SOLUTION: Can someone please help me with this word problem: The area of a rectangle with the perimeter 100 in. is given by the formula: A=50w-w^2 where w is the width. Find the value o

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Question 271986: Can someone please help me with this word problem:
The area of a rectangle with the perimeter 100 in. is given by the formula: A=50w-w^2 where w is the width. Find the value of w that produces the maximum area.
the only way i could try to figure this out is to make columns with the length and width dimensions that equal 100 in perimeter and plug the width into the equation. I started using the width of 45 and worked down to 30 which has the area of 600. But since a square is also a rectangle, do I use 25? Is there an easier way to figure this? Thanks
Found 2 solutions by ankor@dixie-net.com, Earlsdon:
Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
The area of a rectangle with the perimeter 100 in.
is given by the formula: A=50w-w^2 where w is the width.
Find the value of w that produces the maximum area.
:
There is an easy way, this is a quadratic equation,
then put the given equation A = 50w - w^2 in the form y = ax^2 + bx + c
y = -w^2 + 50w where: a=-1, b=50
:
Find the axis of symmetry using the formula x = -b/(2a), in this equation
w =
w =
w = 25 in will produce max area
:
The max area of any rectangle is a square, a fact to remember.

Answer by Earlsdon(6294) (Show Source):
You can put this solution on YOUR website!
Yes, there is an easier way using algebra.
Graph the given function which that shows the area (A) as a function of the width (w). A (area) is the vertical axis while w (width) is the horizontal axis.

You can see from the graph that this is a parabola opening downward so there is a maximum (area) at (w, A) of (25, 625).
You can find the w coordinate of the vertex by:
where: b = 50 and a = -1, so...

The vertex (maximum area in this case) occurs at w = 25.