Question 1168554: (a) A company has determined that its profit for a product can be described by a linear function. The profit from the production and sale of 150 units is $455, and the profit from 250 units is $895.
(i) What is the average rate of change of the profit for this product when between 150 and 250 units are sold?
(ii) Write the equation of the profit function for this product.
(iii) How many units give break-even for this product?
(b) You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q2where q
is the number of lightweight compasses produced in millions. It costs the company $15 to make a lightweight compass.
(i) Write an equation giving profit as a function of the number of lightweight compasses produced.
(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the slope intercept form of a linear equation is y = mx + b
the slope is the change in the value of y divided by the correspond change in the value of x.
you have:
x = 150, y = 455
x = 250, y = 895
the change in the value of y is 440.
the corresponding change in the value of x is 100.
slope is 440 / 100 = 4.4
equation becomes y = 4.4 * x + b
solve for b by replacing x and y with their corresponding value from one of the points.
when x is 150 and y is 455 and m is 4.4, the equation becomes 455 = 4.4 * 150 + b
solve for b to get b = -205
your equation becomes y = 4.4 * x - 205
when x is 150, y is 4.4 * 150 - 205 = 455.
when x is 250, y is 4.4 * 250 - 205 = 895.
break even is when the profit is 0.
anything above 0 and you make a profit.
anything below 0 and you take a loss.
set y = 0 and your equation becomes 0 = 4.4 * x - 205
add 205 to both sides of the equation to get 205 = 4.4 * x.
solve for x to get x = 205 / 4.4 = 46.59090909.
then x = 46.59090909, the equation becomes y = 4.4 * that - 205 = 0
profit is 0.
when x < that, you lose money.
when x > that, you make money.
answers to your questions from your first problem are:
(i) What is the average rate of change of the profit for this product when between 150 and 250 units are sold?
the average rate of change is 4.4 dollars profit per unit produced and sold.
(ii) Write the equation of the profit function for this product.
the profit equation is y = 4.4 * x - 205
based on this equation:
when 0 units are produced and sold, you lose 205 dollars.
when 150 units are produced and sold, you gain 455 dollars.
when 250 units are produced and sold, you gain 895 dollars.
(iii) How many units give break-even for this product?
breakeven is when profit = 0.
you neither gain money or lose any money.
this occurs when y = 0
equation becomes 0 = 4.4 * x - 205
add 205 to both sides of the equation to get 205 = 4.4 * x.
solve for x to get x = 205/4.4 = 46.590909...
if you produce and sell less than that number of units, you lose money.
if you produce and sell more than that number of unist, you gain money.
i haven't answered your second problem because i'm not sure i understand it.
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