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Answer any one of the following.
A utensil manufacturer produces 'x' dinner sets per week and sells each set at ₹ p, where x = (600…
A utensil manufacturer produces '$x$' dinner sets per week and sells each set at ₹ $p$, where $x = \frac{600 - p}{8}$. The cost of production of '$x$' sets is ₹ $\left(\frac{x^2}{2} + 78x + 2000\right)$.
(a)[1]
Write the revenue function.
(b)[1]
Write the profit function.
(c)[2]
Calculate the number of dinner sets to be produced and sold per week to ensure maximum profit.
Answer
Answer (a)
$p=600-8x$, so $R(x)=px=600x-8x^2$.
Final answer: $R(x)=600x-8x^2$
Answer (b)
$P(x)=R(x)-C(x)=600x-8x^2-\frac{x^2}{2}-78x-2000=-\frac{17}{2}x^2+522x-2000$.
Final answer: $P(x)=-\frac{17}{2}x^2+522x-2000$
Answer (c)
$P'(x)=-17x+522=0\Rightarrow x=\frac{522}{17}\approx30.7$; $P''(x)=-17<0$, so profit is maximum there.
Sets are whole numbers: $P(30)=6010$, $P(31)=6013.5$, so 31 is better.
Final answer: 31 (x = 522/17 ≈ 30.7) dinner sets