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!
The general formula for an exponential function is
In this case, b is the base which is any real number greater than 0 and is not 1. Additionally, the domain of an exponential function is all real numbers due to the fact that the exponential function is defined by real numbers. While the range is all real numbers greater than zero.
Area=legnth times width
42=legnth times width
a quilt is normally for a big bed probably at leas 3 feet wide
factor 42
42=2 times 3 tiesm 7
possible values are
2 by 21
3 by 14
6 by 7
I would go with the 6 by 7 format
Answer:
3x^2 + 3xy + 6y
Step-by-step explanation:
Answer:
72 parts
Steps:
Convert the 4 hours into minutes. There are 60 minutes in an hour. Therefore, 4 hours = 240 minutes.
Divide 240 minutes by 40 minutes. This equals 6. Therefore, the machinist can produce the 12 parts 6 times in 4 hours.
Finally, multiply the 12 parts by the 6 times he can produce. Therefore, the machinist can produce 72 parts in 4 hours.
