Graph Proportional Relationships Part 3- Slope and Application
After reviewing the process of slope intercept form, we can finally apply this to real life scenarios. The first example is the speed of a sloth, which is probably more interesting than all of these math tutorials. However, it is important to understand the units involved in real cases.
The first thing one will notice is that sloths are incredibly slow, and second the x-axis and y-axis actually have units of measurement to describe just how slow the sloth really is. We can determine the speed by using the slope intercept form y=mx+b. Since the y-intercept is actually the midpoint (0,0), this will be much easier because the slope is simply y/x: In this case, we will select the point (1.5,9). This would be read as 9 feet per one and half minute. Or we can use (0.5,3) and say the sloth equally moves at 3 feet per half of a minute. Therefore, the slope is 6 and y=6x.
This graph shows the opposite relationship compared to example 1. A table is provided with the graph to show how has time (x-axis) becomes larger, the height (y-axis) becomes smaller. But the table can be deceiving because even though both the x and y coordinates are positive, we know that the line is not. How do we show this? Use the slope formula m=(y2-y1)/(x2-x1): m=(8-4)/(2-6) =(4)/(-4) =-1. When the slope equals -1, we can write the slope intercept form as y=-x. We can say that this candle loses an inch of height every hour.
This example is similar to example 1, but shows a business model that is used more often than sloth speed. This graph has a slope of 10 because we can see that it cost this company 20 dollars per 2 hours for electricity or 10 dollars per hour.
Bright Lights, a company, has determined they need to cut back on their electricity in order to make the budget. Currently, they spend 1,000 dollars per work day (8 hours). The company is open 300 days per year.
1. What is the slope of the graph in terms of cost per hour?
2. How many hours will it take for the company to spend 97,000 dollars?
3. For some reason, the cost per day changed from 125 per day to 140 per day on day 60 in the year. If the cost continues at this rate, how much will be spent by day 111?
1. 1000/8hrs =125/hr
2. Since the midpoint is (0,0) we can assume y=mx and therefore 97,000=125x –> x=776hrs
3. First, we need understand that since the slope has changed we need to use m=(y2-y1)/(x2-x1). Next, convert the time: since we started on day 60 this is actually 480 hours into the year and day 111 will be 888 hours into the year, giving us our x-coordinates. Remember: time is on the x-axis! Next, we need to know how much was already spent by day 60 to give us our y-coordinate: (125)(480)= 60,000. So: 140=(y2-60,000)/(888-480) –> y2=140(888-480)+60000 =$117,120
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