Linear Algebra word problem

Question

A local organization is willing to rent out its hall space for you to use to throw a party. A representative tells you that a party of 50 people would cost $1000, and that a party of 100people would cost $1750. We will assume that the cost to rent the hall, y, is linear function of the number of people who will attend the party, x

a. use the information provided to create 2 ordered pairs for this relationship.

b. use the ordered pairs from (a) to find the cost as a function of the number of people in slope-intercept (y=mx+b) form.

c. What is the slope of your linear function from (b)? What does it mean in the real-world context of the problem? (2 points slope, 2 points explanation)

d. What is the y-intercept of your linear function from(b)? What do you think it means in the real-world context of the problem?

Bonus

Graph y=[x-5], where y=[x] s the greatest integer function (i.e.[x] is the greatest integer less than or equal to x).

Answer 1

You are told "x" is the number of people who will attend the party ; "y" is the cost to rent the hall and that there is a linear relationship (equation for a line).

 

Ordered pairs (x,y) would be:

(50, 1000) and (100, 1750)

 

Use these to determine the equation for a line: y = mx + b

starting with the slope, m

 

m = (y₂-y₁)/(x₂-x₁)

 

m = (1750-1000)/(100-50)

m = 750/50

m = 15

 

The slope is a constant in your equation for a line now. Slopes are rates. In this problem it is the cost per person to attend the party. (15$ per person to attend)

 

input one of the ordered pairs to find the y-intercept, b.

 

y = 15x + b

using (50, 1000)

1000 = 15(5) + b

1000 = 750 + b

1000 - 750 = b

250 = b

 

the y-intercept, b is now a constant in the linear equation. final equation:

 

y = 15x + 250

 

Bonus: Greatest integer functions are step functions, they look like stairs when graphed. Y is the greatest integer less than or equal to x. You will have a solid point on the left where x=y and an open circle point where x approaches the next integer. Looks like this:

 

http://www.mathwarehouse.com/step-function/images/picture-greatest-integer-function-graph.gif