Algebra Help?

Question

Determine the maximum perimeter of a rectangle inscribed in a parabolic arch modeled by the equation y = 7 - 3x^2 so that the base of the rectangle is along the x-axis.

I have no idea what this problem is asking. Any help?
I don't need the answer, just help on getting it started.
(What is parabolic arch?)

Thanks Michael S.
Now I need to know what it means to be inscribed in a parabolic arch. I guess I should have said that in the first place.

If someone could simplify what one of the bottom two answerers said into simpler mathematics, please do so.

Answer 1

A parabola is a curve described by an equation of the form ax² + bx + c = y. You're given y = 7 - 3x², which looks like half the McDonald's golden arches, top of curve at (0,7), x intercepts at ±√(7/3). If (x,0) is the bottom right corner of your rectangle, (-x,0) is the bottom left, (-x, 7-3x²) is top left, and (x, 7-3x²) is top right.

Length of bottom side is 2x, top side is 2x, left and right sides are 7-3x² each, so perimeter is 14 - 6x² + 4x, and that's what you have to maximize, which you do by finding the derivative and equating that to 0. That's early calculus, so what's a nice kid who doesn't know what a parabolic arch is doing in a course like that? That's like wearing the wrong color shirt on the wrong street in the wrong neighborhood, if you know what I mean.

Answer 2

Ok...first off we know that the equation of the perimeter of a rectangle is P = 2w + 2y. From the way the question is worded we can assume that one of the axis of the rectangle has the coords of (x,y). We can find the equation of the perimeter in terms of x by using the y equation they have already given us.

P=2x + 2y
P(x)=2x + 2(7-3x^2)
P(x)=2x + 14 - 6x^2
or
P(x) = -6x^2 + 2x + 14.

Now we are trying to determine the maximum perimeter, or maximum y-value of this graph. Since the slope is negative we can infer that the maximum y value will be the y-coord in the vertex of the graph.
To find the vertex of this equation you could use the maximum feature in a graphing calc. or you could do it the old-fashioned way.
We can first find the x-coord of the vertex (x,y) by finding "h"
h = -b/2a
In the equation P(x) = -6x^2 + 2x + 14, a = -6, b = 2, and c = 14.
Therefore,
h = -2/(2*-6)
h=-2/(-12)
h=1/6
Now to find the y-coord of the vertex we plug the h value we got into the P(x) function.
P(1/6) = -6(1/6)^2 + 2(1/6) +14
P(1/6) = -6(1/36) + 1/3 + 14
P(1/6) = -1/6 +1/3 + 14
P(1/6) = 1/6 + 14
P(1/6) = 14 1/6 or 14.166~
Therefore we found the y-coord of the vertex to be around 14.167 and therefore the maximum value of the perimeter of the rectangle is 14.167 ft.

Answer 3

a paropolic arch is like x^2 or x^3 or x^n