Question

Hi can someone please help me with this? I've been stuck on it for like an hour now

Also please make sure the answer is in terms of pi, thank you so much!!

Answer 1

19.5π = 39π/2 cubic units

Explanation:

You want the volume of the pentagonal shape shown in the figure, assuming cross sections parallel to the y-z plane are semicircles.

Shape

Consider a cylinder with a cone on top of the same diameter. Now, consider that figure to be cut in half along its axis of symmetry. The resulting solid is exactly what we have here.

Volume

We know that a cone has a volume equivalent to that of a cylinder 1/3 its height. Here, the height of the tapered portion of the figure is 1 unit. This means the volume of interest here is half that of a cylinder with radius 3 and height 4 +1/3.

The volume of the figure is ...

V = 1/2(πr²h)

V = 1/2π(3²)(4 1/3) = 39/2π = 19.5π

The volume of interest is 19.5π cubic units.

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Additional comment

If you write the integral for the volume, you find it works out to the same value. The differential of volume is the area of the semicircle multiplied by the differential of thickness dx.

`\displaystyle V=\int_0^1{(\pi)/(2)(3x)^2}\,dx+\int_1^5{(\pi)/(2)(3^2)}\,dx=(\pi)/(2)(3+36)`

Sometimes fractions are preferred for "exact" answers.

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