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

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
`= (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).`