Having a bit of a struggle over here...
Grain is ejected from a chute at the rate of 0.1 metres cubed a minute and is forming a heap on a flat horizontal floor. The heap is in the form of a circular cone of semi-vertical angle 45 degrees. Find the rate, in metres per minute, at which the height of the cone i s increasing at the instant 3 minutes after the opening of the chute. Answer is 0.0732.
Any help would be appreciated
Edit: Jake just beat me to a solution, but I finished it, so I'll post it anyway. Maybe it will be equally useful, we used the same method
Hey there! This is a
chain rule question. These questions can be gnarly, but they all have a very similar approach.
First, let's take the question and break it down into some equations. We are told that the cone is being formed from a flow of 0.1 cubic metres per minute. That is, if we take V to be in cubic metres and t to be in minutes:
Now, we need a second formula to apply the chain rule. Remember the formula for a volume of a cone:
Now this would be great, except we don't want the radius in there. So, to remove it, we use the angle that they gave us. Draw the cone, and you'll be able to see that through some right-angled trigonometry, we can deduce:
Or, you might just be able to tell from the info they give you, both are fine! So, that volume formula becomes:
Now we apply the chain rule:
Now to find the final answer, we require the height at 3 minutes. At 3 minutes, we know the volume is 0.03, therefore:
Substituting this into the formula above will give you the answer!