Ahemispherical bowl of radius r contains water to a depth h. give a formula that you can use to measure the volume of the water in the bowl.

V = π (rh² − ⅓h³)

Step-by-step explanation:

Draw a cross section of the bowl.  Cut a thin, horizontal slice of the water.  This slice is a circular disc of radius x and thickness dy.  It is position a distance of y from the bottom of the bowl.  The volume of this slice is:

dV = πx² dy

By drawing a right triangle, we can define x in terms of y:

x² + (r−y)² = r²

x² + r² − 2ry + y² = r²

x² = 2ry − y²

Substitute:

dV = π (2ry − y²) dy

The total volume of the water is the sum of all the slices from y=0 to y=h.

V = ∫₀ʰ π (2ry − y²) dy

V = π ∫₀ʰ (2ry − y²) dy

V = π (ry² − ⅓y³) |₀ʰ

V = π (rh² − ⅓h³)

answer: c

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