Question

the table below gives the quantity of magnetic inductors demanded and the quantity supplied for selected prices. (a) find the linear equation that gives the price as a function of the quantity demanded. (b) find the linear equation that gives the price as a function of the quantity supplied. (c) use these equations to find the market equilibrium price. price ($) 20 30 40 50 100 quantity demanded (thousands) 384 344 304 264 64 quantity supplied (thousands) 0 200 400 600 1600 (a) what is the price as a function of the quantity demanded? p = \square (type an expression using q as the variable. type your answer in slope - intercept form.)

Explanation:

Step1: Calculate demand slope

Slope $m_d = \frac{\Delta p}{\Delta q_d} = \frac{30-20}{344-384} = \frac{10}{-40} = -\frac{1}{4}$

Step2: Find demand equation

Use point $(q_d,p)=(384,20)$:
$p - 20 = -\frac{1}{4}(q - 384)$
$p = -\frac{1}{4}q + 96 + 20$
$p = -\frac{1}{4}q + 116$

Step3: Calculate supply slope

Slope $m_s = \frac{\Delta p}{\Delta q_s} = \frac{30-20}{200-0} = \frac{10}{200} = \frac{1}{20}$

Step4: Find supply equation

Use point $(q_s,p)=(0,20)$:
$p - 20 = \frac{1}{20}(q - 0)$
$p = \frac{1}{20}q + 20$

Step5: Equate for equilibrium

Set demand = supply:
$-\frac{1}{4}q + 116 = \frac{1}{20}q + 20$
Multiply by 20 to eliminate denominators:
$-5q + 2320 = q + 400$
$-6q = -1920$
$q = 320$
Substitute $q=320$ into supply equation:
$p = \frac{1}{20}(320) + 20 = 16 + 20 = 36$

Answer:

(a) $p = -\frac{1}{4}q + 116$
(b) $p = \frac{1}{20}q + 20$
(c) $\$36$