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Problem 81 Hard Difficulty

Three months after cleanup began at a dump site, 800 cubic yards of toxic waste had yet to be removed. Two months later, that number had been lowered to 720 cubic yards.
a. Find an equation that describes the linear relationship between the length of time $m$ (in months) the cleanup crew has been working and the number of cubic yards $y$ of toxic waste remaining. Write the equation in slope-intercept form.
b. Use your answer to part (a) to predict the number of cubic yards of waste that will still be on the site one year after the cleanup project began.

Answer

a. $y=-40 m+920$
b. $440 \mathrm{yd}^{3}$

Related Courses

Algebra

Elementary and Intermediate Algebra

Chapter 3

Graphing Linear Equations and Inequalities in Two Variables; Functions

Section 6

Point–Slope Form

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Jocelyn P.

October 29, 2020

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Video Transcript

and this problem, we're told three months after clean that began at a dump site. 100 cubic yards, a toxic waste had yet to be removed. So if you know this at the top, I wrote three months is associated with 100 yards. Cute. Then we're told two months later that number had been lowered to 720 cubic yards. So because it's two months later, that means we're five months total into this project's why of five months, uh, connected to 720 cubic yards. So now we want to write an equation to describe the linea of relationship between the length of Time M, which is months the cleanup crew has been working and the number of pubic yards why of task of remaining waste. So what we want to do is right. A question to represent the situation. Well, the different information we have essentially leads to our order pairs. So we have the order pairs 3 800 and the order pair five and 7 20 we want to write a linear regression ship. So the first thing we need to do is find the slope will remember to find slow be used to formula M equals Y sub two minus y sub one over X of two minus except one. So I'm gonna go back to my order pears and label. I'm gonna start with the 1st 1 I'm gonna label three except one and 100 Weiss of one. And then the second would a pair. I'm gonna label five except two and 7 20 Weiss of Do. And now we can substitute those values into our equation. We're gonna have m equals Y sub two, which is 7 20 minus weiss of one, which is 100 divided by, except to which is five minus what except one, which is three. And now we simplify while 7 20 minus 800 is negativity and five ministries to and negative 80 divided by two is negative. 40. So now we have to slope off our line. Well, we're not given the why interest up. So that means we have to use point slope form, which remember, is why minus weiss of one equals m times the quantity of X minus except one where M is our slope and ex of one. And why someone Our points on our line won't remember. We already labeled our first ordered pair except one and Weiss of one so we can use those values. So let's substitute these values into our equation. Well, that means we're gonna have why, minus weiss of one, which is 100 equals M, which we found to be negative 40 times the quantity of X minus x of one, which is three. And now, to put this in slope intercept form, we're gonna first distribute. We're gonna multiply both terms and that prophecy by negative 40. So we're gonna have y minus 100 equals, Will. Negative. 40 times X is negative for the X and negative. 40 times negative three is positive. 1 20 Now to get why by itself, we're gonna add 100 to both sides. We're gonna add it to it's like term of 1 20 So negative 80 plus 80 cancel. And we're left with y equals negative for the X Plus 1 20 plus 100 is positive. 9 20 Now, the only everything I'm going to do is I'm gonna go back and notice the variables they asked us to use. They wanted to use us to use em for months. Well, our months, in this case of representing our ex, so I'm gonna really substitute m in for X and for a while, values we are still using. Why? So our final equation for part a is y equals negative for the M plus 9 20 So this is all the work for partner now for part B. I'm gonna go to a new page here, but I'm gonna rewrite our equation that we just found. So we have y equals negative for the M plus 9 20 Now, Part B is asking us to find to use this equation in part A to predict the number of cubic yards of waste that will still be on the site after one year. Well, we have to keep in mind the units for our problem. Member M stood for months. Well, because they're asking us vote after one year. That would mean how much is still left after 12 months. So we simply need to substitute 12 in for M. So we're gonna have y equals negative 40 in place of M. We're gonna substitute 12 in plus 9 20 and now we just simplify. Well, negative. 40 times 12 is negative for 80 plus 9 20 and negative for a D plus 9 20 is 440 now. If we went back to our units, I'll go back to the first page. Remember, our units for our toxic waste was in terms of yards, Cube. So our final answer for Part B is 440 cubic yards.

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