## Reyjuanx10 3 years ago A flower vase in the form of a hexagonal prism , is to be filled with 512 cubic in of water . Find the height of the water if the wet portion of the flower vase and its volume area numerically equal

1. mayankdevnani

It is a hexagonal prism, and I presume a regular hexagon. The "wet portion" of the prism is the base and the inner sides (not the lid nor anywhere on the outside) Let the sides of the hexagon be length t, and its height be h The area of the hexagon is A = (3√3 / 2) * t^2 The volume of the prism is therefore hA = h(3√3 / 2) * t^2 The perimeter around the prism is equal to 6t The area of the "wet portion" is therefore 6th + (3√3 / 2) * t^2 We require the volume to equal this area, thus h(3√3 / 2) * t^2 = 6th + (3√3 / 2) * t^2 Divide through by t(3√3 / 2) to get h * t = (2 / 3√3) * 6h + t (4h / √3) + t - ht = 0 = (4h / √3) + t(1 - h) (4h / √3) = -t(1 - h) (4h / √3) = t(h - 1) (4h / √3) / (h - 1) = t We know the volume is 512, so hA = h(3√3 / 2) * t^2 = 512 h(3√3 / 2) * ((4h / √3) / (h - 1))^2 = 512 h(3√3 / 2) * (4h / √3)^2 / (h - 1)^2 = 512 h(3√3 / 2) * (16h^2 / 3) / (h^2 - 2h + 1) = 512 h(√3) * (8h^2) / (h^2 - 2h + 1) = 512 √3.8h^3 = 512 (h^2 - 2h + 1) √3.8h^3 - 512h^2 + 1024h - 512 = 0 √3.h^3 - 64h^2 + 128h - 64 = 0 http://ph.answers.yahoo.com/question/index?qid=20100505051316AAXv8po

2. Reyjuanx10

How can I get 38.8 as the answer

3. sauravshakya

But none of the solution of √3.h^3 - 64h^2 + 128h - 64 = 0 is h=38.8

4. sauravshakya

Solution of √3.h^3 - 64h^2 + 128h - 64 = 0 are h=0.87 , h=1.22 and h=34.9

5. Reyjuanx10

I meant is 34.88 haha

6. Reyjuanx10

typoo how did you get the sol'ns ?

7. sauravshakya

u need to solve for h √3h^3 - 64h^2 + 128h - 64 = 0

8. Hares333

so @mayankdevnani is right.. @sauravshakya

9. sauravshakya

I guess yes...

10. Hares333

guess?? but answer is coorect

11. Hares333

*correct

12. Reyjuanx10

√3h^3 - 64h^2 + 128h - 64 = 0 how did you arrive to three answers ? quadratic eqn?

13. sauravshakya

I see no error in the logic...

14. sauravshakya
15. sauravshakya

And since, t=(4h / √3) / (h - 1) h=0.87 is rejected because t cant be negative