Question

you are running a business selling homemade bread. your weekly revenue from the sale of $q$ loaves of bread is $r(q) = 67q - 0.1q^2$ dollars, and the weekly cost of making $q$ loaves of bread is $c(q) = 30 + 23q$ dollars. (note that $p(q) = r(q) - c(q)$).
a. find the weekly profit function $p(q)$.
$p(q) = $
b. find the derivative $p(q)$ of the weekly profit function.
$p(q) = $
c. find the production level $q$ that maximizes the weekly profit.
$q = $ loaves of bread

Explanation:

Step1: Define profit function

$P(q) = R(q) - C(q)$
Substitute $R(q)=67q-0.1q^2$ and $C(q)=30+23q$:
$P(q) = (67q - 0.1q^2) - (30 + 23q)$

Step2: Simplify profit function

Combine like terms:
$P(q) = -0.1q^2 + (67q-23q) - 30 = -0.1q^2 + 44q - 30$

Step3: Differentiate profit function

Apply power rule $\frac{d}{dq}(q^n)=nq^{n-1}$:
$P'(q) = \frac{d}{dq}(-0.1q^2 + 44q - 30) = -0.2q + 44$

Step4: Maximize profit (set $P'(q)=0$)

Solve for $q$:
$-0.2q + 44 = 0$
$0.2q = 44$
$q = \frac{44}{0.2} = 220$

Answer:

a. $P(q) = -0.1q^2 + 44q - 30$
b. $P'(q) = -0.2q + 44$
c. $q = 220$ loaves of bread