Area Increasing as a linear dimension increases -- Looking for intuition on this

In summary, the practical significance of problems like these is that they can be used to help translate equations into physical terms.
  • #1
NoahsArk
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I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like these. In the example typically taught initially in calculus, something is moving and distrance is changing with the equation ## y = x^2 ##. If it's a car moving, then the velocity of the car at any moment is 2x. The way I can translate this into the physical world is that at any given x point, the car at that instant is moving at certain constant speed. The first explanation I read about why this might be important is that, for example, we may need to know how fast the car was going at a given moment when it crashed into something.

I can't think of any similar practical reason why we'd want to know how much the area of a shape is increasing at any given moment. The only application I can think of is if it's a maximization problem (although I've never seen a maxmization problem involving increasing area). Please let me know some examples of when we'd want to know how much area of a shape is increasing at a given moment. I know there must be examples, I just don't know what they are, and knowing the applications makes it easier for me to remember the math. Thanks.
 
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  • #2
NoahsArk said:
I was wondering about the practical significance of problems like these. In the example typically taught initially in calculus, something is moving and distrance is changing with the equation ## y = x^2 ##. If it's a car moving, then the velocity of the car at any moment is 2x. The way I can translate this into the physical world is that at any given x point, the car at that instant is moving at certain constant speed.

Remember that velocity is a time derivative of position, so with your variables, ##y## is position and ##x## would have to be time [which is, perhaps a confusing naming convention].

Furthermore, I'm not sure if it makes sense to say that it travels at a constant speed for an instant. Would probably better to say its instantaneous speed is ##v##, or something to the effect of "in a small interval ##\delta t## (or in your case, ##\delta x##...), ##v## is approximately constant".
 
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  • #3
NoahsArk said:
I can't think of any similar practical reason why we'd want to know how much the area of a shape is increasing at any given moment.
There are lots of practical examples, and not just how the area of a certain shape is changing. One example is how quickly the volume of some fluid in a tank changes relative to how fast the depth is changing. Another is the relationship between volume and radius of a melting snowball. Yet another is a balloon into which air is pumped at a constant volume per minute, and how fast the radius is changing.

One that involves area is a question about tossing a pebble into a still pond, causing circular waves to ripple out. If the wave fronts are moving at a constant speed, by how much does the area inside a circular ripple change?

There are lots more.
 
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  • #4
etotheipi said:
so with your variables, y is position and x would have to be time

In the case of a car driving at an accellerating rate of ## y = x^2 ## x is the time axis, and you could tell the driver at any moment to stop accellerating and to continue driving at the speed he was driving at when you yelled stop. In contrast, though, when the length of the side of a square is our x axis, and the area is our y axis, although we get the same relationship of ## y = x^2 ##, in the physical world it's impossible to have area be increasing linearly with respect to the length of the side. For every unit the side increases by 1, area will increase by a greater and greater amount. For every 1 minute increase a car could travel 3 miles, but for every 1 unit of length the side increases, the area increase of the square won't be the same.

@Mark44, in the examples you gave, which are helpful, I see how area/volume increases relate to the increase of the radius, depth, etc. Area, though, is always increasing at an accellerating rate in these examples. In the circle in the pond example, the area of the circle increases more and more for each unit of increase in the radius. We can find the instantaneous rate of change of the area of the circle with respect to time. For what purposes would it be helpful to know that?
 
  • #5
NoahsArk said:
Area, though, is always increasing at an accellerating rate in these examples. In the circle in the pond example, the area of the circle increases more and more for each unit of increase in the radius. We can find the instantaneous rate of change of the area of the circle with respect to time. For what purposes would it be helpful to know that?
I don't usually see many problems like this, but it's not too hard to come up with an example, albeit somewhat contrived.
Suppose a farmer is plowing a field in such a way that the plowed portion forms a square. Because his tractor moves at a constant speed, the rate of change of the plowed area is constant. At what rate does a side of the square change?
Again, this is a bit contrived, because when he's plowing a new side, the plowed portion is not quite a perfect square.
 
  • #6
NoahsArk said:
For what purposes would it be helpful to know that?
You can look at these elementary problems as building your mental muscles for harder problems. The concept of rate of change of different variables is everywhere in physics, but if you start at real practical problems, then you are jumping in the deep end.

If you wanted to you could start by learning linear and quadratic air resistance, for example. But, what's likely to happen is that basic mathematical ideas will trip you up. You won't understand how quantities change in relation to one another or see how to manipulate equations.

These artificial elementary problems are designed to develop your knowledge and expertise of the building blocks - but, almost by definition, they are too simple to be genuine applications in their own right.
 
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  • #7
Thank you for the responses.
 
