# Revenue and Cost and Profit Problem

• MHB
• needOfHelpCMath
In summary: Good job!In summary, to find the maximum profit given the revenue and cost functions of R(x) = 71x - 2x^2 and C(x) = 23x + 108, we can use the profit function P(x) = -2x^2 + 48x - 108. By finding the axis of symmetry or using vertex form, we can determine that the maximum profit occurs at x = 12, with a profit of \$180.
needOfHelpCMath
Given the following revenue and cost functions, find the maximum profit.

R(x) = 71x - 2x^2; C(x) = 23x + 108

P(x) = Revenue - Cost

P(x) = 71x-2x^2 - (23x+108)

P(x) = -2x^2 + 48x - 108 *I am stuck right here. Using quadratic formula but cannot seem to solve it what I am doing
wrong or have I miss any step?

$$\displaystyle P(x)=-2x^2+48x-108$$

Now, what I would do is find the axis of symmetry (since the vertex lies on this axis)...recall that for the general quadratic:

$$\displaystyle y=ax^2+bx+c$$

The axis of symmetry is the line:

$$\displaystyle x=-\frac{b}{2a}$$

This is the arithmetic mean of the two roots. What is the axis of symmetry for your profit function?

Another approach would be to express your profit function in vertex form:

$$\displaystyle P(x)=-a(x-h)^2+k$$

where:

$$\displaystyle P_{\max}=k$$

The axis of symmetry is:

$$\displaystyle x=-\frac{48}{2(-2)}=12$$

We see that the profit function opens downward, and so the vertex is the global maximum, given by:

$$\displaystyle P_{\max}=P(12)=-2(12)^2+48(12)-108=12(-24+48-9)=12\cdot15=180$$

If we use the vertex form approach, we may write:

$$\displaystyle P(x)=-2(x^2-24x)-108=-2(x^2-24x+144)-108+2\cdot144=-2(x-12)^2+180$$

Hence:

$$\displaystyle P_{\max}=180$$

Completing the square: -2x^2 + 48x - 108= -2(x^2- 24x)- 108.
24/2= 12 and 12^2= 144. Add and subtract 144.

-2x^2+ 48x- 108= -2(x^2- 24x+ 144- 144)- 108= -2(x^2- 24x+ 144)- 288- 108

= -2(x- 12)^2- 396.

That is a parabola opening downward. For every x, it is -396 minus something. Its maximum value, -396, occurs when x= 12.

HallsofIvy said:
Completing the square: -2x^2 + 48x - 108= -2(x^2- 24x)- 108.
24/2= 12 and 12^2= 144. Add and subtract 144.

-2x^2+ 48x- 108= -2(x^2- 24x+ 144- 144)- 108= -2(x^2- 24x+ 144)- 288- 108

= -2(x- 12)^2- 396.

That is a parabola opening downward. For every x, it is -396 minus something. Its maximum value, -396, occurs when x= 12.

The part highlighted above in red has the wrong sign. You are subtracting 288 twice.

Yes, thank you for catching that.

So the parabola is $$\displaystyle -2(x- 12)^2+ 288- 108= -2(x- 12)^2+ 180$$.

That is a parabola, opening downward with vertex (12, 180). The maximum is 180 and occurs at x= 12.

Okay, that's a lot better- you have a positive​ profit!

Last edited by a moderator:

## 1. What is the difference between revenue, cost, and profit?

Revenue refers to the total income generated by a business through sales or services. Cost refers to the expenses incurred by a business in producing goods or services. Profit is the difference between revenue and cost, and represents the financial gain or loss of a business.

## 2. How do you calculate revenue, cost, and profit?

Revenue is calculated by multiplying the quantity of goods or services sold by the price per unit. Cost is calculated by adding up all the expenses involved in producing goods or services. Profit is calculated by subtracting the total cost from the total revenue.

## 3. What is the importance of understanding revenue, cost, and profit in business?

Understanding revenue, cost, and profit is crucial for businesses to make informed decisions about pricing, budgeting, and overall financial health. It allows businesses to identify areas for improvement and make strategic plans to increase profit and reduce costs.

## 4. How can a business increase its profit?

A business can increase its profit by either increasing revenue or reducing costs. This can be achieved through strategies such as increasing prices, expanding into new markets, improving efficiency, and cutting unnecessary expenses.

## 5. What are some common challenges businesses face in managing revenue, cost, and profit?

Some common challenges include accurately forecasting demand and sales, managing production costs and inventory, balancing pricing to remain competitive, and adapting to changing market conditions. It is also important for businesses to regularly review and analyze their financial data to identify potential issues and make adjustments as needed.

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