To determine the linear function that predicts the number of calculators sold per week at a price of $x$ dollars, we need to find the equation of a line in the form:

$n(x) = mx + b$

where:

• $n(x)$ represents the number of calculators sold at a price of $x$ dollars,
• $m$ is the slope of the line, and
• $b$ is the y-intercept, representing the number of calculators sold when the price is $0 (if extended).

Step 1: Find the slope $m$

The problem provides two points on the line:

• At a price of $85, 10,000 calculators are sold: $(85, 10000)$,
• At a price of $80, 14,000 calculators are sold: $(80, 14000)$.

The slope $m$ is calculated as the change in the number of calculators sold divided by the change in price:

$m = \frac{\text{change in } n}{\text{change in } x} = \frac{14000 - 10000}{80 - 85} = \frac{4000}{-5} = -800$

So, the slope $m$ is $-800$.

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

Now, we use one of the given points to find $b$. Let’s use the point $(85, 10000)$ and substitute into the equation $n(x) = mx + b$:

$10000 = -800(85) + b$ $10000 = -68000 + b$ $b = 10000 + 68000 = 78000$

Step 3: Write the final equation

Now that we have the slope $m = -800$ and the intercept $b = 78000$, the linear function is:

$n(x) = -800x + 78000$

This equation predicts the number of calculators sold per week at a price of $x$ dollars.

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Do you want further details or have any questions?

Here are 5 related questions:

1. How would the function change if the price increased to $90?
2. What is the significance of the slope in this problem?
3. How can we use this function to find the price that maximizes revenue?
4. How many calculators will be sold if the price is set at $75?
5. What would happen to the number of calculators sold if the price reached $0?

Tip: The slope of a linear function in economics often represents how sensitive demand is to price changes (elasticity).