Question

A company's profit is linearly related to the number of items the company sells. Profit is a function of the number of items sold. If the company sells 2000 items, its profit is $24,000. If the company sells 3000 items, its profit is $30,000.
a) Find the slope and explain what it means in words.
b) find the vertical (profit) intercept and explain what it mean in words.
c) Find f(4000) and explain what it means in words.

Answer 1

Hey kweb! Let's do dis.

With this problem, it's important to remember the formula for finding the slope of a line, which is:

y2 - y1
_______
x2 - x1

This means that, if you have two ordered pairs and you want to find the slope of a line using these values, you'd take the 'y' value from the first ordered pair and subtract it from the 'y' value of the second ordered pair. For x, the operation is the same.

So if I have these ordered pairs: (1,2) and, mm, let's say (4, 5), and I want to find the slope, then I would plug these numbers into the slope formula.

(1 , 2) (4 , 5)
x1 y1 x2 y2

So:

5 - 2 3
---- = --- = 1
4 - 1 3


"Nick. You're not answering the question."

Actually, you can apply this formula directly to the problem! Watch this!

"If the company sells 2000 items, its profit is $24,000." Immediately, I see an ordered pair. (2000, 24,000). 2000 = x1, and 24,000 = y1

"If the company sells 3000 items, its profit is $30,000." There's another pair!
(3000, 30,000). x2 = 3000, y2 = 30,000.

Because this is said to be a LINEAR equation, we can rest easy and know that our slope formula will come to the rescue.

30,000 - 24,000 6,000
---------------- = -------- = 6
3000 - 2000 1000

So now we have our slope, and we can begin buildin' our linear equation!

y = mx + b ---> Format for linear equation.
y = 6x + b ---> Plugged in slope.

Now, to finish off our dreaded enemy, we find the value of 'b' and seize our equation!

Take one of your ordered pairs - (2,000, 24,000) or (3,000, 30,000) - and plug the x and y values of the pair into our equation. Imma go with (2,000, 24,000) because these numbers make me feel warm and fuzzy inside.

(24,000) = 6(2,000) + b
b = 12,000.

Now we can plug 'b' into our equation!

y = 6x +12,000 is our final result. Don't believe me? Let's test it by going back to the problem!

"If the company sells 2000 items, its profit is $24,000." Since y is the value in our equation that results from operations being done to x, it is 24,000. x will be the number of items sold, or in this case 2,000. To test our equation, ONLY plug in y and see what happens.

24,000 = 6x + 12,000.

Now, according to the problem, x should be 2000. If this is our output for x, we'll know we have the right equation.

12,000 = 6x
x = 2,000

Ha-ha.

So! We've solved for a):

Slope is 6, and we have an explanation for how we found it.

We've solved for b):

The VERTICAL intercept must occur when x = 0, or the line is never actually vertical. So, making x = 0, our equation becomes: y = 6(0) + 12,000. y = 12,000. That's our intercept.

And, we have c):
Just remember that f(x) is the original form, and this is a cinch! f(x) --> f(4,000), so x, or the number of items sold, is 4,000. They want us to find the profit with this information. Easy! Let's get our equation, y = 6x + 12,000.
y = 6(4,000) + 12,000
y, or the company's profit, is: $36,000.


Rock on!

P.S. If, for some reason, you are confused or you see and error in my reasoning, please
feel free to come to me. I don't like leaving anything on the table!