We are given that the sales of a small company were $46,000 in its third year of operation (i.e., when $x = 3$) and $88,000 in its sixth year of operation (i.e., when $x = 6$). We are asked to:

1. Find the slope of the sales line and give an equation for the line in the form $y = mx + b$.
2. Use the equation to find how many years must pass before the sales surpass $110,000.

Part (a): Find the slope and equation of the line.

We have two points on the line:

• $(x_1, y_1) = (3, 46,000)$
• $(x_2, y_2) = (6, 88,000)$

Step 1: Find the slope.

The formula for the slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the values: $m = \frac{88,000 - 46,000}{6 - 3} = \frac{42,000}{3} = 14,000$ So, the slope is $m = 14,000$.

Step 2: Find the equation of the line.

The general form of a line is: $y = mx + b$ We already know $m = 14,000$. Now, we need to find $b$, the y-intercept. To do this, substitute one of the given points (let's use $(3, 46,000)$) into the equation:

$46,000 = 14,000(3) + b$ $46,000 = 42,000 + b$ $b = 46,000 - 42,000 = 4,000$

Thus, the equation of the sales line is: $y = 14,000x + 4,000$

Part (b): Find how many years must pass before the sales surpass $110,000.

We need to find the value of $x$ such that $y = 110,000$.

Using the equation $y = 14,000x + 4,000$, set $y = 110,000$ and solve for $x$:

$110,000 = 14,000x + 4,000$ $110,000 - 4,000 = 14,000x$ $106,000 = 14,000x$ $x = \frac{106,000}{14,000} = 7.57$

Thus, the sales will surpass $110,000 after approximately 7.57 years. Since we can't have a fraction of a year in this context, we round up to the next whole year, meaning the sales will surpass $110,000 after 8 years.

Final Answers:

• (a) The slope of the sales line is 14,000, and the equation is $y = 14,000x + 4,000$.
• (b) The sales will surpass $110,000 after 8 years.

Would you like further details or clarifications on any part?

Here are some related questions to consider:

1. What would the sales be in the 10th year using the same equation?
2. How many years would it take for sales to reach $150,000?
3. What does the slope of the line represent in the context of this problem?
4. How would the equation change if sales in the first year were $20,000?
5. Can we estimate the company’s sales growth if a quadratic function better represented the data?

Tip: When interpreting a linear equation, the slope represents the rate of change, while the intercept represents the starting point or initial value.