A factory produces x calculators per day. The total daily cost in Shillings incured is 5x^2-800x+500. If the calculators are sold for sh (100-10x) each,
i, Determine the profit function
ii, Find the number of calculators that would maximize the daily profit
iii, What is the daily maximum profit?
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profit = revenue - expenses
P(x) = x(100-10x) - (5x^2 - 800x + 500)
= 900x - 15x^2 - 500
P ' (x) = 900 - 30x
= 0 for a max of P
30x=900
x=30
P(30) = 13000
To determine the profit function, we need to subtract the total cost from the total revenue.
i. Profit Function:
The revenue is given by the formula: Revenue = Number of calculators * Sale Price
Revenue = x * (100 - 10x)
The cost function is given as: Cost = 5x^2 - 800x + 500
To calculate the profit function, we need to subtract the cost from the revenue:
Profit = Revenue - Cost
Profit = x * (100 - 10x) - (5x^2 - 800x + 500)
Profit = -5x^2 + 300x - 500
ii. To find the number of calculators that would maximize the daily profit, we need to find the value of x that corresponds to the maximum value of the profit function.
To do this, we can take the derivative of the profit function with respect to x, set it equal to zero, and solve for x.
d(Profit)/dx = -10x + 300
Setting -10x + 300 = 0 and solving for x, we get:
-10x = -300
x = 30
iii. To find the daily maximum profit, we substitute the value of x = 30 into the profit function:
Profit = -5(30)^2 + 300(30) - 500
Profit = -5(900) + 9000 - 500
Profit = -4500 + 9000 - 500
Profit = 4000
Therefore, the daily maximum profit is 4000 Shillings.
To solve this problem, we will follow these steps:
i) Determine the profit function:
Profit is calculated by subtracting the cost from the revenue. The revenue depends on the number of calculators sold and the selling price of each calculator. The cost depends on the number of calculators produced and the total daily cost incurred. Therefore, the profit function can be calculated as:
Profit = Revenue - Cost
Profit = (Number of calculators sold) x (Selling price per calculator) - (Total daily cost incurred)
Profit = (x) x (100-10x) - (5x^2-800x+500)
Profit = 100x - 10x^2 - 5x^2 + 800x - 500
Profit = -15x^2 + 900x - 500
ii) Find the number of calculators that would maximize the daily profit:
To find the number of calculators that would maximize the daily profit, we need to find the derivative of the profit function with respect to x and set it equal to zero. Then, solve the resulting equation for x.
Profit'(x) = d/dx(-15x^2 + 900x - 500)
Profit'(x) = -30x + 900
Setting Profit'(x) equal to zero:
-30x + 900 = 0
30x = 900
x = 30
Therefore, the number of calculators that would maximize the daily profit is 30 calculators.
iii) Find the daily maximum profit:
To find the daily maximum profit, we substitute the value of x (30) into the profit function:
Profit = -15x^2 + 900x - 500
Profit = -15(30)^2 + 900(30) - 500
Profit = -15(900) + 27000 - 500
Profit = -13500 + 27000 - 500
Profit = 13500
Therefore, the daily maximum profit is 13,500 Shillings.