We are given data for worldwide sales of cell phones and their average selling prices for the years 2013 and 2017:

• 2013: Price = $325, Sales = 1,140 million phones
• 2017: Price = $215, Sales = 2,240 million phones

Part (a): Obtain a linear demand function

The demand function $q(p)$ relates the quantity $q$ (sales in millions) to the price $p$. Since we're assuming a linear demand function, it will take the form:

$q(p) = m p + b$

where $m$ is the slope and $b$ is the y-intercept.

To find $m$ and $b$, we use the two points provided: $(p_1, q_1) = (325, 1140)$ and $(p_2, q_2) = (215, 2240)$.

Step 1: Find the slope $m$

The slope of the line is given by:

$m = \frac{q_2 - q_1}{p_2 - p_1} = \frac{2240 - 1140}{215 - 325} = \frac{1100}{-110} = -10$

So, $m = -10$.

Step 2: Find the y-intercept $b$

Now, use one of the points to solve for $b$. Using $(325, 1140)$:

$1140 = -10(325) + b$

$1140 = -3250 + b$

$b = 1140 + 3250 = 4390$

Thus, the linear demand function is:

$q(p) = -10p + 4390$

Predict Sales at a Price of $385

Now, we can use the demand equation to predict sales if the price is raised to $385:

$q(385) = -10(385) + 4390$

$q(385) = -3850 + 4390 = 540$

So, the predicted sales would be 540 million phones.

Part (b): Sales decrease per $1 increase in price

From the slope of the linear demand function, we can interpret that for every $1 increase in price, sales decrease by 10 million units (since $m = -10$).

Summary of Answers:

• (a) The demand function is: $q(p) = -10p + 4390$.
• Predicted sales if price = $385: 540 million phones.
• (b) For every $1 increase in price, sales decrease by 10 million units.

Would you like more details or have any questions?

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Here are 5 related questions to expand the topic:

1. How would the sales prediction change if the price were reduced to $150?
2. What assumptions are we making by using a linear demand model?
3. How can we use elasticity to measure the sensitivity of sales to price changes?
4. What would happen to sales if the price dropped to $0 according to this model?
5. How would you calculate the break-even point where sales drop to zero?

Tip: Always check if a linear model is appropriate by analyzing if the trend is constant over time or if non-linear relationships might exist.