The demand function for a type of portable radio is given by the model d=85−6x2, where d is measured in dollars and x is measured in millions of units. The production cost is $25.00 per radio. Given that profit = revenue - costs answer the following parts. (Hint: Revenue is demand times number sold. Costs are productions costs per number sold.)
Part A: Write an equation giving profit as a function of x million radios sold. Part B: The company currently produces 3 million radios and makes a profit of $18,000,000, but would like to scale back production. What lesser number of radios could the company produce to yield the same profit?
Part C: Give a graph for the profit and label the parts of the graph.

Given : The demand function for a type of portable radio is given by the model [tex]d=85-6x^2[/tex], where d is measured in dollars and x is measured in millions of units. The production cost is $25.00 per radio.

Note: Given that profit = revenue - costs

Revenue is demand times number sold.

Costs are productions costs per number sold.

To find :

Part A: Write an equation giving profit as a function of x million radios sold.

Solution : Let x be the number of radios.

Revenue is demand times number sold

[tex]\text{Revenue}= x\times (85-6x^2)[/tex]

[tex]\text{Revenue}= 85x-6x^3[/tex]

Let the cost of x number of radios is 25x.

Profit = Revenue - Costs

[tex]\text{Profit}= 85x-6x^3-25x[/tex]

[tex]P(x)= 60x-6x^3[/tex]

The required Profit equation : [tex]P(x)= 60x-6x^3[/tex]

Part B : The company currently produces 3 million radios and makes a profit of $18,000,000, but would like to scale back production. What lesser number of radios could the company produce to yield the same profit?

Solution : It is shown graphically which is attached in part c

When we scale back production we get that,

When we sold 0.303 million radios the profit became 18 million.

Part C : Give a graph for the profit and label the parts of the graph.

The graph is attached below which shows the profitable radios points.

Lanuel Lanuel · Answer 2 · 2022-03-28T15:46:23+02:00

An equation giving profit as a function of x million radios sold is [tex]Profit = 60x-6x^3[/tex]

Given the following data:

Production cost = $25.00 per radio.

[tex]d=85-6x^2[/tex]

Where:

d is measured in dollars.

x is measured in millions of units.

How to write a profit equation.

Since revenue is calculated as demand times the number of units sold, we have:

[tex]Revenue =(85-6x^2)\times x\\\\Revenue =85x-6x^3[/tex]

Also, profit is calculated by subtracting costs from revenue. Thus, we have:

[tex]Profit = revenue -costs\\\\Profit = 85x-6x^3-25x\\\\Profit = 60x-6x^3[/tex]

To calculate the lesser number of radios.

Profit = $18,000,000.

Quantity = 3 million radios.

From the polynomial equation for the company's profit, we have:

[tex]Profit = 60x-6x^3\\\\18=60x-6x^3\\\\60x-6x^3-18=0[/tex]

Note: x = 3 is one of the solution to the above polynomial equation. Thus, (x - 3) is a factor.

Dividing it, we have:

[tex]-6(x-3)(x^2+3x-1)=0[/tex]

Solving the quadratic equation by using the quadratic formula, we have:

x = 0.303.

Therefore, the company can make the same profit by selling 3,030,000 units as shown by the graph in the image attached below.

Read more on quadratic equation here: brainly.com/question/1214333