$ Based on the demand identified in part $($b$),$ what is the breakeven selling price per building?

$($d$)$ How much profit will the company make in $2025$ using the demand identified in part $($b$)?$