Problem

Find the equations of the lines that describe th…

06:39

Problem 78 Hard Difficulty

An automated production line uses distilled water at a rate of 300 gallons every 2 hours to make shampoo. After the line had run for 7 hours, planners noted that $2,500$ gallons of distilled water remained in the storage tank. Find a linear equation relating the time $t$ in hours since the production line began and the number $g$ of gallons of distilled water in the storage tank. Write the equation in slope-intercept form.

Answer

$\mathrm{g}=(-150) \mathrm{t}+3550$

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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Video Transcript

in this problem, we're told at a production line uses distilled water at a rate of 300 gallons every two hours to make shampoo. So would have know that here is that we're told the re at which the gallons is being used Well, because we're using up two gallons. That means the rate at which we have the number of gallons is getting decreased by 300 and it's getting decreased by 300 gallons every two hours. Well, because this is representing the read of change that's going to tell us that are slow and how it released to the number of gallons that are lost over time. Well, like I said, we're losing 300 gallons every two hours. Well, that can be reduced to negative 150 gallons per hour. Well, then they go on, tell us that after the line had ran for seven hours, they know this that 2500 gallons of distilled water remained in the storage. So what we want to figure out is a linear equation relating the time. In hours since the production line began, a number of gallons, which is G distilled water in the storage tank Well taken, though when we did our slope. No, this in the numerator We had our units for gallons. Well, remember, when we find slow the changing why is our numerator And so therefore are gallons represents our wide values And then denominator of our slope is represented by the units for hours which typically when we find Slope exes in the denominator Therefore, ours represents our X values. So no, we were X and y represent. We can now come up with ordered pair to represent that after seven hours there was 2500 gallons that remains. Remember, our X value represents our hours, which is seven on our why values represent the gallons which is 2500. So now we're being asked to write in the linear equation. Well, because we have a slope and the point we can use point slope form which remember is why minus y sub one equals m times the quantity of ex mind sex of one where Mr Slope An excess of one and why someone represent the point on our line, which we have. So in our particularly pair seven is going to be our ex of one value and 2500 will be our wise of one value. Now, the other thing to know is they want us to use tea for X value and G for a Y value. So now we're gonna substitute all of these values into our formula. It's like I said, our why value is going to represented by G minus Weiss of one, which is 2500 equals, um, which was our slope, which is negative 150 times acts. But remember, we're using T for time minus x of one, which is seven. And now we need to simply put this equation and slope intercept form. Well, to do that, we first need to distribute negative 1 52 both terms in the prophecies. So we'll have g minus 2500 equals will negative 1 50 times t is negative. 1 50 t and negative. 1 50 times negative. Seven. It's positive. 10 50. Now, I'm gonna go over here to the right to write my equation. Are they getting because I've kind of ran out of room So you have G minus 20. 500 equals negative. 1 50 50. There we go T plus 10 50. So to get G by itself, we need to move that negative 2500 to the right hand side. Well, to do that, we simply add 2500 to both sides of the equation, and we're gonna add it to it's like term of 10. 50. So the negative 2500 and positive 2500 cancel. So we're left with G equals negative. 1 52 and 10 50 plus 25 50 is equal to positive. 35 50. And now we've written over in Europe equation relating the time to the number of gallons that remain.

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