Question

A manufacturer produces bolts of a fabric with a fixed width. The quadtity q of this fabric (measured in yeards) that is sold with a function of the selling price p (in dollars per yard), so we can write \(q = f\left( p \right)\). Then the total revenue earned with selling price p is \(R\left( p \right) = pf\left( p \right)\).

(a) What does it mean to say that \(f\left( {{\bf{20}}} \right) = {\bf{10}},{\bf{000}}\) and \(f'\left( {{\bf{20}}} \right) = - {\bf{350}}\)?

(b) Assuming the values in part (a), find \(R'\left( {{\bf{20}}} \right)\) and interpret your answer.

Short answer

(a) The function \(f\left( {20} \right) = 10,000\) shows that at the price of $20 /yard, 10,000 yards will be sold.

Similarly, the equation \(f'\left( {20} \right) = - 350\) represents that the amount of fabric sold is decreasing at the rate of 350 yards per dollar.

(b) \(R'\left( {20} \right) = \$ 3,000\). It shows as the fabric increases by $20 /yard, the total revenue is $3,000 ($)/yard. The product rule also indicates that \( - \$ 7,000\) loss due to selling less fabric. The additional revenue more than makes up for the loss due to the increase in price.

Step-by-step solution

(a) Step 1: Find answers to part (a)

The equation \(f\left( {20} \right) = 10,000\) shows that at a price of $20 /yard, 10,000 yards will be sold.

Similarly, the equation \(f'\left( {20} \right) = - 350\) represents that the amount of fabric sold is decreasing at a rate of 350 yards per dollar.

(b) Step 2: Find answers to part (b)

Differentiate the function \(R\left( p \right) = pf\left( p \right)\) using the product rule:

\(R'\left( p \right) = pf'\left( p \right) + f\left( p \right)\left( 1 \right)\)

For \(p = 20\);

\(\begin{aligned}R'\left( {20} \right) &= 20f'\left( {20} \right) + f\left( {20} \right)\\ &= 20\left( { - 350} \right) + 10,000\\ &= - 7,000 + 10,000\\ &= 3,000\end{aligned}\)

This means that as the fabric increases $20 /yard, the total revenue at $3,000 ($)/yard.

The product rule also indicates that \( - \$ 7,000\) lost due to selling less fabric. The loss is more than made up for by the additional revenue due to increase in price.