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Triangular Prism

  • #1
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A trinagular prism is 10 m long and has an equilaterlal trinagle for its base. WAter is added at a rate of 2m^3/min. Determine the rate of change of the water level when the water is [itex] \sqrt{3} [/itex] m deep.

ok so the volume of a prism is
[tex] V = \frac{1}{2} lwh [/tex] .... (1)
l is the length
w is the width
h is the height

now dl/dt = 0 because the length of the column of water is constant

to find a relation between h and w i got this because the triangle is an equilaterla triangle

[tex] h = \frac{\sqrt{3}}{2} w[/tex] ... (2)

and it follos that
[tex] \frac{dh}{dt} = \frac{\sqrt{3}}{2}\frac{dx}{dt} [/tex] ... (3)

now subbing 1 into 2

[tex] V = \frac{1}{\sqrt{3}} lh^2 [/tex]

[tex] \frac{dV}{dt} = \frac{l}{\sqrt{3}} 2h \frac{dh}{dt} [/tex]

now here's the problem ... what is h??
h does not represent the depth of the water, does it??
it reprsnts the height of the remainder of the prism that has not been filled iwht water. so really

[tex] \frac{dh_{water}}{dt} = \frac{dh_{empty part}}{dt} [/tex]

is it reasonable to say that??

thank you for all your input!!
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
955
A trinagular prism is 10 m long and has an equilaterlal trinagle for its base. WAter is added at a rate of 2m^3/min. Determine the rate of change of the water level when the water is [itex] \sqrt{3} [/itex] m deep.

ok so the volume of a prism is
[tex] V = \frac{1}{2} lwh [/tex] .... (1)
l is the length
w is the width
h is the height

now dl/dt = 0 because the length of the column of water is constant

to find a relation between h and w i got this because the triangle is an equilaterla triangle

[tex] h = \frac{\sqrt{3}}{2} w[/tex] ... (2)

and it follos that
[tex] \frac{dh}{dt} = \frac{\sqrt{3}}{2}\frac{dx}{dt} [/tex] ... (3)

now subbing 1 into 2

[tex] V = \frac{1}{\sqrt{3}} lh^2 [/tex]

[tex] \frac{dV}{dt} = \frac{l}{\sqrt{3}} 2h \frac{dh}{dt} [/tex]

now here's the problem ... what is h??
h does not represent the depth of the water, does it??
it reprsnts the height of the remainder of the prism that has not been filled iwht water. so really

[tex] \frac{dh_{water}}{dt} = \frac{dh_{empty part}}{dt} [/tex]

is it reasonable to say that??

thank you for all your input!!
h is the height of the triangle so it is the depth of the water- you triangle has its vertex downward, remember?
 
  • #3
1,444
2
h is the height of the triangle so it is the depth of the water- you triangle has its vertex downward, remember?
ahhh true

the book was rather deceptive in taht it drew the triangle right side up

but that wouldnt make a difference??
 
  • #4
11
0
Yes, the depth of water is the height of the prism and which is root 3 as given.
 
  • #5
1,331
45
I doubt stunner5000pt is still working on this problem after 5 years.
 

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