Rates of Change; Help required!!! (1 Viewer)
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Makematics
Well-Known Member
just cause i'm bored.
}~$V$=\pi\int_{0}^{h}x^2~dy\\\\~~~~~~~=\pi\int_{0}^{h}(20y-y^2)~dy\\\\~~~~~~~=\pi[10y^2-\frac{y^3}{3}]_{0}^{h}\\\\~~~~~~~=\pi(10h^2-\frac{h^3}{3})\\\\~~~~~~~=10\pi h^2-\frac{\pi h^3}{3}\\\\~~~~~~~=\frac{\pi h^2}{3}(30-h)~$cm$^3\\\\$In order to find the surface area,~we need to use the formula$~~A=\pi r^2\\\\$When$~y=h~$,~let$~x=r\\\\r^2+h^2-20h=0\\\\r^2=20h-h^2\\\\S=\pi r^2\\\\~~~=\pi(20h-h^2)\\\\~~~=\pi h(20-h)~$cm$^2)
(\alpha )}~$We are given that$~\frac{dV}{dt}=50\\\\V=10\pi h^2-\frac{\pi h^3}{3}\\\\\frac{dV}{dh}=20\pi h-\pi h^2\\\\h=5,~\frac{dV}{dh}=100\pi-25\pi\\\\~~~~~~~~~~~~~~~=75\pi\\\\\frac{dh}{dV}= \frac{1}{75\pi}\\\\\frac{dh}{dt}=\frac{dV}{dt} \times \frac{dh}{dV}\\\\~~~~~=50\times\frac{1}{75\pi}\\\\~~~~~=\frac{2}{3\pi}\\\\~~~~~\approx 0.21~$cm$/$sec.$)
}~S=20\pi h-\pi h^2\\\\\frac{dS}{dh}=20\pi-2\pi h\\\\h=5,~\frac{dS}{dh}=20\pi-10\pi\\\\~~~~~~~~~~~~~~~=10\pi\\\\$From$~\mathbf{(\alpha )}$,we know that$~\frac{dh}{dt}=\frac{2}{3\pi}\\\\\frac{dS}{dt}=\frac{dS}{dh} \times \frac{dh}{dt}\\\\~~~~~=10\pi\times\frac{2}{3\pi}\\\\~~~~~=\frac{20}{3}\\\\~~~~~\approx 6.7~$cm$^2/$sec.$)
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HeroicPandas
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40)
(i) Alternate angles somewhere
Then apply a specific trig ratio that involves x and 200
(ii)Differntiate theta with respect to x (utilising equation found in part (i))
^2})
We want TIME, and to get rid of x...
Apply chain rule:
^2})
We know that the aeroplane is approaching a speed of 80m/s
Since its apporaching in the negative direction (left) (always think mathematically)
Sub in,
^2})
Simplfiy and u will get the answer if u make d(theta)/dt the subject!
(iii)
"Find the rate in which THETA is CHANGING when x = pi/4 "
Simply means, change of theta in terms of time
 = ?)
This is hard to find... and we did part (ii) for a specific reason
All we have to do is sub in theta = pi/4 and we will complete this question
But there is no theta........, there is a DIRTY x.........
Let us try and eliminate it!
So we must find a relationship of THETA and X
*Looks at part (i)*, oh there it is...
Make 'x' the subject of this relation, square it, sub into
Then finally sub in theta
OR u can sub in theta as pi/4 into and this value of x will correspond to that value of theta (pi/4)
Therefore when u sub this 'x' into ur, its like subbing in theta = pi/4
(i) Alternate angles somewhere
Then apply a specific trig ratio that involves x and 200
(ii)Differntiate theta with respect to x (utilising equation found in part (i))
We want TIME, and to get rid of x...
Apply chain rule:
We know that the aeroplane is approaching a speed of 80m/s
Since its apporaching in the negative direction (left) (always think mathematically)
Sub in,
Simplfiy and u will get the answer if u make d(theta)/dt the subject!
(iii)
"Find the rate in which THETA is CHANGING when x = pi/4 "
Simply means, change of theta in terms of time
This is hard to find... and we did part (ii) for a specific reason
All we have to do is sub in theta = pi/4 and we will complete this question
But there is no theta........, there is a DIRTY x.........
Let us try and eliminate it!
So we must find a relationship of THETA and X
*Looks at part (i)*, oh there it is...
Make 'x' the subject of this relation, square it, sub into
Then finally sub in theta
OR u can sub in theta as pi/4 into
Therefore when u sub this 'x' into ur
Makematics
Well-Known Member
looks like you too are bored Heroic 
haha i like how you guide him through
haha i like how you guide him through
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HeroicPandas
Heroic!
- Joined
- Mar 8, 2012
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lollooks like you too are bored Heroic