A small pyramid (pointing up) has a square base that measures 15 feet on each side and is 9 feet tall. and is pouring into the pyramid at a rate of .5 ft^3/sec. Express the volume V of the sand in the pyramid as a function of h, the height of the sand in the pyramid.
I know that the volume of a pyramid is V = 1/3(h)(b^2). How would I find the relationship between the volume of the sand and the height of the sand given the volume of the pyramid?
If I understand what you mean, sand is being poured into the pyramid. And you want to find the volume of the sand in the pyramid as a function of the height, h, of the sand in the pyramid at any time t.
That means the rate of pouring is not necessary.
I assume you are treating the sand as a liquid, meaning its top or surface inside the pyramid is flat or horizontal.
Okay, the sand in the pyramid then is a frustum or truncated pyramid, whose bottom base, B, is 15^2, whose height is h, whose top base, b, is, say, x^2.
V = (h/3)[B +sqrt(Bb) +b] -------volume of a frustum
Draw the figure on paper.
Cut a cross section of the pyramid vertically from the apex to half of the 15^2 base such that the base is cut into two (15ft by 7.5ft)'s. So we have an isosceles triangle whose base is 15ft and whose height is 9ft.
Draw the 9ft altitude. Two congruent right triangles are formed. Any of which has a base 7.5ft and an altitude 9ft.
Draw the height of the sand, h.
We concentrate on one of those congruent right triangles.
There is a smaller right triangle inside, whose base is (x/2) and whose altitude is (9-h).
The two mentioned right traiangles are similar, so proportional
By proportion,
9/7.5 = (9-h)/(x/2)
We want to retain h, so we solve for x in terms of h,
Cross multiply,
9(x/2) = 7.5(9-h)
x = (7.5)(9-h)/(4.5)
x = 15 -1.667h ft.
So, top base, b = x^2 = (15 -1.667h)^2.
And so, volume of the sand,
V = (h/3){15^2 +sqrt[(15^2)(15 -1.667h)^2] +(15 -1.667h)^2}
V = (h/3){225 +15(15 -1.667h) +(225 -50h +2.779h^2)}
V = (h/3){225 +225 -25h +225 -50h +2.779h^2}
V = (h/3){675 -75h +2.779h^2}
V = 0.926h^3 -25h^2 +225h cu.ft. -------------------answer.
Check, when h=9ft, or when the pyramid is full of sand,
V of sand = 0.926(9^3) -25(9^2) +225(9) = 675.054 cu.ft.
V of pyramid = (9/3)(15^2) = 675 cu.ft
Almost the same, so the answer is Ok.