Answer:
the equation in the point slope form is
![y-(-2)=-\frac{7}{3} (x-8)](https://tex.z-dn.net/?f=y-%28-2%29%3D-%5Cfrac%7B7%7D%7B3%7D%20%28x-8%29)
and reducing the equation (slope-intercept form)
![y=-\frac{7}{3} x+\frac{50}{3}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B50%7D%7B3%7D)
Step-by-step explanation:
first we calculate the slope of the line with the formula:
![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
where
is a point where the line passes, and
is another point where the line passes.
Since we have the following points:
(8, -2)
(5,5)
we conclude that
![x_{1}=8\\y_{1}=-2\\x_{2}=5\\y_{2}=5](https://tex.z-dn.net/?f=x_%7B1%7D%3D8%5C%5Cy_%7B1%7D%3D-2%5C%5Cx_%7B2%7D%3D5%5C%5Cy_%7B2%7D%3D5)
now we substitute this values to find the slope:
![m=\frac{5-(-2)}{5-8}\\ \\m=\frac{5+2}{-3} \\m=-\frac{7}{3}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B5-%28-2%29%7D%7B5-8%7D%5C%5C%20%5C%5Cm%3D%5Cfrac%7B5%2B2%7D%7B-3%7D%20%5C%5Cm%3D-%5Cfrac%7B7%7D%7B3%7D)
to find the equation now that we know the slope we use the point-slope equation:
![y-y_{1}=m(x-x_{1})](https://tex.z-dn.net/?f=y-y_%7B1%7D%3Dm%28x-x_%7B1%7D%29)
and we subtitute the slope and the values of
and
:
![y-(-2)=-\frac{7}{3} (x-8)](https://tex.z-dn.net/?f=y-%28-2%29%3D-%5Cfrac%7B7%7D%7B3%7D%20%28x-8%29)
we reduce this equation:
![y+2=-\frac{7}{3} x+\frac{7}{3}*8 \\\\y+2=-\frac{7}{3} x+\frac{56}{3}\\\\y=-\frac{7}{3} x+\frac{56}{3}-2\\ \\y=-\frac{7}{3} x+\frac{50}{3}](https://tex.z-dn.net/?f=y%2B2%3D-%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B7%7D%7B3%7D%2A8%20%5C%5C%5C%5Cy%2B2%3D-%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B56%7D%7B3%7D%5C%5C%5C%5Cy%3D-%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B56%7D%7B3%7D-2%5C%5C%20%5C%5Cy%3D-%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B50%7D%7B3%7D)
the equation in the point slope form is
![y-(-2)=-\frac{7}{3} (x-8)](https://tex.z-dn.net/?f=y-%28-2%29%3D-%5Cfrac%7B7%7D%7B3%7D%20%28x-8%29)
and reducing the equation (slope-intercept form)
![y=-\frac{7}{3} x+\frac{50}{3}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B7%7D%7B3%7D%20x%2B%5Cfrac%7B50%7D%7B3%7D)
From your information
Let shorter base = h +3 and longer base = h + 7 where h is the height.
Formula for area of a trapezoid is
![A = \frac{1}{2} (b_1 +b_2)h ](https://tex.z-dn.net/?f=A%20%3D%20%20%5Cfrac%7B1%7D%7B2%7D%20%28b_1%20%2Bb_2%29h%0A)
Substituting into the formula we get the following equation.
![150= \frac{1}{2}(h+3+h+7)h](https://tex.z-dn.net/?f=150%3D%20%5Cfrac%7B1%7D%7B2%7D%28h%2B3%2Bh%2B7%29h%20)
![150 = \frac{1}{2} (2h+10)h](https://tex.z-dn.net/?f=150%20%3D%20%20%5Cfrac%7B1%7D%7B2%7D%20%282h%2B10%29h)
![150=h^2+5h](https://tex.z-dn.net/?f=150%3Dh%5E2%2B5h)
This is a quadratic equation which I will solve by factoring.
![h^2+5h-150 =0](https://tex.z-dn.net/?f=h%5E2%2B5h-150%20%3D0)
![(h-10)(h+15)=0](https://tex.z-dn.net/?f=%28h-10%29%28h%2B15%29%3D0)
![h-10= 0 ..... h=10](https://tex.z-dn.net/?f=h-10%3D%200%20.....%20h%3D10)
this is a useable answer (it's positive and length must be nonnegative)
![h+15=0 .... h=-15](https://tex.z-dn.net/?f=h%2B15%3D0%20....%20h%3D-15)
can't use this answer
so shorter base = 10 + 3 = 13 yards and longer base = 10 + 7 = 17 yards
Answer:
Step-by-step explanation:
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