Question

The linear-quadratic system of equations represents the supply and demand for running shoes where x represents the price and y represents the number (in thousands) of pairs of shoes produced.

Supply: y = 1.25x – 50
Demand: y = –0.01x^2 + 2x – 40
At what price will the supply equal the demand? Express your answer as a decimal rounded to the nearest hundredth.

How many pairs of shoes should be produced? Express your answer as a decimal rounded to the nearest hundredth.

Answer 1

approximately 37.86 thousand pairs of shoes should be produced at the price of $30.29 to achieve equilibrium between supply and demand.

Explanation:

To find the price at which supply equals demand, we need to set the supply equation equal to the demand equation and solve for x:

1.25x - 50 = -0.01x^2 + 2x - 40

First, let's rearrange the equation to set it to zero:

0 = -0.01x^2 + 2x - 40 - 1.25x + 50

Next, combine like terms:

0 = -0.01x^2 + 0.75x + 10

Now, we need to solve this quadratic equation for x. We can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = -0.01, b = 0.75, and c = 10.

x = (-(0.75) ± √((0.75)^2 - 4(-0.01)(10))) / 2(-0.01)

x = (-0.75 ± √(0.5625 + 0.4)) / -0.02

x = (-0.75 ± √0.9625) / -0.02

Now, calculate the two possible values of x:

x₁ = (-0.75 + √0.9625) / -0.02 ≈ 30.29

x₂ = (-0.75 - √0.9625) / -0.02 ≈ -2.79

Since the price cannot be negative in this context, we discard the negative value of x. Thus, the price at which supply equals demand is approximately $30.29.

Next, to find the number of pairs of shoes produced (y) at this price, we can use either the supply or demand equation since they will be equal at equilibrium.

Let's use the supply equation:

y = 1.25x - 50

y = 1.25 * 30.29 - 50

y ≈ 37.86