Answer:
$45.5
Explanation:
For a firm, profit is equal to total revenue minus total cost. This can be represented mathematically as follows:
π = R - C ................................................................................................... (1)
Where π represents profit, R represents total revenue and C represents total cost.
Total revenue (R) is unit selling price (P) multiply by the sales volume (S). This is represented mathematically as follows:
R = P * S ...................................................................................................... (2)
From the question, sales volume is given as follows:
S = $100 – P .............................................................................................. (3)
We then substitute for S in equation (2) and solve as follows:
R = P($100 – P) = $100P – 
............................................................... (2a)
From the question, C is given as:
C = $917 + 9S .............................................................................................. (4)
We then substitute for S in equation (4) and solve as follows:
C = $917 + 9($100 – P) = $917 + $900 – 9P = $17 – 9P .................. (4a)
To derive the profit function, the solution for R in equation (2a) and the solution for C in equation (4a) are substituted into equations (1) above and then solved as follows:
π = ($100P – ) – ($17 – 9P) = $100P – – $17 + 9P
= $100P + 9P – – $17
π = $109P – – $17 ................................................................................ (1a)
Equation (1a) is the profit function. The calculation of the derivative of the profit function (1a) with respect to the price is obtained as follows:
dπ/dP = $109 – 2P = 0 ................................................................................. (5)
From equation (5), we can solve for P as follows:
$109 – 2P = 0
$109 = 2P
2P = $109
P = $109/2
P = $54.5 ............................................................................................................ (6)
To determine the sales volume, we substitute 54.5 in equation (6) for P in the product’s sales volume (S) in equation (3) and we then solve as follows:
S = $100 – $54.5 = $45.5
Therefore, the sales volume (S) at which the firm’s profit is a maximum is $45.5.
