Start by sketching a graph of the function that represents the bowl. I recommend sketching the lower half of the graph of x2 + y2 = a2 to make your computations a little simpler.htoor9 said:Homework Statement
A hemispherical bowl of radius "a" contains water to a depth "h"
Fine the volume of water in the bowl.
Homework Equations
Pi * Integral(R(x))^2
basically the disk method
The Attempt at a Solution
The answer is (Pi*h2*(3a-h))/3...i just have no idea how to set up the integral to get that. Thanks guys
Mark44 said:Start by sketching a graph of the function that represents the bowl. I recommend sketching the lower half of the graph of x2 + y2 = a2 to make your computations a little simpler.
Draw a line across the bowl representing a level of h units. Since h is measured from the bottom of the bowl, if you have located the bowl as I suggested, the water level crosses the y-axis at (0, h - a). For example, if the bowl's radius is 6'' and the water is 2" deep, the water level on the y-axis is at (0, 2 - 6) = (0, -4).
Draw a typical "slice" of the water. What is its volume? That's what you will use for your integrand.
Mark44 said:The bottom point of the bowl is at (0, -a).
The formula for calculating the volume of a bowl using variables is V = πr²h, where V is the volume, π is the mathematical constant pi, r is the radius of the bowl, and h is the depth or height of the bowl.
Yes, the volume of a bowl can still be calculated even if the depth is unknown. The only requirement is knowing the radius of the bowl. If the depth is unknown, it can be represented as a variable in the formula, and the volume can still be calculated.
If the radius of a bowl is doubled, the volume will increase by a factor of 4. This is because the volume is directly proportional to the square of the radius. So if the radius is doubled, the volume will be multiplied by 2² = 4.
The units used for the radius and depth of the bowl should be consistent. For example, if the radius is measured in inches, then the depth should also be measured in inches. The resulting volume will then be in cubic inches.
No, this formula can only be used to calculate the volume of a bowl with a circular base. For bowls with non-circular bases, a different formula must be used, depending on the shape of the base.