The Simmons family and the Wright family each used their sprinklers last summer. The water output rate for the Simmons family's
sprinkler was 20l per hour. The water output rate for the Wright family's sprinkler was 35l per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output 1475l of . How long was each sprinkler used?
2 answers:
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The rate of change, or slope, is the change in y over change in x. To find the slope, pick two points. The slope will be the change in y-coordinates over the change in x-coordinates.
As an equation, this is
To find the slope for function 1, pick any two points (x,y) in the table.
Let's choose (-1, 3) and (-2, 5).
The slope is
.
Let's choose (-1, 0) and (0, 4) for the two points for function 2.
The slope is
.
You want to compare the magnitude (without negatives) of the slope when comparing the greater rate of change/slope. In other words, you take the absolute value of each slope.
The magnitude of function 1's slope is 2.
The magnitude of function 2's slope is 4.
As an equation, this is
To find the slope for function 1, pick any two points (x,y) in the table.
Let's choose (-1, 3) and (-2, 5).
The slope is
Let's choose (-1, 0) and (0, 4) for the two points for function 2.
The slope is
You want to compare the magnitude (without negatives) of the slope when comparing the greater rate of change/slope. In other words, you take the absolute value of each slope.
The magnitude of function 1's slope is 2.
The magnitude of function 2's slope is 4.
First off, let's solve <span>x−5y=15 for "y".
</span>
<span>
now, notice the function in slope-intercept form, well, it has a slope of 1/5.
now, a perpendicular line to that one, will have a negative reciprocal to that, let's check what that is.
</span>
<span>
so, we're looking for the equation of a line whose slope is -5 and goes through -2,5.
</span>![\bf \begin{array}{lllll} &x_1&y_1\\ % (a,b) &({{ -2}}\quad ,&{{ 5}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies -5 \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y-{{ y_1}}={{ m}}(x-{{ x_1}})}\implies y-5=-5[x-(-2)] \\\\\\ y-5=-5(x+2)\implies y-5=-5x-10\implies y=-5x-5](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Barray%7D%7Blllll%7D%0A%26x_1%26y_1%5C%5C%0A%25%20%20%20%28a%2Cb%29%0A%26%28%7B%7B%20-2%7D%7D%5Cquad%20%2C%26%7B%7B%205%7D%7D%29%0A%5Cend%7Barray%7D%0A%5C%5C%5C%5C%5C%5C%0A%25%20slope%20%20%3D%20m%0Aslope%20%3D%20%7B%7B%20m%7D%7D%3D%20%5Ccfrac%7Brise%7D%7Brun%7D%20%5Cimplies%20-5%0A%5C%5C%5C%5C%5C%5C%0A%25%20point-slope%20intercept%0A%5Cstackrel%7B%5Ctextit%7Bpoint-slope%20form%7D%7D%7By-%7B%7B%20y_1%7D%7D%3D%7B%7B%20m%7D%7D%28x-%7B%7B%20x_1%7D%7D%29%7D%5Cimplies%20y-5%3D-5%5Bx-%28-2%29%5D%0A%5C%5C%5C%5C%5C%5C%0Ay-5%3D-5%28x%2B2%29%5Cimplies%20y-5%3D-5x-10%5Cimplies%20y%3D-5x-5)
<span>
</span>
</span>
<span>
now, notice the function in slope-intercept form, well, it has a slope of 1/5.
now, a perpendicular line to that one, will have a negative reciprocal to that, let's check what that is.
</span>
<span>
so, we're looking for the equation of a line whose slope is -5 and goes through -2,5.
</span>
</span>