Mathematics, 20.12.2019 06:31 kirstenturkington

Amachine starts dumping sand at the rate 20 m^3/min, forming a pile in the shape of a cone. the height of the pile is always twice the length of the base diameter. after 5 minutes, how fast is the base radius increasing? include steps.

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Answer from: yazzyv05

Base radius is increasing by 2.88 m after 5 minutes

Step-by-step explanation:

Given:-

Height of cone (h) =2 $\times$ base diameter(d)

$h=2\times d$

Machine starts dumping sand at the rate of 20 $m^{3}/min$

So,

Volume of cone in 1 min = 20 $m^{3}$

Now, Volume of cone in 5 mins = Volume of cone in 1 min $\times 5$

Volume of cone in 5 mins = 20 $\times 5$

Volume of cone in 5 mins = 100 $m^{3}$ ---------(equation 1)

Now, Let base diameter of cone = b

height of the cone = h

formula for volume of cone is,

Volume of cone (V) = $\pi\times r^{2} \times\frac{h}{3}$

$V=\pi\times( \frac{d}{2} )^{2}\times(\frac{2\times d}{3})$ ----(since r= $\frac{d}{2}$ )

$V=\pi\times\frac{d^{2} }{4} \times \frac{2d}{3}$

$V=\frac{\pi\times d^{3} }{6}$ ---------(equation 2)

Now substituting equation 1 in equation 2,

$100=\frac{\pi\times d^{3} }{6}$

$d^{3}=\frac{100\times 6}{3.14}$ ---------(as $\pi=3.14$ )

$d^{3} =\frac{600}{3.14}$

$d^{3} =191.08$

By cube rooting both the sides we get,

$\sqrt[3]{d} =\sqrt[3]{191.08}$

$d=5.76 \ m$ ----------------(diameter)

$as\ r=\frac{d}{2}$

$r=\frac{5.76}{2}$

$r=2.88\ m$ ---------------(radius of base at 5 mins)

Therefore base radius is increasing by 2.88 m after 5 minutes

Answer from: Quest

3, 12, 48, 384, 1536, 6144, 24576, 98304

step-by-step explanation:

just keep timing it by 4.

Answer from: Quest

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

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