A real estate developer would like to analyze his revenue,cost and profit for a specific hosing project that is under construction.last month he sold 20 large houses for m$ 250000 and 10 small houses for $150000.The cost build large houses was c(x)=56000x+50000x+1000x^2
and small homes was c(x)=36000+36000x+750x^2
a) write down the revenue equation for large homes and find the profit equation
b)find the marginal profit for both large and small homes if they were to sell one additional homes
c)graph both cost and revenue for large homes
If there are x large and y small homes, then
R(x,y) = 250000x + 150000y
The cost is
C(x,y) = (56000x+50000x+1000x^2)+(36000+36000y+750y^2)
The profit is thus
P(x,y) = R(x,y)-C(x,y)
If you just want figures for large homes, set y=0.
a) The revenue equation can be calculated by multiplying the number of houses sold (x) with the selling price of each type of house. Given that the developer sold 20 large houses for $250,000 each and 10 small houses for $150,000 each, the revenue equation for large houses can be written as:
R(x) = 20x * $250,000
The profit equation can be obtained by subtracting the cost equation from the revenue equation. For large houses, the cost equation is given as:
C(x) = 56,000x + 50,000x + 1,000x^2
Therefore, the profit equation for large houses is:
P(x) = R(x) - C(x)
b) To find the marginal profit for both large and small homes if one additional home is sold, we need to calculate the change in profit (ΔP) when x increases by 1.
For large houses:
ΔP(x) = P(x + 1) - P(x)
Substituting the equations for P(x), P(x + 1), C(x), and R(x):
ΔP(x) = [R(x + 1) - C(x + 1)] - [R(x) - C(x)]
Simplifying the equation, we get:
ΔP(x) = [R(x + 1) - R(x)] - [C(x + 1) - C(x)]
By substituting the revenue and cost equations for large houses into the above equation, you can calculate the marginal profit for large houses.
Similarly, you can follow the same steps to find the marginal profit for small houses.
c) To graph the cost and revenue for large homes, you can plot the cost equation (C(x)) and the revenue equation (R(x)) on a graph. The x-axis represents the number of homes sold, and the y-axis represents the cost and revenue values. By substituting different values of x into the equations, you can plot multiple points and connect them to create a graph of the cost and revenue curves for large homes.