Question

A manufacturer of kitchen appliances is preparing to set the price on a new blender. Given functions for Demand and Total Cost, develop a mathematical model for the total profit TP in terms of the price, P.

D = 2500 - 3P
TC = 5000 + 5D
A) TP = -17500 + 2515 P - 3 P²
B) TP = 8 P^2 + 2500 P - 7500
C) TP = 7,515 + 500 P + 3 P²
D) TP = -4.5 P^2 + 2500 P - 5000

Answer 1

The mathematical model for the total profit TP in terms of the price P is TP = -4.5P^2 + 2500P - 5000.

Step-by-step explanation:

To develop a mathematical model for the total profit TP in terms of the price, P, we need to find the equation for total profit. Total profit (TP) is calculated by subtracting the total cost (TC) from the total revenue (TR). The total revenue is obtained by multiplying the price (P) by the quantity demanded (D), which can be expressed as TP = TR - TC = P * D - TC.

Using the given functions for demand (D = 2500 - 3P) and total cost (TC = 5000 + 5D), we can substitute these in the equation for total profit: TP = P * (2500 - 3P) - (5000 + 5(2500 - 3P)). Simplifying this equation yields: TP = -4.5P^2 + 2500P - 5000.

Therefore, option D) TP = -4.5P^2 + 2500P - 5000 is the mathematical model for the total profit TP in terms of the price P.