Question

Celeste has just started her own dog-grooming business.on the first day,she groomed 4 dogs for a profit of $26.8.on the second day,she groomed 15 dogs for a profit of $416.2.she thinks that she will maximize her profit if she grooms 11 dogs per day.Assuming that her profit can be modelled by a quadratic relation,calculate her maximum profit.

Answer 1

Suppose that grooming of x dogs per day, Celeste will earn a profit of $ p(x).
Given that the profit function is quadratic, we let,
\[p(x)=ax ^{2}+bx+c.\]
We know that, \[p(4)=26.8, p(15)=416.2.\]
\[p(4)=26.8 \rightarrow 16a+4b+c=26.8...........[1].\]
\[p(15)=416.2 \rightarrow 225a+15b+c=416.2...................[2].\]
For maximum profit, we must have, \[p \prime(x)=0, p \prime \prime(x)<0................[3].\]
It is known that grooming of 11 dogs yields maximum profit, we get from [3],
\[p \prime(11)=0, p \prime \prime(11)<0...........[4].\]
\[p \prime(x)=2ax+b, and, [4] \rightarrow 22a+b=0.............[5].\]
Solving eqns. [1], [2], and, [5], we get, \[a=-11.8, b=259.6, c=-822.8.\]
\[:. p(x)=-11.8x ^{2}+259.6x-822.8.\]
So, Celeste's maximum profit \[p(x)=$605.\]
Enjoy Maths.!