A rectangle has its two bottom corners lying on the x-axis and the two top corners on the parabola , where . If the area of the rectangle is to be maximized, then what are its dimensions?
A) 7 33
B) 8 32
C) 10 35
D) 4 32

Question

A rectangle has its two bottom corners lying on the x-axis and the two top corners on the parabola  , where  . If the area of the rectangle is to be maximized, then what are its dimensions?
A) 7 33 
B) 8 32 
C) 10 35 
D) 4 32

Quizplus · Accepted Answer

Let the rectangle have width $2x$ and height $y$. Then the area of the rectangle is $2xy$.

Let the distance from the origin to the point where the rectangle intersects the parabola be $d$. Then, since the parabola is given by $y=11-d^2$, we have $y=11-d^2$ for both of the top corners of the rectangle. Since the bottom corners lie on the x-axis, we have $y=0$ for both of the bottom corners. This gives us the coordinates of the four corners of the rectangle: $(\pm x, 0)$ and $(\pm x, 11-d^2)$.

The length of the diagonal of the rectangle is equal to the distance between the two top corners, which is $\sqrt{(2x)^2 + (11-d^2)^2}$. By the Pythagorean Theorem, this is equal to $\sqrt{4x^2 + d^4 -22d^2 + 121}$.

Since the rectangle has two bottom corners on the x-axis, the distance between these corners is $2x$. Therefore, $2x$ must be less than or equal to the distance between the two points on the parabola where the rectangle intersects it. This distance is equal to twice the distance from the origin to the point where the parabola intersects the x-axis at $x=d$. Therefore, $2x\leq 2d$, or $x\leq d$.

Now, we want to maximize the area of the rectangle, which is $2xy$. Using the equation $y=11-d^2$, we have $$2xy=2x(11-d^2)=22x-2xd^2.$$ So, we want to maximize $22x-2xd^2$ subject to the constraint that $x\leq d$. We can use calculus to find that this maximum occurs when $x=\frac{8}{3}$ and $d=\frac{16}{3}$, and the maximum area is $\frac{256}{3}$.

Therefore, the dimensions of the rectangle are  $\boxed{	extbf{(B) }8\sqrt{2}	imes 4\sqrt{2}}$.

A Rectangle Has Its Two Bottom Corners Lying on the X-Axis