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?

Width =

Height =

Answer 1

Width =√10 and Height =
`(10)/(4)`

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 ,
`(dA)/(dh) =0 = 10h-4h^(3)`

⇒
`h[10-4h^(2) ]=0`

⇒
`h^(2) =(10)/(4)` {Since, h≠ 0}

⇒ h = ±
`(√(10) )/(2)`

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

⇒
`k=5-(10)/(4) =(10)/(4)`

Hence, Width = 2h =√10 and Height = k =
`(10)/(4)`. (Answer)