Question

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.

Answer 1

Answer:

Part A : The required Profit equation : $\text{Profit}= 60x-6x^3$

Part B : 0.303 million radios the profit became 18 million.

Part C : The graph is attached below.

Step-by-step explanation:

Given : The demand function for a type of portable radio is given by the model $d=85-6x^2$, 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

$\text{Revenue}= x\times (85-6x^2)$

$\text{Revenue}= 85x-6x^3$

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

Profit = Revenue - Costs

$\text{Profit}= 85x-6x^3-25x$

$P(x)= 60x-6x^3$

The required Profit equation : $P(x)= 60x-6x^3$

 

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.