Question

Through many focus groups, Hasbro determined they could sell 80,000 furbies at a price of $29.99. However, if they lowered their price to $21.99, they could sell 60,000 more furbies. Find the linear demand equation (price function, y) as a function of the quantity, x, sold.

Answer 1

Explanation:

To find the linear demand equation as a function of the quantity sold, you can use the point-slope form of a linear equation:

\(y - y_1 = m(x - x_1)\)

Where:

- \(y\) is the price (\(P\)),

- \(x\) is the quantity sold (\(Q\)),

- \(m\) is the slope of the demand curve, and

- \((x_1, y_1)\) is a point on the demand curve.

You have two points to work with: (80,000, $29.99) and (140,000, $21.99).

First, calculate the slope (m) using these two points:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

\(m = \frac{21.99 - 29.99}{140,000 - 80,000}\)

\(m = \frac{-8}{60,000}\)

Now, choose one of the points (let's use the first one, (80,000, $29.99)) and plug it into the point-slope form:

\(P - 29.99 = \frac{-8}{60,000}(Q - 80,000)\)

Now, simplify the equation:

\(P - 29.99 = \frac{-8}{60,000}Q + \frac{8}{60,000}(80,000)\)

\(P - 29.99 = \frac{-8}{60,000}Q + \frac{8}{7}\)

Now, isolate \(P\) (price) on the left side:

\(P = \frac{-8}{60,000}Q + \frac{8}{7} + 29.99\)

To make it more readable, you can multiply all terms by 60,000 to clear fractions:

\(60,000P = -8Q + \frac{8}{7} \cdot 60,000 + 60,000 \cdot 29.99\)

\(60,000P = -8Q + \frac{480,000}{7} + 1,799,400\)

Finally, you can express this equation in a more standard form:

\(60,000P + 8Q = \frac{480,000}{7} + 1,799,400\)

This is your linear demand equation as a function of quantity sold, where \(P\) is the price and \(Q\) is the quantity sold.