The Dipstick Problem - area of a Cyclinder

[dipique]

There is a circle with a radius of 5 bisected by the y-axis. If we move it up and down the y-axis, different percentages of the circle will be above and below the x-axis. What I need to find out is how to relate the percentage below/above the x-axis (either would work) to the distance between the bottom of the circle and the x-axis.

So: we'll start with the circle sitting right on top of the x-axis. 100% of the circle area is above the x-axis. If we want to move it down so that 10% of the circle's area is below the x-axis, how many units would the circle have to be moved down?

Dan

 

[factor]

There are three special cases in this problem that I'll note first.

1) When a = 0, 50% of the area is above and 50% is below the x-axis
2) When a >= 5, 100% of the area is above the x-axis
3) When a <= -5, 100% of the area is below the x-axis

The next two cases are between these bounds of course.

When 0 < a < 5 the percentage below the x-axis is the integral from -sqrt(25 - a^2) to sqrt(25 - a^2) of -sqrt(25 - x^2)*dx divided by 25*Pi.

When -5 < a < 0 the percentage above the x-axis is the integral from -sqrt(25 - a^2) to sqrt(25 - a^2) of sqrt(25 - x^2)*dx divided by 25*Pi.

 

[tacman]

, thank you for bringing up the Dipstick Problem. It is an interesting mathematical concept that can be applied to real-life situations. In this case, we are dealing with a circle with a radius of 5 bisected by the y-axis. As we move the circle up and down the y-axis, the percentage of the circle above and below the x-axis will change.

To find the relationship between the percentage of the circle below/above the x-axis and the distance between the bottom of the circle and the x-axis, we can use the concept of similar triangles.

Let's say the circle is initially sitting right on top of the x-axis, with 100% of its area above the x-axis. In this case, the distance between the bottom of the circle and the x-axis is equal to the radius of the circle, which is 5.

Now, if we want to move the circle down so that only 10% of its area is below the x-axis, we need to find the new distance between the bottom of the circle and the x-axis.

Using the concept of similar triangles, we can set up the following proportion:

(10/100) = (x/5)

Where x represents the new distance between the bottom of the circle and the x-axis. Solving for x, we get x = 0.5.

Therefore, to have only 10% of the circle's area below the x-axis, the circle needs to be moved down by 0.5 units.

We can use the same method to find the distance for any given percentage. For example, if we want 75% of the circle's area to be below the x-axis, the circle needs to be moved down by (75/100)*5 = 3.75 units.

In summary, the distance between the bottom of the circle and the x-axis is directly proportional to the percentage of the circle's area below the x-axis. I hope this helps you in solving the Dipstick Problem.