Answer:
The volume is increasing at a rate of 16286 in³/min.
Step-by-step explanation:
a) The volume of a cone is given by:
Where:
r: is the radius
h: is the height
The rate of change of the volume can be calculated by using the chain rule:
![\frac{dV}{dt} = \frac{\pi}{3}[\frac{dh}{dt}r^{2} + h\frac{d(r^{2})}{dt}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B%5Cpi%7D%7B3%7D%5B%5Cfrac%7Bdh%7D%7Bdt%7Dr%5E%7B2%7D%20%2B%20h%5Cfrac%7Bd%28r%5E%7B2%7D%29%7D%7Bdt%7D%5D%20)
Since h = 4/3 r we have:
![\frac{dV}{dt} = \frac{\pi}{3}[\frac{d(\frac{4r}{3})}{dt}r^{2} + \frac{4r}{3}\frac{d(r^{2})}{dt}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B%5Cpi%7D%7B3%7D%5B%5Cfrac%7Bd%28%5Cfrac%7B4r%7D%7B3%7D%29%7D%7Bdt%7Dr%5E%7B2%7D%20%2B%20%5Cfrac%7B4r%7D%7B3%7D%5Cfrac%7Bd%28r%5E%7B2%7D%29%7D%7Bdt%7D%5D%20)
![\frac{dV}{dt} = \frac{4\pi}{9}[\frac{dr}{dt}r^{2} + r\frac{d(r^{2})}{dt}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B4%5Cpi%7D%7B9%7D%5B%5Cfrac%7Bdr%7D%7Bdt%7Dr%5E%7B2%7D%20%2B%20r%5Cfrac%7Bd%28r%5E%7B2%7D%29%7D%7Bdt%7D%5D%20)
![\frac{dV}{dt} = \frac{4\pi}{9}[\frac{dr}{dt}r^{2} + 2r^{2}\frac{dr}{dt}]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B4%5Cpi%7D%7B9%7D%5B%5Cfrac%7Bdr%7D%7Bdt%7Dr%5E%7B2%7D%20%2B%202r%5E%7B2%7D%5Cfrac%7Bdr%7D%7Bdt%7D%5D%20)
(1)
With:
= 3 in/min
And by entering the above values into equation (1) we have:
![\frac{dV}{dt} = \frac{4\pi}{9}[(3 in/min)*(36 in)^{2} + 2*(36 in)^{2}*3 in/min]](https://tex.z-dn.net/?f=%20%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B4%5Cpi%7D%7B9%7D%5B%283%20in%2Fmin%29%2A%2836%20in%29%5E%7B2%7D%20%2B%202%2A%2836%20in%29%5E%7B2%7D%2A3%20in%2Fmin%5D%20)
Therefore, the volume is increasing at a rate of 16286 in³/min.
I hope it helps you!
Answer:
the first one
Step-by-step explanation:
To solve the equation for x, first, we add 16 from both sides of the equation:
Now, we divide by 6:
Finally, we get that:
Answer:
Answer:
smart dot company would be c=.50t+12
communication plus would be c=2.50t
Step-by-step explanation:
smart dot starts with $12 a month and adds $0.50 for each hour of internet use. communication plus has no starting fee so it just adds $2.50 for each hour of internet use.
So then for the table you would put the number given for time in for the letter t and find what c would be. This would then find your cost for that amount of hours. To get you started smart dot company starts off with 0 hours so put that in for t being c=0.50(0)+12 and c=12. Communication plus also starts with 0 so it would be c=2.50(0) and c=0
Then you graph once you have both values you can graph having x=time and y=cost
Let me know if this helps!
