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finlep [7]
3 years ago
15

For the cost function c equals 0.1 q squared plus 2.1 q plus 8​, how fast does c change with respect to q when q equals 11​? Det

ermine the percentage rate of change of c with respect to q when q equals 11.
Mathematics
1 answer:
Soloha48 [4]3 years ago
7 0

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

Given that c changes with respect to q that is, \dfrac{dc}{dq}. So differentiating given function,  

\dfrac{dc}{dq}=\dfrac{d}{dq}\left (0.1\:q^{2}+2.1\:q+8 \right)

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)

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

\dfrac{dc}{dt}=0.1\left(2\:q\right)+2.1

\dfrac{dc}{dt}=0.2\left(q\right)+2.1

Substituting the value of q=11,

\dfrac{dc}{dt}=0.2\left(11\right)+21.

\dfrac{dc}{dt}=2.2+2.1

\dfrac{dc}{dt}=4.3

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

Rewriting in terms of cost C,

Percentage\:rate\:of\:change=\dfrac{C'\left(q\right)}{C\left(q\right)}\times 100

Calculating value of C\left(q \right)

C\left(q\right)=0.1\:q^{2}+2.1\:q+8

Substituting the value of q=11,

C\left(q\right)=0.1\left(11\right)^{2}+2.1\left(11\right)+8

C\left(q\right)=0.1\left(121\right)+23.1+8

C\left(q\right)=12.1+23.1+8

C\left(q\right)=43.2

Now using the formula for percentage,  

Percentage\:rate\:of\:change=\dfrac{4.3}{43.2}\times 100

Percentage\:rate\:of\:change=0.0995\times 100

Percentage\:rate\:of\:change=9.95%

Percentage rate of change of c with respect to q is 9.95%

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