The cost and revenue functions for producing and selling x units of a product are given. Cost and revenue are expressed in dollars.
Upper C left parenthesis x right parenthesis equals 36 comma 048 plus 11 x
Upper R left parenthesis x right parenthesis equals 35 x
a.
Find the number of units that must be produced and sold to break even. At this level, what is the dollar amount coming in and going out?
b.
Write the profit function from producing and selling x units of the product.
a. To find the number of units that must be produced and sold to break even, we need to equate the cost function (C(x)) with the revenue function (R(x)).
The cost function is given as: C(x) = 36,048 + 11x
The revenue function is given as: R(x) = 35x
At the break-even point, the cost and revenue will be equal. So, we can set up the equation:
C(x) = R(x)
Substituting the given functions, we have:
36,048 + 11x = 35x
Now, we can solve this equation for x:
36,048 = 35x - 11x
36,048 = 24x
x = 36,048 / 24
x = 1,502 units (approximately)
Therefore, the number of units that must be produced and sold to break even is 1,502 units.
To find the dollar amount coming in and going out at the break-even point, we need to substitute this value of x into the cost and revenue functions.
Cost at break-even: C(1,502) = 36,048 + 11(1,502)
Cost at break-even = $53,090
Revenue at break-even: R(1,502) = 35(1,502)
Revenue at break-even = $52,570
Therefore, at the break-even point, the dollar amount coming in (revenue) is $52,570, and the dollar amount going out (cost) is $53,090.
b. The profit function can be calculated by subtracting the cost function from the revenue function:
Profit function (P(x)) = R(x) - C(x)
Substituting the given cost function and revenue function, we have:
P(x) = 35x - (36,048 + 11x)
P(x) = 35x - 36,048 - 11x
P(x) = 24x - 36,048
Therefore, the profit function from producing and selling x units of the product is P(x) = 24x - 36,048.