Humidifier physics homework

In summary, in this conversation, the goal is to determine the height, h, at which to fix the center of a circular disk in a humidifier, in order to maximize the exposed wet surface area. The dry area is given by pi*h^2, the wet area is pi*r^2 - pi*h^2, the area of the sector created by radii intersecting the water is (acos(h/r))/2 * r^2, and the area of the triangle above the water is sqrt(r^2-h^2)*h. To find the desired area, the wet area above water is subtracted from the total wet area. Finally, in order to find the absolute
  • #1
jstew2189
6
0
A circular disk of fixed radius r is used in a humidifier, vertically. It is partially submerged in water and rotates on its center. The goal is to determine the height above the water, h, at which to fix the center so that the exposed wet surface area is maximized.
1. Find the surface area that is dry, i.e. never is submerged as the disk rotates, as a function of r and/or h.
2. Find the total surface area that is wet as a function of r and/or h. (i.e. either submerged or exposed)
3. Find the area of the sector of the circle that is created by radii that intersect the water as function of r and/or h. (HINT: Use theta as half of the angle from radius to radius of your sector.)
4. Find the area of the triangle that is in this sector, but above the water, as a function of r and/or h.
5. Find the surface area that is submerged at any time as a function of r and/or h.
6. Subtract to find the desired area, area that is exposed and wet, as a function of r and/or h.
7. Define this function in Derive and follow our standard method for finding maxima and minima. Remember that r is the fixed radius and h is the height, which we can change. (Hint: Recall 0 < h < r so you know the endpoints and are trying to find the absolute max on this interval. )
8. Note your final answer should be a way to find h as a function of r. If Derive will not give an expression of the form h =, with only a function of r on the right side, you may need to do some algebra yourself to find a nice form.
9. For disks of radii 1, 2, 3, 4, and 5 give the value for h that should be used.


Well basically i want to know if my approach thus far is correct...
Also I need some help getting going on the last few parts of the question so.. any help would be wonderful. here's what i have so far:

1. Dry Area = pi*h^2
2. Wet area = pi*r^2 - pi*h^2
3. (acos(h/r))/2 * r^2 = sector area
4. sqrt(r^2-h^2)*h = area of triangle

im working on 5 being the wet area - (sector area - area of triangle)...
 
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  • #2
3. acos(h/r) is theta, ie half the angle of the sector at the centre. Need to double your result.

2 is submerged area + wet area above water

5 is submerged area

for 6 you are working out wet area above water
 
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1. What is the purpose of a humidifier?

A humidifier is a device that increases the moisture content in the air. It is commonly used to alleviate symptoms of dryness, such as dry skin, congestion, and sore throat. It can also help to prevent damage to wooden furniture and floors caused by dry air.

2. How does a humidifier work?

A humidifier works by adding water vapor into the air through various methods, such as evaporation, ultrasonic vibrations, or a fan. The added moisture increases the humidity level in the room and makes the air more comfortable to breathe.

3. What is the ideal humidity level for a room?

The ideal humidity level for a room is between 30-50%. This range is considered the most comfortable for humans and can also help to prevent the growth of mold and bacteria. However, the optimal humidity level may vary depending on personal preferences and the climate of the area.

4. Can a humidifier be harmful?

If not properly maintained, a humidifier can potentially be harmful. If the water in the humidifier is not changed regularly, it can become a breeding ground for bacteria and mold, which can then be released into the air and cause respiratory issues. It is important to follow the manufacturer's instructions for cleaning and maintenance to ensure the safe use of a humidifier.

5. How do I calculate the size of humidifier I need for a room?

The size of a humidifier needed for a room depends on the square footage of the room and the desired humidity level. As a general rule, a humidifier should be able to add 1 gallon of water per day for every 500 square feet of the room. However, it is best to consult the manufacturer's recommendation for the specific model of humidifier.

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