MHB What is the solution to the Petal Area Problem?

[DeusAbscondus]

I've posted an attachment with the problem set forth,
but I can't seem to get the next part loaded within my set upload limit.

Any help would be appreciated.

D'Abs

 

[MarkFL]

I think you need to write the volume V in cm³ as:

$\displaystyle V=15\cdot55^2-4\cdot100\cdot15\int_0^{\frac{1}{4}}\frac{\sqrt{x}}{2}-4x^2\,dx$

Do you see why?

 

[DeusAbscondus]

I think you need to write the volume V in cm³ as:

$\displaystyle V=15\cdot55^2-4\cdot100\cdot15\int_0^{\frac{1}{4}}\frac{\sqrt{x}}{2}-4x^2\,dx$

Do you see why?

But would that not be equivalent to multiplying the entire volume of the block by the volume of the cut out 3d flower?

$$1. 15\cdot 55^2 \text {is the volume of the uncut prism and the integrand is the curve }y=\frac{\sqrt{x}}{2}-y=4x^2$$

I will have to take the current screenshot down and put the next one up; awkward, but space won't allow for both at the same time.

thanks heaps for helping,
D'abs

 

[MarkFL]

I am taking the volume of the prism and subtracting from it the volume of the leaf design which is cut through it.

 

[DeusAbscondus]

I am taking the volume of the prism and subtracting from it the volume of the leaf design which is cut through it.

Ah, that is my problem: I can 't see why one may not derive the volume of design without reference to the volume of the block from which it is "hewn", except that datum of depth: in this case 15cm.
The references to width and length, to the volume of entire block, seem to be entirely beside the point:
surely all you need is:
- the area of one leaf;
- knowledge that each leaf is identical AND that the front and back of the design are identical; and finally,
- depth of block

then it is simply:
$$4\cdot 15cm \int^{1/4}_0 [f(x)-g(x)] dx $$

Isn't it?

 

[MarkFL]

I need to amend my previous statement to say

$\displaystyle V=15\cdot55^2-4\cdot100^2\cdot15\int_0^{\frac{1}{4}}\frac{\sqrt{x}}{2}-4x^2\,dx$

I get 32,875 cm³.

 

[MarkFL]

Ah, that is my problem: I can 't see why one may not derive the volume of design without reference to the volume of the block from which it is "hewn".

You could, but it would be much more difficult I think.

 

[DeusAbscondus]

You could, but it would be much more difficult I think.

My notes say:

"The volume of concrete in the block:
= face area x depth
= 0.03288
Therefore, the concrete required is about 0.03288 m^3 or 33000cm^3"

This is what i mean: "face area x depth" should mean, on the face of it, the result one obtains from integrating the difference between the given functions by the depth ie: 15.

Bringing in the volume of the entire square just confuses me.
Still confused.
Thanks anyway,
D'abs

 

[MarkFL]

If you take the area of the leaf, multiply it by the depth of the block, you have the volume removed from the block. This is why I took the volume of the block, then subtracted that volume which was removed to get the remaining volume.