SOLUTION: a park planner has sketched a rectanular park in the first quadrent of a coordinate grid. wo sides of the park lie on the x- and y-axes. a trapezoidal flower bed will be bounded by

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: a park planner has sketched a rectanular park in the first quadrent of a coordinate grid. wo sides of the park lie on the x- and y-axes. a trapezoidal flower bed will be bounded by      Log On


   

Question 120877: a park planner has sketched a rectanular park in the first quadrent of a coordinate grid. wo sides of the park lie on the x- and y-axes. a trapezoidal flower bed will be bounded by the line y=x+7, the x-axis, and the vertical lines x=1 and x=a, where a is greater than 1. The areaA of the trapezoid is modeled by A=1/2a squared+7a-15/2. assume the lengthsamong the axes are measured in meters. For what value of a will the trapezoid have an area of 25 meters? use the quadratic formula to find the answer
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
A=%28%28a-1%29%28%28a%2B7%29%2B8%29%29%2F2
A=%28%28a-1%29%28a%2B15%29%29%2F2
A=%28a%5E2%2B14a-15%29%2F2


You have an expression in terms of the variable a that represents the area, so all you need to do is set that expression equal to 25 and solve.

A=%281%2F2%29a%5E2%2B7a-%2815%2F2%29

%281%2F2%29a%5E2%2B7a-%2815%2F2%29=25

Since fractional coefficients will be a pain when you evaluate the quadratic formula, multiply through by 2:

a%5E2%2B7a-15=50

Now add -50 to both sides to get the equation in standard form:

a%5E2%2B14a-65=0

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-14+%2B-+sqrt%28+14%5E2-4%2A1%2A%28-65%29+%29%29%2F%282%29+
x+=+%28-14+%2B-+sqrt%28+456+%29%29%2F%282%29+
x+=+%28-14+%2B-+2sqrt%28+114+%29%29%2F%282%29+
x+=+-7%2B-+sqrt%28+114+%29+

-7-sqrt%28114%29 can be excluded because it is clearly less than zero and the problem conditions demand a > 1. So the correct and only answer is:

x=-7%2Bsqrt%28114%29, roughly 3.7

Should you believe me? Not without checking, you shouldn't.

A=%28%28%28-7%2Bsqrt%28114%29%29-1%29%28%28%28-7%2Bsqrt%28114%29%29%2B7%29%2B8%29%29%2F2
A=%28sqrt%28114%29-8%29%28sqrt%28114%29%2B8%29%2F2=%28114-64%29%2F2=50%2F2=25 Answer checks