11-gopal patil

2022-10-08

The solid lies between planes perpendicular to the x-axis at
x = -1 and x = 1. The cross-sections perpendicular to the
x-axis between these planes are squares whose bases run from the
semicircle y = - 21 - x2 to the semicircle y = 21 - x2
.

Nick Camelot

The given information describes a solid that is bounded by two perpendicular planes, $x=-1$ and $x=1$, and the cross-sections perpendicular to the x-axis between these planes are squares. The bases of these squares are defined by the semicircles $y=-21-{x}^{2}$ and $y=21-{x}^{2}$.
In order to visualize this solid, we can imagine slicing it perpendicular to the x-axis at various x-values between -1 and 1. At each x-value, the cross-section formed is a square. The length of each side of the square is determined by the difference between the y-values of the two semicircles at that x-value.
To find the length of each side, we subtract the y-values of the semicircles at a given x-value:
Side length = $\left(21-{x}^{2}\right)-\left(-21-{x}^{2}\right)$
Simplifying the expression:
Side length = $21-{x}^{2}+21+{x}^{2}$
Side length = $42$
Hence, the side length of each square cross-section is 42 units.
To summarize, the solid is bounded by the planes $x=-1$ and $x=1$, and the cross-sections perpendicular to the x-axis between these planes are squares with a side length of 42 units.

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