  • #8
NoahsArk said:
I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like these. In the example typically taught initially in calculus, something is moving and distrance is changing with the equation ## y = x^2 ##. If it's a car moving, then the velocity of the car at any moment is 2x. The way I can translate this into the physical world is that at any given x point, the car at that instant is moving at certain constant speed. The first explanation I read about why this might be important is that, for example, we may need to know how fast the car was going at a given moment when it crashed into something.

I can't think of any similar practical reason why we'd want to know how much the area of a shape is increasing at any given moment. The only application I can think of is if it's a maximization problem (although I've never seen a maxmization problem involving increasing area). Please let me know some examples of when we'd want to know how much area of a shape is increasing at a given moment. I know there must be examples, I just don't know what they are, and knowing the applications makes it easier for me to remember the math. Thanks.

Oil spill mitigation and damage reporting would be a good practical application.
 
  • #9
31.4mp/s2
 
  • #10
Thank you. What's got me stuck is, in a problem like an oil spill where the area of the oil spill is a function of the radius, why do we care about finding the derivitive of that function? Say the radius of the oil spill is increasing at 1 meter p/s. I can find the deriviative: ## A = \pi r^2 ## where r is a function of t.
## \frac {dA} {dt} = \pi 2r \frac {dr} {dt} ##
## \frac {dA} {dt} = \pi 2r (1) ##
So, when, for example, when the radius is 5 meters, the area of the oil spill is increasing at
## 3.14(2)(5) ## = ## 31.4 m p/s^2 ##
Why do we care about how fast the area is increasing at a certain moment? I could see how we might want to know what the total area of the oil spill is at a certain moment in order to assess how much damage has been done. For total area, though, we don't need calculus.
 
  • #11
NoahsArk said:
Thank you. What's got me stuck is, in a problem like an oil spill where the area of the oil spill is a function of the radius, why do we care about finding the derivitive of that function? Say the radius of the oil spill is increasing at 1 meter p/s.
Any time something is given as a rate of change, that's a derivative. The immediately preceding sentence can be translated to ##\frac{dr}{dt} = 1 \frac m s##
BTW, don't include 'p' -- that threw me off for a bit. Although it shows up in abbreviations like mph and kph, in any sort of mathematical or scientific setting, we write mi/hr or km/hr, and so on.
NoahsArk said:
I can find the deriviative: ## A = \pi r^2 ## where r is a function of t.
## \frac {dA} {dt} = \pi 2r \frac {dr} {dt} ##
## \frac {dA} {dt} = \pi 2r (1) ##
So, when, for example, when the radius is 5 meters, the area of the oil spill is increasing at
## 3.14(2)(5) ## = ## 31.4 m p/s^2 ##
Why do we care about how fast the area is increasing at a certain moment?
Maybe we have some sort of machine that can vacuum up oily water at a certain number of meters2/sec. It might be handy to know whether our machine can keep up when the spill is a certain size.
NoahsArk said:
I could see how we might want to know what the total area of the oil spill is at a certain moment in order to assess how much damage has been done. For total area, though, we don't need calculus.
Well, if you want to be one of the guys standing on the side, holding a shovel, then certainly you don't need calculus.
 
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  • #12
NoahsArk said:
I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like these.

Possibly interesting...
http://galileo.phys.virginia.edu/classes/609.ral5q.fall04/LecturePDF/L14-GALILEOSCALING.pdf
http://ww3.haverford.edu/physics-astro/Amador/links/documents/HCBiomechanicsLab2011.pdf

https://philosophy.unc.edu/wp-content/uploads/sites/122/2013/10/dimensional-explanations.pdf
http://www.physics.drexel.edu/~bob/Manuscripts/group.pdf (just look at the first 4 pages... II-A to II-C)
 
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1. How does increasing the linear dimension of an area affect its overall size?

As the linear dimension of an area increases, the overall size of the area also increases. This is because the area is directly proportional to the square of its linear dimension. So, if the linear dimension doubles, the area will quadruple.

2. Why does the area increase at a faster rate than the linear dimension?

This is because the area is a two-dimensional measurement, while the linear dimension is only one-dimensional. So, as the linear dimension increases, it is multiplied by itself to calculate the area, resulting in a faster rate of increase.

3. Can you provide an example to illustrate this concept?

Sure, let's say we have a square with a side length of 2 cm. The area of this square would be 2 cm x 2 cm = 4 cm². Now, if we double the side length to 4 cm, the area would become 4 cm x 4 cm = 16 cm², which is four times the original area.

4. Does this concept apply to all shapes?

Yes, this concept applies to all shapes, as long as they have a linear dimension and an area. For example, a circle's area is directly proportional to the square of its radius, so as the radius increases, the area increases at a faster rate.

5. How is this concept relevant in real-world applications?

This concept is relevant in many fields, such as engineering, architecture, and physics. It helps in understanding the relationship between different dimensions and areas, which is crucial in designing structures and calculating measurements accurately.

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