Parabolas and rectangles

Question

An arch has the shape of the parabola y = 11 − x². A rectangle is fit under the arch by selecting a point (x, y) on the parabola.

The parabola's vertex is at (0,11) and it is sloping down. The rectangle reaches to from (0,0) to (0,9.25) and (-1.5,0) to (1.5,0). The area is 22x-2x³. 

 

 

If x = 1, the rectangle has base 2 and height 10. Find the base of a second rectangle that has the same area. (Round your answer to two decimal places.)


Please help me solve this. Thanks!

Answer 1

A(x) = 22x-2x^3

A(1) = 22-2=20

A(2.7) = about 20.034 = about 20 when base = 5.4 and height = about 3.7

5.4 times 3.7 = 999/50 = 19.98=about 20

solve the cubic equation for x

-2x^3 +22x -20 =0

or divide by -2 and solve the cubic

x^3-11x+10= 0

x=1 is one solution

x = about 2.7 is a second solution

by descartes rule of sign there is a maximum 2 positive real solutions

that's 1 and about 2.7

and there is a maximum 1 negative real solution