Grain is falling from a chute onto the ground, forming a conicalpile whose diameter is always three times its height. How high isthe pile (to the nearest hundredth of a foot) when it contains 1000ft^3 of grain?

Adrianna Macias

Adrianna Macias

Answered question

2022-07-25

Grain is falling from a chute onto the ground, forming a conicalpile whose diameter is always three times its height. How high isthe pile (to the nearest hundredth of a foot) when it contains 1000 f t 3 of grain?

Answer & Explanation

penangrl

penangrl

Beginner2022-07-26Added 17 answers

You'll need the equation for the volume of a cone, which is:
V = 1 / 3 π r 2 h where h is the height of the cone,and r is the radius.
We know that the volume V = 1,000 and that diameter = 3h. Now diameter also = 2 r.
so 3h =2r
Divide both sides by 2 to get: (3/2) h =r.
Now, wherever you see an r in the volume equation,put 3/2 h there instead.
And instead of the V put in 1,000
1 , 000 = 1 / 3 π ( 3 / 2 h ) 2 h
Now you can simplify by squaring the 3/2 andthe h.
1 , 000 = 1 / 3 π ( 9 / 4 ) h 2 h or 1 , 000 = 1 / 3 π ( 9 / 4 ) h 3
Now you can solve for height h.
If you multiply both sides by 3, you have 3 , 000 = π ( 9 / 4 ) h 3
Then divide by π to get 3 , 000 / π = ( 9 / 4 ) h 3
Now multiply both sides by 4 / 9.
3 , 000 ( 4 ) / ( 9 π ) = h 3
Now you have to type the left side into yourcalculator, and then take the cube root of the answer, to findh.
424 = h 3
or (424)(1/3) = h = 7.5
valtricotinevh

valtricotinevh

Beginner2022-07-27Added 3 answers

Volume of a cone is represented by V = 1 3 π r 2 h
You are given the final volume of 1000 f t 3 so you know that.
You are told that the diameter is always 3 times the height. If you think of the cone as a triangle, the base would be thediameter and if you draw a dotted line from the tip of the triangleto the middle point of the base, that would be the height. Let's just call the height x. If the height is x then thediameter must be 3x. For the equation above, you need theheight and radius, so just divide the diameter by 2. Plug allthe numbers into the equation and solve for x:
V = 1 3 π ( 3 2 x ) 2 x
Your x should be something around 7.515 ft.
Round that to the nearest hundredth to get 7.52 ft.

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