Answer
a) $\$3$; $7000$ hot dogs
b) $p\ge 3$
c) The price decreases close to the equilibrium price of $\$3$.
Work Step by Step
We are given:
$S(p)=-2000+3000p$
$D(p)=10,000-1000p$
a) Find the equilibrium price:
$S(p)=D(p)$
$-2000+3000p=10,000-1000p$
$-2000+3000p+1000p=10,000-1000p+1000p$
$-2000+4000p=10,000$
$-2000+4000p+2000=10,000+2000$
$4000p=12,000$
$p=\dfrac{12,000}{4000}$
$p=3$
The equilibrium price is $\$3$. The equilibrium quantity is:
$S(3)=D(3)=-2000+3000(3)=7000$ hot dogs
b) Solve the inequality:
$D(p)\lt S(p)$
$10,000-1000p\lt -2000+3000p$
$10,000-1000p+1000p\lt -2000+3000p+1000p$
$10,000\lt -2000+4000p$
$10,000+2000\lt -2000+4000p+2000$
$12,000\lt 4000p$
$\dfrac{12,000}{4000}\lt 3$
$3\lt p$
$p\gt 3$
c) If the demand is less than the supply, the price tends to decrease, getting close to the equilibrium price of $\$3$.
