\(\begin{aligned}TR &= P \times Q\\ &= (100 - 3Q + 4{A^{\frac{1}{2}}})Q\\ &= 100Q - 3{Q^2} + 4{A^{\frac{1}{2}}}Q\end{aligned}\)
Profit refers to the difference between the total revenue and cost incurred on fixed and variable factors.
Profit = TR – C
\({\rm{Pr = (100Q - 3}}{{\rm{Q}}^{\rm{2}}}{\rm{ + 4}}{{\rm{A}}^{\frac{{\rm{1}}}{{\rm{2}}}}}{\rm{Q) - (4}}{{\rm{Q}}^{\rm{2}}}{\rm{ + 10Q + A)}}\)
First-order condition:
\(\begin{aligned}\frac{{\partial \Pr }}{{\partial Q}} &= 0\\100 - 6Q + 4{A^{\frac{1}{2}}} - 8Q - 10 &= 0\\90 - 14Q + 4{A^{\frac{1}{2}}} &= 0{\rm{ - - - - - - - - - - - - - - - - - - equation (i)}}\end{aligned}\)
\(\begin{aligned}\frac{{\partial \Pr }}{{\partial A}} &= 0\\\left( {\left( {\frac{1}{2}} \right)\frac{4}{{{A^{\frac{1}{2}}}}}} \right)Q - 1 &= 0\\{A^{\frac{1}{2}}} &= 2Q{\rm{ - - - - - - - - - - - - - - - - - equation (ii)}}\end{aligned}\)
From equation (i) and (ii)
\(\begin{aligned}90 - 14Q + 4{A^{\frac{1}{2}}} &= 0\\90 - 14Q + 4\left( {2Q} \right) &= 0\\6Q &= 90\\Q &= 15\end{aligned}\)
Putting the value of Q in equation (ii)
\(\begin{aligned}{A^{\frac{1}{2}}} &= 2(15)\\A &= {\left( {2 \times 15} \right)^2}\\A &= \left( {4 \times 225} \right)\\A &= 900\end{aligned}\)
Putting the value of A and Q in the demand curve, the value of P is calculated:
\(\begin{aligned}P &= 100 - 3\left( {15} \right) + 4{\left( {900} \right)^{\frac{1}{2}}}\\ &= 100 - 45 + {\rm{120}}\\ &= 220 - 45\\ &= 175\\\end{aligned}\)
The value of A is 900, the value of Q is 15, and the value of P is 175.
