the total revenue and the total profit functions (in Ringgit) of a product are given as follows:
total revenue :R(x)= x^3-95-800x
total profit: P(x)=1/3x^3+800x-120
where x is the number of units produced and sold.
find the level of production in which the total cost is a minimum.
how to get answer?
To find the level of production at which the total cost is a minimum, we need to find the minimum point of the total cost function.
The total cost function is given by:
Total Cost = Total Revenue - Total Profit
Given that the total revenue function is R(x) = x^3 - 95x - 800x and the total profit function is P(x) = (1/3)x^3 + 800x - 120, we can express the total cost function as:
Total Cost = R(x) - P(x)
Substituting the given functions into the total cost equation:
Total Cost = (x^3 - 95x - 800x) - ((1/3)x^3 + 800x - 120)
Simplifying the equation:
Total Cost = x^3 - 95x - 800x - (1/3)x^3 - 800x + 120
Total Cost = (x^3 - (1/3)x^3) - 95x - 800x - 800x + 120
Total Cost = (2/3)x^3 - 1895x + 120
To find the level of production where the total cost is at a minimum, we need to find the critical points of the total cost function. The critical points occur where the derivative of the function is equal to zero.
Taking the first derivative of the total cost function:
d(Total Cost)/dx = d/dx((2/3)x^3 - 1895x + 120)
d(Total Cost)/dx = 2x^2 - 1895
Setting the derivative equal to zero:
2x^2 - 1895 = 0
Rearranging the equation:
2x^2 = 1895
Dividing both sides by 2:
x^2 = 1895/2
Taking the square root of both sides:
x = sqrt(1895/2)
x ≈ 31.04
Therefore, the level of production at which the total cost is minimum is approximately 31.04 units.