Question

A flower vase, in the form of a hexagonal prism, is to be filled with 512 cubic inches of water. Find the height of the water if the wet portion of the flower vase and its volume are numerically equal.

I need a solution for this problem, kindly help me. The correct answer in our book is 34.88 inches

:)))

Answer 1

@Kainui

Answer 2

@robtobey

Answer 3

@phi

Answer 4

Can you draw a picture and label it and use those to try to make equations. Try breaking it up into triangles possibly if you think that'll help you.

Answer 5

The problem states that the volume of the vase is 512, and the "wetted" surface area is also 512.

If we call the area of the base B and the height h
the volume is Bh, and Bh= 512

The surface area will the the area of the base plus the 6 sides. Call the length of 1 side of the hexagon "s". Then the surface area will be
6*s*h (6 sides with width s and height h) plus the area of the base (which we call B)

the area of a hexagon is B= sqrt(27)/2 s^2

Answer 6

we can set up a messy cubic equation to solve for h.
From Bh=512 we find
\[ B= \frac{512}{h}\]
From B= sqrt(27)/2 s^2, we find
\[ s= \sqrt{\frac{2B}{\sqrt{27}}}= \sqrt{\frac{1024}{h\sqrt{27}}}\]
surface area = 512
6 s h + B = 512
replace s and B to get
\[ 6 h\sqrt{\frac{1024}{h\sqrt{27}}}+\frac{512}{h}=512\]
You can simplify that to
\[ \sqrt[4]{3} h^{\frac{3}{2}} -8h +8=0\]
you will need a graph calculator or wolfram to solve it.

Answer 7

The attachment calculates h to twenty digits.
h = 34.86094741770796904

Answer 8

Yes, and the "book answer" 34.88 seems a little bit off. Also, I guess we rule out the other solution h=1.22 inches by saying this is too short to be a flower vase.

Using geogebra, here is a plot that gives the possible answer. It gives a rough idea of the solution.

Answer 9

@phi

A series of plots is attached.