Answer:
Rate of change of c with respect to q is 4.3
Percentage rate of change c with respect to q is 9.95%
Step-by-step explanation:
Cost function is given as, ![c=0.1\:q^{2}+2.1\:q+8](https://tex.z-dn.net/?f=%20c%3D0.1%5C%3Aq%5E%7B2%7D%2B2.1%5C%3Aq%2B8)
Given that c changes with respect to q that is,
. So differentiating given function,
![\dfrac{dc}{dq}=\dfrac{d}{dq}\left (0.1\:q^{2}+2.1\:q+8 \right)](https://tex.z-dn.net/?f=%20%5Cdfrac%7Bdc%7D%7Bdq%7D%3D%5Cdfrac%7Bd%7D%7Bdq%7D%5Cleft%20%280.1%5C%3Aq%5E%7B2%7D%2B2.1%5C%3Aq%2B8%20%5Cright%29)
Applying sum rule of derivative,
![\dfrac{dc}{dq}=\dfrac{d}{dq}\left(0.1\:q^{2}\right)+\dfrac{d}{dq}\left(2.1\:q\right)+\dfrac{d}{dq}\left(8\right)](https://tex.z-dn.net/?f=%20%5Cdfrac%7Bdc%7D%7Bdq%7D%3D%5Cdfrac%7Bd%7D%7Bdq%7D%5Cleft%280.1%5C%3Aq%5E%7B2%7D%5Cright%29%2B%5Cdfrac%7Bd%7D%7Bdq%7D%5Cleft%282.1%5C%3Aq%5Cright%29%2B%5Cdfrac%7Bd%7D%7Bdq%7D%5Cleft%288%5Cright%29)
Applying power rule and constant rule of derivative,
![\dfrac{dc}{dt}=0.1\left(2\:q^{2-1}\right)+2.1\left(1\right)+0](https://tex.z-dn.net/?f=%20%5Cdfrac%7Bdc%7D%7Bdt%7D%3D0.1%5Cleft%282%5C%3Aq%5E%7B2-1%7D%5Cright%29%2B2.1%5Cleft%281%5Cright%29%2B0%20)
![\dfrac{dc}{dt}=0.1\left(2\:q\right)+2.1](https://tex.z-dn.net/?f=%5Cdfrac%7Bdc%7D%7Bdt%7D%3D0.1%5Cleft%282%5C%3Aq%5Cright%29%2B2.1)
![\dfrac{dc}{dt}=0.2\left(q\right)+2.1](https://tex.z-dn.net/?f=%5Cdfrac%7Bdc%7D%7Bdt%7D%3D0.2%5Cleft%28q%5Cright%29%2B2.1)
Substituting the value of
,
![\dfrac{dc}{dt}=0.2\left(11\right)+21.](https://tex.z-dn.net/?f=%5Cdfrac%7Bdc%7D%7Bdt%7D%3D0.2%5Cleft%2811%5Cright%29%2B21.)
![\dfrac{dc}{dt}=2.2+2.1](https://tex.z-dn.net/?f=%5Cdfrac%7Bdc%7D%7Bdt%7D%3D2.2%2B2.1)
![\dfrac{dc}{dt}=4.3](https://tex.z-dn.net/?f=%5Cdfrac%7Bdc%7D%7Bdt%7D%3D4.3)
Rate of change of c with respect to q is 4.3
Formula for percentage rate of change is given as,
![Percentage\:rate\:of\:change=\dfrac{Q'\left(x\right)}{Q\left(x\right)}\times 100](https://tex.z-dn.net/?f=Percentage%5C%3Arate%5C%3Aof%5C%3Achange%3D%5Cdfrac%7BQ%27%5Cleft%28x%5Cright%29%7D%7BQ%5Cleft%28x%5Cright%29%7D%5Ctimes%20100)
Rewriting in terms of cost C,
![Percentage\:rate\:of\:change=\dfrac{C'\left(q\right)}{C\left(q\right)}\times 100](https://tex.z-dn.net/?f=Percentage%5C%3Arate%5C%3Aof%5C%3Achange%3D%5Cdfrac%7BC%27%5Cleft%28q%5Cright%29%7D%7BC%5Cleft%28q%5Cright%29%7D%5Ctimes%20100)
Calculating value of ![C\left(q \right)](https://tex.z-dn.net/?f=C%5Cleft%28q%20%5Cright%29)
![C\left(q\right)=0.1\:q^{2}+2.1\:q+8](https://tex.z-dn.net/?f=C%5Cleft%28q%5Cright%29%3D0.1%5C%3Aq%5E%7B2%7D%2B2.1%5C%3Aq%2B8)
Substituting the value of
,
![C\left(q\right)=0.1\left(11\right)^{2}+2.1\left(11\right)+8](https://tex.z-dn.net/?f=C%5Cleft%28q%5Cright%29%3D0.1%5Cleft%2811%5Cright%29%5E%7B2%7D%2B2.1%5Cleft%2811%5Cright%29%2B8)
![C\left(q\right)=0.1\left(121\right)+23.1+8](https://tex.z-dn.net/?f=C%5Cleft%28q%5Cright%29%3D0.1%5Cleft%28121%5Cright%29%2B23.1%2B8)
![C\left(q\right)=12.1+23.1+8](https://tex.z-dn.net/?f=C%5Cleft%28q%5Cright%29%3D12.1%2B23.1%2B8)
![C\left(q\right)=43.2](https://tex.z-dn.net/?f=C%5Cleft%28q%5Cright%29%3D43.2)
Now using the formula for percentage,
![Percentage\:rate\:of\:change=\dfrac{4.3}{43.2}\times 100](https://tex.z-dn.net/?f=Percentage%5C%3Arate%5C%3Aof%5C%3Achange%3D%5Cdfrac%7B4.3%7D%7B43.2%7D%5Ctimes%20100)
![Percentage\:rate\:of\:change=0.0995\times 100](https://tex.z-dn.net/?f=Percentage%5C%3Arate%5C%3Aof%5C%3Achange%3D0.0995%5Ctimes%20100)
![Percentage\:rate\:of\:change=9.95%](https://tex.z-dn.net/?f=Percentage%5C%3Arate%5C%3Aof%5C%3Achange%3D9.95%25)
Percentage rate of change of c with respect to q is 9.95%