Asked by sara

Find the volume of the solid whose base is the region in the first quadrant bounded by y=x^5, y=1 and the y-axis and whose cross -sections perpendicular to the y-axis are equilateral triangles.

I don't know what the bounds for the integral would be when I try to calculate the volume. I put everything in terms of x and so I got the volume to be
V=1/8(40/7x^(7/5)-5/4x^(8/5)),
which then needs to be evaluated by the bounds. I am not sure if my process is right and help would be appreciated.
Answered by oobleck
each triangle has a base of length x, making its area √3/4 x^2
since x = y^(1/5) we have
v = ∫[0,1] √3/4 y^(2/5) dy = √3/4 * 5/7 y^(7/5) [0,1] = 5√3/28
Answered by sara
okay thank you so much for the help!
Answered by sara
i just tried to hand this in on my homework site and it said it was incorrect
Answered by sara
also I got that each triangle would have a sidelength of x=5root(y) because I was told to set it in terms of y.
Answered by Anonymous
how about writing that as (5/7) sqrt (3/4)
or maybe sqrt (25/196
Answered by Anonymous
I mean sqrt (75/196)
Answered by sara
if you meant trying the answer as sqrt(75/196 that is wrong
Answered by Anonymous
I did it again and agreed with oobleck
Answered by Anonymous
( 5√3 ) / 28
Answered by Anonymous
It is so easy to get mixed up with top and bottom online :(
Answered by sara
oh no that is okay. let me try it
Answered by sara
yes that is correct. could you explain your process on how you solved the question so I can recheck my own work and see where I went wrong
Answered by oobleck
you should know by now that the area of an equilateral triangle of side s is
(s^2 √3)/4
Break it into two 30-60-90 right triangles to see how.
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