Question

The cost of a business is the amount of money spent to start the business and keep it in operation. The revenue is the amouunt of
money the business earns from the sale of goods or services.
An online business called Muffins A-plenty started on January 1 of a recent year. The lines in the graph represent the cost function and
the revenue function of the first six months that Muffins A-plenty has been in operation.
Part A
Write equations to represent the cost function and the revenue function for this business,
Parts
The profit of a business equals revenue minus cost. First write an equation that represents the profit function for Muffins replenty using
the functions for cost and revenue, Then, predict the profit after 10 months of operation, found your answer to the nearest dollar. Provide
valid mathematical reasoning and calculations to support your answer,

Answer 1

The profit function of the business is the difference between the revenue and the cost functions

• The equations of the cost and the revenue functions are: $y = \frac 43x$  and $y = 0.5x + 3.5$
• The profit function is $y = \frac 56x - 3.5$
• The profit after 10 months of operation is 29/6

Part A: The equations to represent the functions

For the revenue function, we have the following points

(x,y) = (0,0) and (6,8.0)

Since the line passes through the origin, the equation of the line would be:

$y = \frac{y_1}{x_1} * x$

So, we have:

$y = \frac{8.0}{6} * x$

$y = \frac 43x$  

For the cost function, we have the following points

(x,y) = (0,3.5) and (9,36) for x > 5

The equation of the line would be:

$y = \frac{y_1-y_2}{x_1-x_2} * (x - x_1) + y_1$

So, we have:

$y = \frac{6.5-3.5}{6-0} * (x - 0) + 3.5$

$y = 0.5x + 3.5$

Hence, the equations of the cost and the revenue functions are: $y = 0.5x + 3.5$ and $y = \frac 43x$

Part B: The profit function

This is calculated as:

Profit = Revenue - Cost

So, we have:

$y = \frac 43x - 0.5x - 3.5$

Evaluate the difference

$y = \frac 56x - 3.5$

Part C: The profit after 10 months of operation

Substitute 10 for x in $y = \frac 56x - 3.5$

$y = \frac 56 * 10 - 3.5$

$y = \frac {50}6 - 3.5$

Evaluate the difference

$y = \frac {50-21}6$

$y = \frac {29}6$

Hence, the profit after 10 months of operation is 29/6

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