A hole is drilled through the center of a ball of radius r, leav-ing a solid with a hollow cylindrical core of height h. Showthat the volume of this solid is independent of the radius ofthe ball.

Answered: A hole is drilled through the center of…

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter10: Measurement, Area, And Volume

Section10.8: Volumes Of Pyramids And Cones

Problem 13E

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A hole is drilled through the center of a ball of radius r, leav-
ing a solid with a hollow cylindrical core of height h. Show
that the volume of this solid is independent of the radius of
the ball.

Expert Solution

Step 1

According to the question, A hole is drilled through the center of a ball of radius r, leaving a solid with a hollow cylindrical core of height h.

We need to show that, the Volume of this solid is independent of the radius of the ball.

To rotate this figure, we would use a semi circle and rotate it about y axis and should remember, we are drilling a hole, so we need a drill is actually start or region.

Drawing a triangle, we have: hypotenuse = r and perpendicular = L/2

thus, base = $\sqrt{(r^{2}} - {(\frac{L}{2})}^{2})$

Now, Volume = V = $2 \int_{\sqrt{(r^{2} - {(L / 2)}^{2}}}^{r} 2 πx \sqrt{r^{2} - x^{2}} dx$

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