Question

starting a business can be difficult. according to some data, after 5 years, 35 businesses remain open and after 10 years only 20 businesses remain open.
a. pick variables to use in describing this pattern and indicate what each of these variables represent.

variable (letter)	meaning of variable (including units)
dependent	b	business remain open

b. what is the slope of your model?
c. explain what the slope means in the context of the situation?
d. find the vertical intercept of your model.
e. explain what the vertical intercept means in the context of the situation?
f. write an equation to model the data. be careful what variables you use!
g. according to your model, how many businesses are open after 12 years?
h. according to your model, after how many years will 40 businesses be still open?
i. what is the horizontal intercept of your model?
j. explain what the horizontal intercept means in the context of the situation?

Explanation:

Step1: Define two - point form variables

Let $(y_1,b_1)=(5,35)$ and $(y_2,b_2)=(10,20)$.

Step2: Calculate the slope

The slope formula is $m=\frac{b_2 - b_1}{y_2 - y_1}$. Substituting the values, we have $m=\frac{20 - 35}{10 - 5}=\frac{- 15}{5}=-3$.

Step3: Use point - slope form to find the y - intercept

The point - slope form is $b - b_1=m(y - y_1)$. Using the point $(y_1,b_1)=(5,35)$ and $m=-3$, we get $b-35=-3(y - 5)$. Expand to $b-35=-3y + 15$. Then $b=-3y+50$. The vertical intercept (when $y = 0$) is $b = 50$.

Step4: Write the linear equation

The linear equation is $b=-3y + 50$.

Step5: Answer part b

The slope of the model is $-3$.

Step6: Answer part c

The slope of $-3$ means that the number of businesses remaining open decreases by 3 each year.

Step7: Answer part d

The vertical intercept is 50.

Step8: Answer part e

The vertical intercept of 50 means that at the start (year $y = 0$), there were 50 businesses open.

Step9: Answer part f

The equation is $b=-3y + 50$.

Step10: Answer part g

When $y = 12$, $b=-3\times12 + 50=-36 + 50 = 14$.

Step11: Answer part h

Set $b = 40$ in the equation $40=-3y+50$. Then $3y=10$, so $y=\frac{10}{3}\approx3.33$ years.

Step12: Answer part i

To find the horizontal intercept, set $b = 0$. Then $0=-3y + 50$, so $y=\frac{50}{3}\approx16.67$ years.

Step13: Answer part j

The horizontal intercept of $\frac{50}{3}\approx16.67$ years means that after approximately 16.67 years, no businesses will remain open.

Answer:

b. - 3
c. The number of open - businesses decreases by 3 each year.
d. 50
e. At the start, there were 50 businesses open.
f. $b=-3y + 50$
g. 14
h. $\frac{10}{3}$ years
i. $\frac{50}{3}$ years
j. After approximately 16.67 years, no businesses will remain open.