Problem

In the report "Expenditures on Children by Famili…

05:05

Problem 83 Hard Difficulty

There is a linear relationship between the length of the protective pad that wraps around a trampoline and the radius of the trampoline. Use the data in the table to find an equation that gives the length $l$ of pad needed for any trampoline with radius $r .$ Write the equation in slope-intercept form. Use units of feet for both $l$ and $r$
$$
\begin{array}{|c|c|}
\hline \text { Radius } & {\text { Pad length }} \\
\hline 3 \mathrm{ft} & {19 \mathrm{ft}} \\
{7 \mathrm{ft}} & {44 \mathrm{ft}} \\
\hline
\end{array}
$$

Answer

$$l=\frac{25}{4} r+\frac{1}{4}$$

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

this problem. We're told a linear relationship exists between the length of the protective pad that wraps around the trampoline and the radius of the trampling which were shown in our table. So we're told that a trampling that has a radius of three feet would have a path length of 19 feet and its radius was seven feet. The paddling would be 44 feet. So we want to write an equation that gives the length l of the pad needed for any trampling with radius are so because we're given our table, we now can come up with are ordered pairs. So the order pairs that we've fallen this line. If we were the graft, this relationship would be 3 19 and 7 44 So because we're being asked to write an equation, first thing we need is our slow. Well, remember to find the slow we do m equals wife's of two minus Weiss of one over, except to minus X of one. So I'm gonna go back and label my order pairs. So for the first or two pair, I'm gonna label three except one in 19 Weiss of one. And for the second order, pair seven will be my except two and 44 will be my wife. So now I'm going to substitute these values in for my formula. So I'm gonna have m equals Weiss of two, which is 44 minus weiss of one, which is 19 all over, except to which is seven minus x of one, which is three. And now we're going to simplify. Well, 44 minus 19 is 25 and seven. Minus three is four. So we've found the slope is 25 force. Well, now we're not giving a why. Intercept. So we're gonna have to use point slope form, which remember, is why minus y sub one equals m times the quantity of x minus except one where to use this formula, we need to slow, which we just found. And we need our values for X of one. And why so one, which is just a point on the line. But remember, we already labeled that first door repair except one. And why someone so we can use those. So we're gonna substitute these values into our equation. So we're gonna have why, minus why someone which is 19 equals M, which we found to be 25 force times the quantity of X minus x of one, which is three. And now we're gonna simplify. We have to get wide by itself to get this in slope intercept form. So what we're gonna do first is we're going to distribute the 25 4th toe both terms in that prem facie. So we're gonna have y minus 19 equals wont 25 4th times X is just 25 for sex and 25 force times. Negative three is negative. 75 force. So now, to get why by itself, we have to move that 19 negative 19 to the other side. Well, to do that, we're gonna add 19 and we're gonna add it to it's like term, which was negative. 75/4. Well, negative. 19 plus 19 is there? Oh, so those cancel. So we have y equals Walt. The 25 force will come down, and now we need to combine negative 75 force in 19. So I'm gonna come over right here in this middle section here. I'm gonna make a little space for it. It's like I said, we had negative 75 force plus 19 which I'm gonna turn 19 in the fraction by putting a one under it. Well, we're adding fractions. So in the common denominator, which in this case is for so I need to multiply the numerator and denominator of that second fraction by four. So we're gonna have negative 75/4 plus Ah, 76/4. Well, negative. 75/4. Plus 76/4. It's just 1/4. So now we'll have plus 1/4. So were found. Our Lanier relationship is why equals 25 Force X plus 1/4.

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