Monthly demand for a particular product is linear as a function of its unit price $p.$ The manufacturer estimates that if a particular item were free, consumer demand would be $15$ thousand units each month. If the product instead costs $$24,$ the demand would be about $3$ thousand units each month.

a$)$ Write a formula for monthly demand $d$ thousand units as a linear function of unit price $p.$

$p=0,d=15000,amp+c$

$c=15000,$

when $p=24,d=3000,d=-500p+15000$

$3000,=24m+15000,$

$24m,=-i2000,$

$m,=-500,$

b$)$ Write a sentence to interpret the value of $d=f(8)$ within context and including units.

$p=8,d=f(8)+15if$ the product costs

$d=f(8)=-500(8)+15000,8$ dollars then

$f(8)=11000$ the demand will $be11$ thousand

c$)$ The quantity supplied $\text{'}$by the manufacturer of the product is also a linear function of $p.$ When the product costs $$8,$ only $3$ thousand units are supplied each month. When the price increases to $$20,$ instead $12$ thousand units are supplied each month. Write a formula for monthly supply, $s,$ as a linear function of unit price $p.$

$12n+k-9000=k$

$12n-9000=2k$

$17000=20n+k$

$6n-1500=1,0=0n+12+12000$

d$)$ Recall that the market equilibrium is the point at which quantity demanded and quantity supplied are equal. At what price does market equilibrium occur in this case? What is the quantity demanded each month at market equilibrium?