The following table shows worldwide sales of a certain type of cell phone and their average selling prices in 2013 and 2017.
Year 	2013	2017
Selling Price ($)	325	215
Sales (millions)	1,140	2,240
(a)
Use the data to obtain a linear demand function for this type of cell phone.
q(p) =

Use your demand equation to predict sales to the nearest million phones if the price is raised to $385.
million phones
(b)
Fill in the blank.
For every $1 increase in price, sales of this type of cell phone decrease by
million units.

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?

---

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.

The following table shows worldwide sales of a certain type of cell phone and their average selling prices in 2013 and 2017.
Year 2013 2017
Selling Price ($) 325 215
Sales (millions) 1,140 2,240
(a) Use the data to obtain a linear demand function for this type of cell phone.
q(p) =
Use your demand equation to predict sales to the nearest million phones if the price is raised to $385.
(b) For every $1 increase in price, sales of this type of cell phone decrease by million units.

Learn how to derive a linear demand function using sales and price data from 2013 and 2017. Predict cell phone sales when the price is raised to $385. Understand how a $1 increase in price affects sales. Suitable for high school algebra students.

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