When a company produces and sells x thousand units per week, its total weekly profit is p thousand dollars, where upper p equals startfraction 800 x over 100 plus x squared endfraction . the production level at t weeks from the present is x equals 4 plus 2 t. find the marginal profit, startfraction dp over dx endfraction and the time rate of change of profit, startfraction dp over dt endfraction . how fast (with respect of time) are profits changing when tequals8?
(a) Profit function P(x) = 0.02x^2+60x-80
(b) Average profit P(x)/x = P/x = 0.02x+60-80/x
Marginal profit dP/dx = 0.04x+60
Cost function: C(x) = -0.02x^2+40x+80
Price function: p(x) = 100
(a) The profit function P(x) = x*p(x)-C(x) can be expressed as:
(b)Average profit function: P(x)/x
Marginal profit function: dP/dx
(d) See Explanation
Solving (a) Profit function; P(x)
Collect like terms
Solving (b): Average profit function and Marginal profit function
Break down the fraction
Solving (c): Average profit and Marginal profit if x = a
Substitute 500 for x
Solving (d): Interpret the values in (c)
They make a profit of 59.8 for the first 500 items
From the 501st item, the profit is 70
The marginal profit is .
The profit function p (x) is the difference between the revenue and cost function.
The profit function is given as follows:
Determine the marginal profit function as follows:
Thus, the marginal profit is .