Ever so often I get bored during classes and decide to do some math.
I haven't done this kind of math since high school (only 2 years ago, but it's good to keep your mind fresh), so I have the feeling I forget stuff every once in a while. My teachers aren't of much help, they know seemingly less maths than I do.
Now, for the question at hand:
We start with a square pyramid(ABCDE) with base dimensions of 10 cm by 10 cm and a height of 20 cm.
We want to know what the largest possible volume of an inscribed cuboid is (inside the pyramid obviously)
This seemed a bit... too logical... did I do it right?
Edit: Gonna make a drawing with paint soon
I haven't done this kind of math since high school (only 2 years ago, but it's good to keep your mind fresh), so I have the feeling I forget stuff every once in a while. My teachers aren't of much help, they know seemingly less maths than I do.
Now, for the question at hand:
We start with a square pyramid(ABCDE) with base dimensions of 10 cm by 10 cm and a height of 20 cm.
We want to know what the largest possible volume of an inscribed cuboid is (inside the pyramid obviously)
Code:
Pyramid(ABCDE) = 10 cm x 10 cm x 20 cm
f(x) = 4x * (10 - 2x) * x
= 4x * (10x - 2x^2)
= 40x^2 - 8x^3 [color="gray"]= -8x^3 + 40x^2[/color]
f'(x) = -24x^2 + 80x
-24x^2 + 80x = 0
x(-24x + 80) = 0
x = 0 v -24x + 80 = 0
24x = 80
x = 80/24 = 3 + 1/3
f(3 + 1/3) = 4 * (3 + 1/3) * (10 - 2 * (3 + 1/3)) * (3 + 1/3)
= (13 + 1/3) * (3 + 1/3) * (3 + 1/3)
= 148 + 4/27 cm^3
Cube(ABCDEFGH) ≈ 3.33 cm x 3.33 cm x 13.33 cm
Volume(ABCDEFGH) ≈ 148.148 cm^3
Edit: Gonna make a drawing with paint soon
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