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 $r(p)=pf(p)$.
(a) what does it mean to say that $f(25)=10,000$ in the context of this problem?
there are 350 total yards of fabric and $25 to spend on it.
there are 10,000 total yards of fabric and $350 to spend on it.
when the price of fabric is $25/yard, 350 yards will be sold.
when the price of fabric is $25/yard, 10,000 yards will be sold.
when the price of fabric is $350/yard, 25 yards will be sold.
what does it mean to say that $f(25)= - 350$ in the context of this problem?
as the price of the fabric decreases past $350/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 increases past $25/yard, the amount of fabric which will be sold is decreasing at a rate of 350 yards per (dollar per yard)
as the price of the fabric decreases past $25/yard, the amount of fabric which will be sold is increasing at a rate of $350 per (dollar per yard)
as the price of the fabric decreases past $25/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 increases past $350/yard, the amount of fabric which will be sold is increasing at a rate of 25 yards per (dollar per yard)
(b) assuming the values in part (a), find $r(25)$.
$r(25)=$
interpret your answer in the context of this problem.
as the price of fabric decreases past $350/yard, the total revenue is increasing at $1250 per (dollar per yard)
as the price of fabric increases past $350/yard, the total revenue is increasing at $10,000 per (dollar per yard)
as the price of fabric increases past $25/yard, the total revenue is increasing at $1250 per (dollar per yard)
as the price of fabric increases past $25/yard, the total revenue is decreasing at $10,000 per (dollar per yard)
as the price of fabric decreases past $25/yard, the total revenue is decreasing at $350 per (dollar per yard)
as the price of fabric increases past $25/yard, the total revenue is decreasing at $350 per (dollar per yard)

Explanation:

Step1: Interpret $f(25)=10000$

The function $q = f(p)$ gives quantity sold as a function of price. So when $p = 25$, $q=10000$. This means when the price of fabric is $\$25$ per yard, 10,000 yards will be sold.

Step2: Interpret $f'(25)= - 350$

The derivative $f'(p)$ represents the rate of change of the quantity sold with respect to the price. A negative value means the quantity sold is decreasing. So as the price of the fabric increases past $\$25$ per yard, the amount of fabric which will be sold is decreasing at a rate of 350 yards per (dollar per yard).

Step3: Find $R'(25)$

We know that $R(p)=p\cdot f(p)$. Using the product - rule $(uv)' = u'v+uv'$, where $u = p$ and $v = f(p)$. Then $R'(p)=f(p)+p\cdot f'(p)$. Substitute $p = 25$, $f(25)=10000$ and $f'(25)=- 350$. So $R'(25)=f(25)+25\cdot f'(25)=10000+25\times(-350)=10000 - 8750=1250$.

Step4: Interpret $R'(25)$

$R'(25) = 1250$ means as the price of fabric increases past $\$25$ per yard, the total revenue is increasing at $\$1250$ per (dollar per yard).

Answer:

a. When the price of fabric is $\$25$/yard, 10,000 yards will be sold; As the price of the fabric increases past $\$25$/yard, the amount of fabric which will be sold is decreasing at a rate of 350 yards per (dollar per yard)
b. $R'(25)=1250$; As the price of fabric increases past $\$25$/yard, the total revenue is increasing at $\$1250$ per (dollar per yard)