The balloon has a volume
dependent on its radius
:
![V(r)=\dfrac43\pi r^3](https://tex.z-dn.net/?f=V%28r%29%3D%5Cdfrac43%5Cpi%20r%5E3)
Differentiating with respect to time
gives
![\dfrac{\mathrm dV}{\mathrm dt}=4\pi r^2\dfrac{\mathrm dr}{\mathrm dt}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dV%7D%7B%5Cmathrm%20dt%7D%3D4%5Cpi%20r%5E2%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20dt%7D)
If the volume is increasing at a rate of 10 cubic m/s, then at the moment the radius is 3 m, it is increasing at a rate of
![10\dfrac{\mathrm m^3}{\mathrm s}=4\pi (3\,\mathrm m)^2\dfrac{\mathrm dr}{\mathrm dt}\implies\dfrac{\mathrm dr}{\mathrm dt}=\dfrac5{18\pi}\dfrac{\rm m}{\rm s}](https://tex.z-dn.net/?f=10%5Cdfrac%7B%5Cmathrm%20m%5E3%7D%7B%5Cmathrm%20s%7D%3D4%5Cpi%20%283%5C%2C%5Cmathrm%20m%29%5E2%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20dt%7D%5Cimplies%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20dt%7D%3D%5Cdfrac5%7B18%5Cpi%7D%5Cdfrac%7B%5Crm%20m%7D%7B%5Crm%20s%7D)
The surface area of the balloon is
![S(r)=4\pi r^2](https://tex.z-dn.net/?f=S%28r%29%3D4%5Cpi%20r%5E2)
and differentiating gives
![\dfrac{\mathrm dS}{\mathrm dt}=8\pi r\dfrac{\mathrm dr}{\mathrm dt}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dS%7D%7B%5Cmathrm%20dt%7D%3D8%5Cpi%20r%5Cdfrac%7B%5Cmathrm%20dr%7D%7B%5Cmathrm%20dt%7D)
so that at the moment the radius is 3 m, its area is increasing at a rate of
![\dfrac{\mathrm dS}{\mathrm dt}=8\pi(3\,\mathrm m)\left(\dfrac5{18\pi}\dfrac{\rm m}{\rm s}\right)=\dfrac{20}3\dfrac{\mathrm m^2}{\rm s}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dS%7D%7B%5Cmathrm%20dt%7D%3D8%5Cpi%283%5C%2C%5Cmathrm%20m%29%5Cleft%28%5Cdfrac5%7B18%5Cpi%7D%5Cdfrac%7B%5Crm%20m%7D%7B%5Crm%20s%7D%5Cright%29%3D%5Cdfrac%7B20%7D3%5Cdfrac%7B%5Cmathrm%20m%5E2%7D%7B%5Crm%20s%7D)
3x + 3 = 2x
three times a number = 3x ,and 3 = 3x + 3, is equal to two times the number =
3x + 3 = 2x
Answer: c and e
explanation: put the equation in the graphing calculator and look at the graph and table the answer will show
Answer:
The 1st Blank is: A pure substance, or simply a substance
The 2nd blank on the left is: Elements
The 3rd blank on the right is: Compounds
The 4th blank which is the second middle one is: A mixture
The 5th blank which is the bottom left one is: Heterogeneous mixture
The 6th blank which is the last one on the bottom right is: Homogeneous mixture
Look at the step-by-step explanation if you get confused at the bottom.
Step-by-step explanation:
The 1st Blank: I would just put a pure substance.
The 2nd blank on the left: They are three characteristics of elements but just put elements.
The 3rd blank on the right: They are three characteristics of compounds but just put compounds.
The 4th blank which is the second middle one is: The characteristics of a mixture but just put a mixture.
The 5th blank which is the bottom left one: Are the characteristics of a heterogeneous mixture but just put heterogeneous mixture.
The 6th blank which is the last one on the bottom right: Are the characteristics of a homogeneous mixture but just put homogeneous mixture.