Question

The owner of a bike shop that produces custom built bike frames has determined that the demand equation for bike frames is given by the equation

D(q) = –6.10q^2 –5q + 1000
where D(q) is the price in dollars and q is the number of bike frames demanded per week. The supply equation for bike frames is
S(q) = 3.20q^2 + 10q – 80
where q is the quantity the supplier will make available per week in the market when the price is p dollars. Find the equilibrium point (q, p) rounded to the nearest hundredth.

Answer 1

equilibrium point (10,340)

Explanation:

To find the equilibrium point, equal the demand and the supply:

`D(q)=S(q)\\\\-6.10q^2-5q+1000=3.2q^2+10q-80`

Reorganize the terms in one side and reduce similar terms:

`3.2q^2+6.1q^2+5q+10q-80-1000=0\\\\9.3q^2+15q-1080=0`

that's a cuadratic equation, solve with the general formula when:

a=9.3, b=15, c=-1080

`q_(1)=\frac{-b+\sqrt{b^(2)-4ac} }{2a}\\\\q_(2)=\frac{-b-\sqrt{b^(2)-4ac} }{2a}\\\\q_(1)=\frac{-15+\sqrt{(-15)^(2)-4(9.3)(-1080)} }{2(9.3)}\\\\q_(1)=(-15+201)/(18.6)\\\\q_(1)=(186)/(18.6)\\\\q_1=10`

q can't be negative because it is the quantity of bike frames, so:

`q_(2)=\frac{-b-\sqrt{b^(2)-4ac} }{2a}\\\\q_(2)=\frac{-15-\sqrt{(-15)^(2)-4(9.3)(-1080)} }{2(9.3)}\\\\q_(2)=(-15-201)/(18.6)\\\\q_(2)=(-216)/(18.6)\\\\`

This value of q can't be considered.

Then substitute the value of q in D(q) to find the price p:

`D(10) = -6.10(10)^2-5(10) + 1000\\\\D(10)=340=p`

The equilibrium point (q,p) is (10,340).