A hemispherical bowl of radius 8 inches is filled to a depth o...

Question

A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)

Accepted Answer

A hemispherical aquarium has a radius of 6 ft. If the water is filled to a depth of 8 ft, with H between 0 and 6, write the volume of water as a function of H. So, let's create our integral first. Know the depth Goes from 0 to 8 to 6. This means that we can write our bounds, our lower bounds. Will be -6. Which means our upper bounds. Will be -6 plus H. So these two bounds will be the bounds of our integral. Now, we can find our equation. We know this is a hemisphere. So we can use the equation of a sphere. Being X2 + Y2 equals R2, or in this case the equation of a circle. And we can direct this to solve for X. We know R is 6 because the radius is 6 ft. We have X2 + Y2 equals 36. We solve for X, we subtract Y2. And then we take the square root on both sides. X equals the square root of 36 minus Y2. Now, you can go ahead and write our integral. This integral will be given by the volume equation. Pi multiplied by the integral from -6 to -6 plus H. Of the square root 36 minus Y2 squared. This is in terms of why, so we will want DY. We can simplify this to get the square root of 36 minus Y2 squared, which is just 36 minus Y2. Now we can take our integral. We now have the integral, which is pi multiplied by 36 y. Minus Y cube divided by 3. From 6 to 6 plus H. Let's go ahead and simplify. We get pay Multiplied. By 36, -6 plus H. Minus -6 + HQ divided by 3. Minus 36 multiplied by -6, minus -6 cubed divided by 3. We can now simplify this. This will give us pie. Multiply By -216. Plus 36 H. Plus 72. 36 H. Plus 6 H squared. Minus 1/3 H cubes. Plus 216. 72. We can simplify further to get i multiplied. By 6 squared. Minus 1/3 HQs. We see some things canceled out, such as the 216. The 72, and the 36 H. Finally, we can distribute our pi. We get the answer V equals 6 pi H2 minus pi divided by 3 HQs. This will be our final answer. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)

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