Respuesta :
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, [tex] c=0.1\:q^{2}+2.1\:q+8[/tex]
Given that c changes with respect to q that is, [tex]\dfrac{dc}{dq}[/tex]. So differentiating given function,
[tex] \dfrac{dc}{dq}=\dfrac{d}{dq}\left (0.1\:q^{2}+2.1\:q+8 \right)[/tex]
Applying sum rule of derivative,
[tex] \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)[/tex]
Applying power rule and constant rule of derivative,
[tex] \dfrac{dc}{dt}=0.1\left(2\:q^{2-1}\right)+2.1\left(1\right)+0 [/tex]
[tex]\dfrac{dc}{dt}=0.1\left(2\:q\right)+2.1[/tex]
[tex]\dfrac{dc}{dt}=0.2\left(q\right)+2.1[/tex]
Substituting the value of [tex]q=11[/tex],
[tex]\dfrac{dc}{dt}=0.2\left(11\right)+21.[/tex]
[tex]\dfrac{dc}{dt}=2.2+2.1[/tex]
[tex]\dfrac{dc}{dt}=4.3[/tex]
Rate of change of c with respect to q is 4.3
Formula for percentage rate of change is given as,
[tex]Percentage\:rate\:of\:change=\dfrac{Q'\left(x\right)}{Q\left(x\right)}\times 100[/tex]
Rewriting in terms of cost C,
[tex]Percentage\:rate\:of\:change=\dfrac{C'\left(q\right)}{C\left(q\right)}\times 100[/tex]
Calculating value of [tex]C\left(q \right)[/tex]
[tex]C\left(q\right)=0.1\:q^{2}+2.1\:q+8[/tex]
Substituting the value of [tex]q=11[/tex],
[tex]C\left(q\right)=0.1\left(11\right)^{2}+2.1\left(11\right)+8[/tex]
[tex]C\left(q\right)=0.1\left(121\right)+23.1+8[/tex]
[tex]C\left(q\right)=12.1+23.1+8[/tex]
[tex]C\left(q\right)=43.2[/tex]
Now using the formula for percentage,
[tex]Percentage\:rate\:of\:change=\dfrac{4.3}{43.2}\times 100[/tex]
[tex]Percentage\:rate\:of\:change=0.0995\times 100[/tex]
[tex]Percentage\:rate\:of\:change=9.95%[/tex]
Percentage rate of change of c with respect to q is 9.95%
