Writing Linear Functions

Question

A school system is reducing the amount of dumpster loads of trash removed each week. In week 5, there were 45 dumpster loads of waste removed. In week 10, there were 30 dumpster loads removed. Assume that the reduction in the amount of waste each week is linear. Write an equation in function form to show the amount of trash removed each week.

Answer 1

Hi Dootchine,

The plan is to find the slope of the line using the 2 pairs of the given coordinates, use point-slope form for the equation of a line, then solve that result for y.

Let (x₁, y₁) be (5, 45) and (x₂, y₂) be (10, 30).

m = (30 - 45)/(10 - 5); m = -15/5 = -3

y - 45 = -3(x - 5)

y - 45 = -3x + 15

y = -3 x + 60. The slope of the line is -3 and the y-intercept is (0, 60).

Regards,

Anita A.

Answer 2

Hi there! I'd love to help you out!

Linear functions can be written in slope-intercept form which is written as:

y = mx + b

where m is our rate of increase or decrease and b is our y-intercept. In this case, b will be the number of dumpsters removed before week 5.

Let's say x will be our number of weeks and y is the total number of dumpsters removed. Now, from week 5 to 10 there are 5 weeks. Between those weeks the number of dumpsters loads removed reduced from 45 to 30, making our total number reduced 15. Now we know this reduction happened in even intervals through those 5 weeks. So to find the number reduced per week, we can divide 15 by 3. This means our reduction is 3 dumpsters per week. We can plug this in for "m" in our equation but we have to make sure we put it in as a negative number since we are reducing the number of dumsters:

y = -3x + b

Now to figure out what our "b" is. Remember, "b" is the number of dumpsters we start with. We have to take into account that there were 15 MORE dumpsters in week 4 than in week five. So, we are going to add 15 to the number of dumpsters in week 5 (45+15). So our "b" will be 60:

y = -3x + 60

Now we can check our equation to make sure it works. Let's see if our week 10 figures work. Remember, "x" stands for the week number. So we will plug 10 in for x:

y = -3(10) + 60

y = -30 + 60

y = 30

Doing this shows that our equation worked! I hope this explanation was thorough and answered all of your questions. If you have any additional questions please don't hesitate to comment or reach out. Learn on! :)

Answer 3

In 5 weeks it dropped 33%. 33/5=6.66%/wk.