Question

The unit cost, in dollars, to produce tubs of ice cream is $18 and the fixed cost is $11610. The price-demand function, in dollars per tub, is p(x)=374-2x.
Find the cost C(x).
Find the revenue function R(x).
Find the profit function P(x).
At what quantity is the smallest break-even point?

Answer 1

Answer:

Step-by-step explanation:

Let x represent the number of tubs of ice cream that was produced.

The unit cost, in dollars, to produce tubs of ice cream is $18 and the fixed cost is $11610. This means that the total cost of producing x tubs of ice cream would be

C(x) = 18x + 11610

The price-demand function, in dollars per tub, is p(x)=374-2x.

The revenue function is product of the output by the price function

R(x) = x × p(x) = xp(x)

R(x) = x(374 - 2x) = 374x - 2x²

The profit function P(x) = R(x) - C(x)

Therefore,

P(x) = 374x - 2x² - (18x + 11610)

P(x) = 374x - 2x² - 18x - 11610

P(x) = - 2x² + 374x - 18x - 11610

P(x) = - 2x² + 356x - 11610

At the break even point,

Revenue = total cost.

Therefore,

374x - 2x² = 18x + 11610

2x² + 18x - 374x + 11610 = 0

2x² - 356x + 11610 = 0

Dividing through by 2, it becomes

x² - 178x + 5805 = 0

Applying the general formula for quadratic equations,

x = [- b ±√(b² - 4ac)]/2a

x = [- - 178 ±√(-178² - 4 × 1 × 5805)]/2 × 1

x = [178 ±√(31684 - 23220)]/2

x = [178 ±92]/2

x = (178 + 92)/2 or (178 - 92)/2

x = 135 or x = 43

Therefore, the quantity for the smallest break-even point is 43.

Answer 2

Function assigns the value of elements of one set to the other specific element of another set. The quantity is the smallest break-even point is 43.

What is a Function?

A function assigns the value of each element of one set to the other specific element of another set.

Let the number of tubs of ice cream that was manufactured be x.

A.)

As it is given that the cost of making a tub of ice cream is $18, and the fixed cost is $11,610, the total cost of producing the ice cream may be stated as follows:

$C(x) = 18x + 11610$

B.)

As it is given the price-demand function, in dollars per tub is p(x)=374-2x, therefore, the revenue function is the product of the output and the price function, which can be given as:

$R(x) = x \times p(x) = x\ p(x)$

Substitute the function and the values you will get,

$R(x) = x \times( 374-2x)= 374x-2x^2$

C.)

As described that the difference between the revenue generated and the total cost is the profit, therefore, the profit function can be written as,

$P(x) = R(x)-C(x)\\\\P(x) = (374x-2x^2) - (18x+11610)\\\\P(x)= -2x^2+356x-11610$

D.) In order to know the break-even point, you must know that the revenue of the organization is equal to the total cost of the organization. therefore,

$R(x) = C(x)\\\\374 - 2x^2 = 18x +11610\\\\2x^2 - 356x + 11610 = 0\\\\\text{Dividing the equation by 2}\\\\x^2 -178x +5805=0$

Further, finding the roots of the quadratic equation,

x = 135, 43.

Therefore, the quantity is the smallest break-even point is 43.

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