To solve this problem, we'll break it down into two parts: finding the linear demand equation and then using it to estimate the demand if the price is increased.

Part (a): Finding the linear demand equation

We are given two points that represent the relationship between price and quantity sold:

• At $P = 2.50$ (price per gallon), $Q = 15$ (since 15,000 gallons = 15 thousand gallons).
• At $P = 2.75$, $Q = 12$ (since 12,000 gallons = 12 thousand gallons).

We assume a linear relationship between the price $P$ and the quantity sold $Q$, which is in the form:

$Q = mP + b$

Where:

• $Q$ is the quantity demanded (in thousands of gallons),
• $P$ is the price per gallon,
• $m$ is the slope of the line,
• $b$ is the y-intercept.

Step 1: Calculate the slope $m$

The slope $m$ of a linear equation is given by the formula:

$m = \frac{Q_2 - Q_1}{P_2 - P_1}$

Substitute the given values:

$m = \frac{12 - 15}{2.75 - 2.50} = \frac{-3}{0.25} = -12$

Step 2: Use the slope to find the y-intercept $b$

Now that we have the slope $m = -12$, we can substitute one of the points into the linear equation to solve for $b$. Using the point $(P = 2.50, Q = 15)$:

$Q = mP + b$

$15 = -12(2.50) + b$

$15 = -30 + b$

$b = 15 + 30 = 45$

Final linear equation:

The demand equation is:

$Q = -12P + 45$

Part (b): Estimate the demand if the price increases by $0.20

If the price is increased by $0.20, the new price will be:

$P = 2.75 + 0.20 = 2.95$

Now, substitute $P = 2.95$ into the demand equation $Q = -12P + 45$:

$Q = -12(2.95) + 45 = -35.4 + 45 = 9.6$

Therefore, if the price is increased by $0.20, the estimated demand would be 9.6 thousand gallons (or 9,600 gallons).

Summary:

• The linear demand equation is: $Q = -12P + 45$
• If the price is increased by $0.20 to $2.95, the demand would be approximately 9,600 gallons.

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Would you like more details on any part? Feel free to ask!

Here are 5 related questions to expand on this:

1. How would the demand change if the price was decreased by $0.10?
2. What does the slope of the demand equation represent in this context?
3. If a competitor lowers their price to $2.40, how much gas would you expect to sell?
4. How can elasticity of demand be calculated using this equation?
5. What factors could cause this demand model to break down in real-world scenarios?

Tip: When interpreting slopes in demand equations, a negative slope indicates that as prices increase, the demand generally decreases, which is typical in most markets.