A manufacturer produces bolts of a fabric with a fixed width. The quantity q of

Question

A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q = f(p) Then the total revenue earned with selling price p is R(p) = pf(p). What does it mean to say that f(10) = 10,000? There are 10,000 total yards of fabric and $500 to spend on it. When the price of fabric is $500/yard, 10 yards will be sold. There are 500 total yards of fabric and $10 to spend on it. When the price of fabric is $10/yard, 10,000 yards will be sold. When the price of fabric is $10/yard, 500 yards will be sold. What does it mean to say that f '(10) = -500? As the price of the fabric increases past $10/yard, the amount of fabric which will be sold is decreasing at a rate of 500 yards per (dollar per yard). As the price of the fabric increases past $500/yard, the amount of fabric which will be sold is increasing at a rate of 10 yards per (dollar per yard). As the price of the fabric decreases past $10/yard, the amount of fabric which will be sold is increasing at a rate of 10,000 yards per (dollar per yard). As the price of the fabric decreases past $500/yard, the amount of fabric which will be sold is decreasing at a rate of $10,000 per (dollar per yard). As the price of the fabric decreases past $10/yard, the amount of fabric which will be sold is increasing at a rate of $500 per (dollar per yard). Assuming the values in part (a), find R'(10). R'(10) = 12750 Interpret your answer. As the price of fabric decreases past $500/yard, the total revenue is increasing at $5000 per (dollar per yard). As the price of fabric increases past $10/yard, the total revenue is decreasing at $500 per (dollar per yard). As the price of fabric increases past $10/yard, the total revenue is increasing at $5000 per (dollar per yard). As the price of fabric decreases past $10/yard, the total revenue is decreasing at $10,000 per (dollar per yard). As the price of fabric increases past $500/yard, the total revenue is increasing at $10,000 per (dollar per yard).

Explanation / Answer

R (p) = p*f(p) . differentiate using product rule since p itself is the variable

R dash (p) = p* fdash(p) + f(p) *1

R dash(10) = 10*f dash(10) +f(10) =10*-500+10000 = 5000, your interpretation is correct in part C .

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