Square-based pyramids surface area

In summary, the problem is to find the square-based pyramid with the smallest surface area among those with a volume of 1. This is a messy elementary calculus problem that involves defining variables for the length of the base side, the height of the pyramid, and the altitude of the triangle face. By setting the volume equal to 1 and rearranging the equations, we can find the minimum surface area by solving for the value of x. Finally, we can use this value to calculate the exact surface area of the pyramid. It is important to carefully work through the steps to fully understand the process.
  • #1
arcnets
508
0
Among all square-based pyramids which have volume V = 1, which one has the smallest surface area?
 
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  • #2
This is a messy elementary calculus problem. Define:
a= length of base side
h= height of pyramid
t= altitude of triangle face
from this t2=h2+(a/2)2

Then

V (volume) = a2h/3
S (surface area) =a2+2at

Set V=1, use the above formula for t, and let x=a2, we get:

S=x+(36/x+x2)1/2

S'=1+(x-18/x2)/(36/x+x2)1/2

Set S'=0, we get x3=9/2

To verify that this is a minimum (not maximum or horizontal inflection), observe that:
x near 0, S approx 6/x1/2,
x gets large, S approx 2x.

Finally:
S=(9/2)1/3+((9/2)2/3+36(2/9)1/3)1/2

I suggest you work this through to understand how it goes.
 
  • #3
Wow, mathman. Thanks.
I don't seem to understand this step:

S=x+(36/x+x2)1/2
S'=1+(x-18/x2)/(36/x+x2)1/2

Shouldn't it read

S'=1+(x-18/x2)/(36/x+x2)-1/2

...or something?
 
  • #4
You're right. I had the minus when I was doing it (pencil and paper), but I neglected to type it in. However, the final result still stands.
 
  • #5
Er, isn't it right as is?

dividing by z1/2 is the same as multiplying by z-1/2
 
  • #6
Oops! Yes. I overlooked the /.
 
  • #7
Hurkl got it right. When I read arcnets comment, I also forgot I had put in the /.
 

1. What is a square-based pyramid?

A square-based pyramid is a 3-dimensional shape with a square base and four triangular faces that meet at a point, known as the apex.

2. How do you calculate the surface area of a square-based pyramid?

The surface area of a square-based pyramid can be calculated by finding the area of the base (length x width) and adding the areas of the four triangular faces (1/2 x base x height) together.

3. What is the formula for finding the surface area of a square-based pyramid?

The formula for finding the surface area of a square-based pyramid is SA = B + 4(1/2 x base x height), where SA is the surface area, B is the area of the base, and base and height refer to the dimensions of the triangular faces.

4. Can the surface area of a square-based pyramid be calculated using only the length of the edges?

Yes, the surface area of a square-based pyramid can be calculated using only the length of the edges. The formula for this is SA = 2(√3 x edge length)^2.

5. How does the surface area of a square-based pyramid compare to other types of pyramids?

The surface area of a square-based pyramid is typically smaller than other types of pyramids with the same base area, such as rectangular or triangular pyramids. This is because the square base allows for less surface area for the same volume.

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