i might be wrong about this one
94x689=64,766
if you round it to the nearest ten you get 90x690=62100
62100 is closer to 60,000 than it is to 70,000.
so you would round it down to 60,000.
120 = 15% x
15% = 0.15
x = 120 / 0.15 = 800
<span>49 17 g 143 225
</span><span>
you have g in here
is g is 9?
perfect squares
49 = 7^2
225 = 15^2</span>
Answer:
The rate of change of the height is 0.021 meters per minute
Step-by-step explanation:
From the formula
![V = \frac{1}{3}\pi r^{2}h](https://tex.z-dn.net/?f=V%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20r%5E%7B2%7Dh)
Differentiate the equation with respect to time t, such that
![\frac{d}{dt} (V) = \frac{d}{dt} (\frac{1}{3}\pi r^{2}h)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdt%7D%20%28V%29%20%3D%20%5Cfrac%7Bd%7D%7Bdt%7D%20%28%5Cfrac%7B1%7D%7B3%7D%5Cpi%20r%5E%7B2%7Dh%29)
![\frac{dV}{dt} = \frac{1}{3}\pi \frac{d}{dt} (r^{2}h)](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Cfrac%7Bd%7D%7Bdt%7D%20%28r%5E%7B2%7Dh%29)
To differentiate the product,
Let r² = u, so that
![\frac{dV}{dt} = \frac{1}{3}\pi \frac{d}{dt} (uh)](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Cfrac%7Bd%7D%7Bdt%7D%20%28uh%29)
Then, using product rule
![\frac{dV}{dt} = \frac{1}{3}\pi [u\frac{dh}{dt} + h\frac{du}{dt}]](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Bu%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20h%5Cfrac%7Bdu%7D%7Bdt%7D%5D)
Since ![u = r^{2}](https://tex.z-dn.net/?f=u%20%3D%20r%5E%7B2%7D)
Then, ![\frac{du}{dr} = 2r](https://tex.z-dn.net/?f=%5Cfrac%7Bdu%7D%7Bdr%7D%20%3D%202r)
Using the Chain's rule
![\frac{du}{dt} = \frac{du}{dr} \times \frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7Bdu%7D%7Bdt%7D%20%3D%20%5Cfrac%7Bdu%7D%7Bdr%7D%20%5Ctimes%20%5Cfrac%7Bdr%7D%7Bdt%7D)
∴ ![\frac{dV}{dt} = \frac{1}{3}\pi [u\frac{dh}{dt} + h(\frac{du}{dr} \times \frac{dr}{dt})]](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Bu%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20h%28%5Cfrac%7Bdu%7D%7Bdr%7D%20%5Ctimes%20%5Cfrac%7Bdr%7D%7Bdt%7D%29%5D)
Then,
![\frac{dV}{dt} = \frac{1}{3}\pi [r^{2} \frac{dh}{dt} + h(2r) \frac{dr}{dt}]](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Br%5E%7B2%7D%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20h%282r%29%20%5Cfrac%7Bdr%7D%7Bdt%7D%5D)
Now,
From the question
![\frac{dr}{dt} = 7 m/min](https://tex.z-dn.net/?f=%5Cfrac%7Bdr%7D%7Bdt%7D%20%3D%207%20m%2Fmin)
![\frac{dV}{dt} = 236 m^{3}/min](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20236%20m%5E%7B3%7D%2Fmin)
At the instant when ![r = 99 m](https://tex.z-dn.net/?f=r%20%3D%2099%20m)
and ![V = 180 m^{3}](https://tex.z-dn.net/?f=V%20%3D%20180%20m%5E%7B3%7D)
We will determine the value of h, using
![V = \frac{1}{3}\pi r^{2}h](https://tex.z-dn.net/?f=V%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20r%5E%7B2%7Dh)
![180 = \frac{1}{3}\pi (99)^{2}h](https://tex.z-dn.net/?f=180%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%2899%29%5E%7B2%7Dh)
![180 \times 3 = 9801\pi h](https://tex.z-dn.net/?f=180%20%5Ctimes%203%20%3D%209801%5Cpi%20h)
![h =\frac{540}{9801\pi }](https://tex.z-dn.net/?f=h%20%3D%5Cfrac%7B540%7D%7B9801%5Cpi%20%7D)
![h =\frac{20}{363\pi }](https://tex.z-dn.net/?f=h%20%3D%5Cfrac%7B20%7D%7B363%5Cpi%20%7D)
Now, Putting the parameters into the equation
![\frac{dV}{dt} = \frac{1}{3}\pi [r^{2} \frac{dh}{dt} + h(2r) \frac{dr}{dt}]](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Br%5E%7B2%7D%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20h%282r%29%20%5Cfrac%7Bdr%7D%7Bdt%7D%5D)
![236 = \frac{1}{3}\pi [(99)^{2} \frac{dh}{dt} + (\frac{20}{363\pi }) (2(99)) (7)]](https://tex.z-dn.net/?f=236%20%3D%20%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5B%2899%29%5E%7B2%7D%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20%28%5Cfrac%7B20%7D%7B363%5Cpi%20%7D%29%20%282%2899%29%29%20%287%29%5D)
![236 \times 3 = \pi [9801 \frac{dh}{dt} + (\frac{20}{363\pi }) 1386]](https://tex.z-dn.net/?f=236%20%5Ctimes%203%20%3D%20%5Cpi%20%5B9801%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20%28%5Cfrac%7B20%7D%7B363%5Cpi%20%7D%29%201386%5D)
![708 = 9801\pi \frac{dh}{dt} + \frac{27720}{363}](https://tex.z-dn.net/?f=708%20%3D%209801%5Cpi%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20%5Cfrac%7B27720%7D%7B363%7D)
![708 = 30790.75 \frac{dh}{dt} + 76.36](https://tex.z-dn.net/?f=708%20%3D%2030790.75%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%2076.36)
![708 - 76.36 = 30790.75\frac{dh}{dt}](https://tex.z-dn.net/?f=708%20-%2076.36%20%3D%2030790.75%5Cfrac%7Bdh%7D%7Bdt%7D)
![631.64 = 30790.75\frac{dh}{dt}](https://tex.z-dn.net/?f=631.64%20%3D%2030790.75%5Cfrac%7Bdh%7D%7Bdt%7D)
![\frac{dh}{dt}= \frac{631.64}{30790.75}](https://tex.z-dn.net/?f=%5Cfrac%7Bdh%7D%7Bdt%7D%3D%20%5Cfrac%7B631.64%7D%7B30790.75%7D)
![\frac{dh}{dt} = 0.021 m/min](https://tex.z-dn.net/?f=%5Cfrac%7Bdh%7D%7Bdt%7D%20%3D%200.021%20m%2Fmin)
Hence, the rate of change of the height is 0.021 meters per minute.