Answer:
Step-by-step explanation:
Answering a comes from simplification, and answering b and c are done all in one step: completing the square on the quadratic that results from a.
(a) If Martina has 240 m of fencing and is only utilizing one side for the length and 2 sides for the width, the perimeter formula is
240 = x + 2w where x is a length and w is the width. Solving this for w in terms of x:
240 - x = 2w so
[tex]w=120-.5x[/tex] The area for a rectangle is L * W, so our area using the lengths we have is
A(x) = x(120 - .5x) and we simplify:
A(x) = 120x - .5x² That's the answer to a.
Now for b and c, we will complete the square on this to get the vertex.
Begin by factoring out the -.5:
[tex]A(x)=-.5(x^2-240x)[/tex] Now we take half the linear term, square it and add it both inside the parenthesis and outside the parenthesis. Our linear term is 240. Half of 240 is 120, and 120 squared is 14400:
[tex]A(x)=-.5(x^2-240x+14400)+7200[/tex] (The 7200 comes from multiplying the 14400 times the -.5; -.5 times 14400 is -7200 so to balance things out, we have to add 7200).
The perfect square binomial that results from this is
A(x) = -.5(x - 120)² + 7200. From this we determine that our vertex is
(120, 7200). The 120 is the value of x, the length we are asked to find in b; the 7200 is the max area we are asked to find in c.
