Question

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 5 − x 2 . What are the dimensions of such a rectangle with the greatest possible area?

Answer 1

Answer:

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola

y=5−x^2. What are the dimensions of such a rectangle with the greatest possible area?

Width =

Height =

Width =√10 and Height $= \frac{10}{4}$

Step-by-step explanation:

Let the coordinates of the vertices of the rectangle which lie on the given parabola y = 5 - x² ........ (1)

are (h,k) and (-h,k).

Hence, the area of the rectangle will be (h + h) × k

Therefore, A = h²k ..... (2).

Now, from equation (1) we can write k = 5 - h² ....... (3)

So, from equation (2), we can write

$A =h^{2} [5-h^{2} ]=5h^{2} -h^{4}$

For, A to be greatest ,

$\frac{dA}{dh} =0 = 10h-4h^{3}$

⇒ $h[10-4h^{2} ]=0$

⇒ $h^{2} =\frac{10}{4} {Since, h≠ 0}$

⇒ $h = ±\frac{\sqrt{10} }{2}$

Therefore, from equation (3), k = 5 - h²

⇒ $k=5-\frac{10}{4} =\frac{10}{4}$

Hence,

Width = 2h =√10 and

Height = $k =\frac{10}{4}.$