The Dipstick Problem - area of a Cyclinder

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In summary, the problem involves a circle with a radius of 5 bisected by the y-axis. Depending on the position of the circle along the y-axis, different percentages of the circle's area will be above and below the x-axis. To determine the distance between the bottom of the circle and the x-axis in order to have a certain percentage of the circle's area below the x-axis, the bounds and integrals can be used, with special cases for when the radius is 0, greater than or equal to 5, or less than or equal to -5.
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dipique
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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
 
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  • #2
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.
 
  • #3
, 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.
 

1. What is the Dipstick Problem and why is it important?

The Dipstick Problem refers to the challenge of accurately measuring the volume of liquid in a cylindrical container using only a dipstick. It is important because many industries rely on accurate volume measurements for transportation, storage, and production purposes.

2. How can the Dipstick Problem be solved?

The Dipstick Problem can be solved using mathematical formulas, such as the formula for the area of a cylinder (A=πr^2h). By plugging in the radius and height of the cylinder, the area can be calculated and used to determine the volume of liquid in the container.

3. Are there any limitations to using a dipstick for volume measurements?

Yes, there are limitations to using a dipstick for volume measurements. These include the accuracy of the dipstick's measurement markings, the shape of the container (which may not be a perfect cylinder), and the presence of any obstructions or irregularities in the liquid being measured.

4. How do factors such as temperature and density affect the Dipstick Problem?

Temperature and density can affect the Dipstick Problem because they can alter the volume of the liquid being measured. For example, as temperature increases, the volume of a liquid may expand, resulting in an inaccurate measurement. Density also plays a role, as denser liquids will have a different volume than less dense liquids of the same mass.

5. Are there any alternative methods for accurately measuring the volume of a cylindrical container?

Yes, there are alternative methods for measuring the volume of a cylindrical container. These include using a measuring cup or graduated cylinder, which have more precise measurement markings, or using advanced technologies such as ultrasonic sensors or optical scanning devices.

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