Question

A hemispherical 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.

Answer 1

The formula to measure the volume of water in a hemispherical bowl is V = (2/3)πr²h.

Step-by-step explanation:

The formula to measure the volume of water in a hemispherical bowl is V = (2/3)πr2h, where V is the volume, π is a constant approximately equal to 3.14159, r is the radius of the bowl, and h is the depth of the water.

1. The volume of a half of the globe is given by V = 2/3 * π * r³. However, since we are only interested in measuring the water's volume—which represents a portion of the hemisphere—we will need to modify the formula accordingly.

2. Within the larger hemisphere, a smaller one is created by the water's depth, h. To ascertain the volume of this more modest half of the globe, we can take away it from the volume of the bigger side of the equator.

3. The span of the more modest half of the globe is likewise r, since it has a similar ebb and flow as the bigger side of the equator.

4. The level of the more modest side of the equator, h, is equivalent to the profundity of the water.

5. To find the volume of the more modest side of the equator, we utilize the recipe V = 2/3 * π * r³ and substitute the sweep r and level h.

6. The volume of the water is around 50% of the volume of the more modest half of the globe since it just fills one side of the bowl.

7. In this way, the volume of the water in the hemispherical bowl is given by 1/2 * (2/3 * π * r³) = 1/3 * π * r³.

8. To represent the profundity of the water, we increase the volume by the proportion of the level h to the sweep r, giving us 1/3 * π * r² * h.

9. We can further simplify by rewriting the formula as 1/2 * * h2 * (3r - h).

Answer 2

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)