Rate of Change Q (1 Viewer)

rand_althor · Aug 17, 2015

$$Sand is pouring from a chute at the rate of 12 m^3$/$min. The falling sand forms a conical pile on the ground, whose altitude is always equal to the diameter of the base. Show that when the altitude is $h$ m, the volume $V$ m$^3$ of sand in the pile is given by $V=\frac{\pi h^3}{12}$.$

I think the pile is in the shape of a concave down parabola. Haven't managed to find the volume. Can someone help?

Zlatman · Aug 17, 2015

"conical pile"

From that, we can say that the pile is in the shape of a cone, with altitude (height) h and diameter h.

Using the volume of a cone:

$\begin{align*} V &= \frac{\pi r^2 h}{3}\\ V &= \frac{\pi [\frac{h}{2}]^2 h}{3} \\ V &= \frac{\pi h^3}{12} \end{align*}$

kawaiipotato · Aug 17, 2015

rand_althor said:
$$Sand is pouring from a chute at the rate of 12 m^3$/$min. The falling sand forms a conical pile on the ground, whose altitude is always equal to the diameter of the base. Show that when the altitude is $h$ m, the volume $V$ m$^3$ of sand in the pile is given by $V=\frac{\pi h^3}{12}$.$

I think the pile is in the shape of a concave down parabola. Haven't managed to find the volume. Can someone help?

Wouldn't it just be a cone, as specified by the question?
$$ V = $\frac{1}{3}\pi r^2 h,$ where $r = \frac{h}{2}$

rand_althor · Aug 17, 2015

Ah okay, I misinterpreted the question. Thanks for the help!

rand_althor · Aug 18, 2015

Can someone help with this one too?

$$Two men P and Q walk along roads OA, OB at right angles to one another. P walks at 6 km/h along OA and Q at 5 km/h along OB; they start simultaneously from O. Show that the rate at which the distance between the 2 men is increasing is constant, and find this rate.$

braintic · Aug 18, 2015

rand_althor said:
Can someone help with this one too?

$$Two men P and Q walk along roads OA, OB at right angles to one another. P walks at 6 km/h along OA and Q at 5 km/h along OB; they start simultaneously from O. Show that the rate at which the distance between the 2 men is increasing is constant, and find this rate.$

I'm afraid this is not even a calculus question. It is just addition of vectors.

InteGrand · Aug 18, 2015

rand_althor said:
Can someone help with this one too?

$$Two men P and Q walk along roads OA, OB at right angles to one another. P walks at 6 km/h along OA and Q at 5 km/h along OB; they start simultaneously from O. Show that the rate at which the distance between the 2 men is increasing is constant, and find this rate.$

$Let $O$ be the origin, and let $OA$ be along the positive $x$-axis (measured in km), and $OB$ along the positive $y$-axis (measured in km). Let $t\geq 0$ be the time in hours. Then person $P$ has $y(t)=0$ and $x(t)=6t$ for all $t\geq 0$. Person $Q$ has $x(t)=0$ and $y(t)=5t$ for all $t\geq 0$. Then by Pythagoras' theorem, the distance between the two at time $t$ is $D(t) =\sqrt{(6t)^2 + (5t)^2}=\sqrt{36t^2 + 25t^2}=\sqrt{61}t$ (as $t\geq 0$). So we have $\frac{\mathrm{d}D}{\mathrm{d}t} = \sqrt{61}$, i.e. the distance between the two men is increasing at a constant rate of $\sqrt{61}$ km/h.$

rand_althor · Aug 18, 2015

braintic said:
I'm afraid this is not even a calculus question. It is just addition of vectors.

How did you recognise that? I approached it like it was strictly rates of change, so I was stuck for a while on this one.

Thanks for the help InteGrand!

braintic · Aug 18, 2015

rand_althor said:
How did you recognise that? I approached it like it was strictly rates of change, so I was stuck for a while on this one.

Thanks for the help InteGrand!

Because that scenario is exactly what addition of velocity vectors is all about.

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Rate of Change Q (1 Viewer)

rand_althor

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Zlatman

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kawaiipotato

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rand_althor

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rand_althor

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braintic

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InteGrand

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rand_althor

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braintic

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