Finding cost function from linear function word problem!!! Help!!

Question

if 500 people attend the even, te total costs for the event are estimated to be $7200. For each increase of 50 people, the total cost will increase by $400. Use the information to find a formula for the cost function c(q)?

 

table pairs include:

 

(500,7200)

(550,7600)

(600,8000)

(650,8400)

(700,8800)

(750,9200)

(800,9600)

(850,10000)

(900,10400)

Answer 1

This is a step-by-step method to solving this kind of problem. It's a little long but explains each step in detail if you want to understand how to solve this problem.

 

For this problem, imagine the x variable to be the number of people, and the y variable to be the cost. The main reason for this is because you generally want to know how much a certain number of people would cost, but not vice versa.

Generally, the variable you already have would be the x, since you want something where you know the number of people and can find out the cost from there.

 

Seeing as you're given a set point (which in this case would be that 500 people cost $7200) and a way to tell the rate of change (For each increase of 50 people, the cost will increase by $400), we can use what's called the point-slope form. This is ideal when you know the rate of change for a linear problem already, and have one point that fits it.

 

For the point-slope form, you need three things: The x and y coordinates of the given point, and the slope that indicates change.

 

Now you know that the given point, as stated earlier, is 500 people cost $7200. It isn't the 50 people cost $400 because that indicates for increasing by those amounts, and change is counted in the slope. Since we already established that people are x, and cost is y, the point can be written as (500,7200), where x of the point = 500 and y of the point = 7200.

 

Now the slope indicates the change in the linear equation, so anything that talks about change can be used to get the slope. In point-slope form, the slope is usually change in y divided by change in x. This problem says that for every 50 people increase (the change in x), there is an increase of $400 (the change in y). We can thus get that the slope is 400 / 50, or 8.

 

The point-slope form is always y = (slope) * ( x - (x of the point)) + (y of the point) or y- (y of the point) = (slope) * (x-(x of the point)). We'll be using the first format here, although you can use whichever you want since they're both the same essentially. Since we have all of this information, we can tell that y = 8(x-500) + 7200.

 

This problem asked for it to be written as c(q), so this means that they want the x to be written like q (the variable inside the parenteses) and y to be c(q) (the function of x). As such, we can reword the solution to this problem to be

 

c(q) = 8(q-500) + 7200

 

Note that c(q) has to be by itself to define a function, so if you ended up with c(q) - 7200 = 8 (x - 500) by using the other point-slope format, you would have to add 7200 to keep c(q) by itself on one side. This is usually a mathematical convention.

 

Since you were given other points, you could use those to double-check that this equation is correct. To do this, substitute the first value of the point for q and the second value in the point for c(q), and then check that the equation is true. This is important to make sure that you did the problem correctly. If you didn't, try the problem again. In this case, everything checks out.

 

And there you have it. A detailed solution to the problem. If you have any questions on any of the steps, please ask me. I want you to understand in detail how to solve this problem.