Find the average height of a pyramid

[Permanence]

Homework Statement

Homework Equations

V = (1/3) * A * H [Volume of Pyramid]

The Attempt at a Solution

The first thing I did was to calculate the height of pyramid from the volume formula. I got a perpendicular height of 15. I'm not sure where to go from there.

I'm under the impression that average height has something to do with geometric center. I researched the topic and found something about Pappus Theorem, and that the center of mass is 3/4 of the way from the vertex to the mid point of the base. Using that I come up with an average height of 3.75, but I'm not certain if that number is correct or if it even represents average height.

I found a related problem online, but it involved things that I have not discussed in class and I had trouble following along.

I would really appreciate some guidance. I'd like to know if my answer of 3.75 is correct, and how I would go about solving this using multivariable calculus.

 

[Dick]

Homework Statement

Homework Equations

V = (1/3) * A * H [Volume of Pyramid]

The Attempt at a Solution

The first thing I did was to calculate the height of pyramid from the volume formula. I got a perpendicular height of 15. I'm not sure where to go from there.

I'm under the impression that average height has something to do with geometric center. I researched the topic and found something about Pappus Theorem, and that the center of mass is 3/4 of the way from the vertex to the mid point of the base. Using that I come up with an average height of 3.75, but I'm not certain if that number is correct or if it even represents average height.

I found a related problem online, but it involved things that I have not discussed in class and I had trouble following along.

I would really appreciate some guidance. I'd like to know if my answer of 3.75 is correct, and how I would go about solving this using multivariable calculus.

Yes, I think it is correct. I get the same thing. I did it by thinking of square cross sections of the pyramid at height x. Like in your picture. If the area of the square at height x is A(x) then the average height is given by $\frac{\int A(x) x dx}{\int A(x) dx}=\frac{\int A(x) x dx}{volume}$. Can you fill in the missing parts like limits on the integral and the form of A(x) in terms of x? Very resourceful solution to the problem, by the way. The average height is the height of the center of mass. Good job.

 

[Permanence]

Hi, thank you for your reply. I understand what you did, but I was hoping you could clarify something for me

A classmate used the following from the book to solve the problem.
He then set it 80/16 to get the answer 5.

That answer is different than both the method you used and the method I used which is leaving me confused. Do you see anything wrong with using that equation?

Thank youedit:

I thought about it more, and I'm not sure if it's 5 but I see some error in that approach.

Imagine the length of the sides of the base are 1, and the height is 1. You get a volume of 1/3. Solving with that formula you get an average height of 1/3. That is the center for a two dimensional right triangle, but the center of a three dimensional pyramid is 1/4 (according to the slightly related example problem I attached in my original post)

 

[LCKurtz]

I tend to agree with $\bar h = 5$ using the definition $A_b \bar h = V$ where $A_b$ is the area of the base and $V$ is the volume.

 

[Permanence]

This problem has really confused me. I don't understand why different approaches are leading to different answers. I believe my professor is going to want 5, but I see too much conflict to pick one answer.

LCKurtz, can you explain why that method gives an incorrect answer for a base 1, and height 1?

 

[LCKurtz]

No. Offhand I don't see why the answers are different. I will look at it some more tomorrow.

 

[Dick]

I tend to agree with $\bar h = 5$ using the definition $A_b \bar h = V$ where $A_b$ is the area of the base and $V$ is the volume.

Ok, I see what going on here. The answer you are giving is interpreting the 'average' in the question as the average of the height $x$ being $\frac{\int x dA}{A}$. The center of mass answer is interpreting it as being $\frac{\int x dV}{V}$. One's an area average and the other is a volume average. Completely different. But both valid without a context to know which you want.

 

[LCKurtz]

Yeah, after sleeping on it I have another take also. You define the average value of a function $f(x)$ on $[a,b]$ as$$
A=\frac 1 {b-a}\int_a^b f(x)~dx$$This is a completely different calculation than calculating the y coordinate $\bar y$ of the center of area enclosed between $f(x)$ and the $x$ axis on that interval.

For example, if $f(x)\equiv 1$ on $[0,1]$, the $y$ value is always $1$ and its average is $1$. But $\bar y=\frac 1 2$.

 

[Permanence]

Thank you for both for your replies.
My professor said that my approach to the problem was incorrect, and that the centroid has nothing to do with the problem. I didn't entirely understand why he dismissed my approach, he said something about not needing to bring the z variable into this.

 

[LCKurtz]

Thank you for both for your replies.
My professor said that my approach to the problem was incorrect, and that the centroid has nothing to do with the problem. I didn't entirely understand why he dismissed my approach, he said something about not needing to bring the z variable into this.

Read my post #8 for a 1 variable example. The centroid does have nothing to do with it. It's the same for two variables.

 

[Ray Vickson]

Thank you for both for your replies.
My professor said that my approach to the problem was incorrect, and that the centroid has nothing to do with the problem. I didn't entirely understand why he dismissed my approach, he said something about not needing to bring the z variable into this.

I think the appropriate definition of "average" depends on context---what you want to do with it. The volume-oriented definition has something to do with centroids (centers of gravity), but other types of averages are also possible. For example, suppose you stand back from the pyramid and just trace the visible outline; what you see looks like a one-variable graph of a "triangular" function, something like $y = 1-a|x|$ for $|x| \leq 1/a$. Now the average value of $y$ would be like the 1-variable average in your thumbnail from the book. This average would be much more relevant than the centroid (the 2-dimensional average) to people who are interested in climbing the pyramid. They would go along surface, so the interior of the pyramid is of little relevance to them.

 

[Permanence]

Thank you again for the replies. Ray your example about the climbing has helped me understand the differences.

 

[SammyS]

I tend to agree with $\bar h = 5$ using the definition $A_b \bar h = V$ where $A_b$ is the area of the base and $V$ is the volume.

Yes. I agree with 5 also.

 

[existent]

5 is the height of the entire triangle, how could that be the average height when most of the heights lie below 5?

 

[Dick]

5 is the height of the entire triangle, how could that be the average height when most of the heights lie below 5?

What triangle are you talking about and why do you think it's height is 5?

 

[mathwonk]

Perhaps what they should have said is to consider the height function as a function of two variables, with domain the base of the pyramid, and whose value at a point of the base equals the perpendicular height of the pyramid above that point; i.e. the distance from the base at which a line perpendicular to the base at the given point will intersect the surface of the pyramid. Then find the average of this height function, averaged over the base (hence averaging wrt area). Thus the sides of the pyramid form the graph of the height function. Without some such precise statement, I myself would have thought we were averaging the heights of all points in the pyramid, both inside it and on the surface, and would have assumed the answer was the location of the centroid. I could still be wrong, but at least you are not the only person confused by the statement of this problem.

By the way this is an excellent discussion and illustrates the value of understanding a problem rather than just worrying about getting the "right" answer.

 

[mathwonk]

And by the way, the concept of centroid is relevant in the following sense. Because the height function is piecewise linear, the average height over each face is the height of the average point, i.e. of the centroid of the face. Since the centroid of a triangle is 1/3 of the way up we get the result.