Sarah has been running a dog-walking business since 2010. She walks dogs twice a day, takes them to the park, and returns them to their homes. Each year, she has increased her fee by the same amount. The table shows what Sarah charged each customer for two given years of her business:
Year Annual Dog-walking Fee
2010 $350
2014 $750
A. What is the rate of change and initial value for Sarah’s business? How do you know?
B. Write an equation in slope-intercept form to represent the fees that Sarah charges each year.
A. To determine the rate of change and initial value for Sarah's business, we need to find the slope and y-intercept of the relation between the year and the annual dog-walking fee.
First, let's calculate the change in fees from 2010 to 2014:
Change in fees = $750 - $350 = $400
Next, calculate the change in years:
Change in years = 2014 - 2010 = 4 years
Now, to find the rate of change (slope), divide the change in fees by the change in years:
Rate of change = Change in fees / Change in years = $400 / 4 years = $100 per year
The rate of change is $100 per year. This means that Sarah increases her fees by $100 each year.
The initial value is the fee charged in the first year, which is $350.
B. To write an equation in slope-intercept form (y = mx + b) to represent the fees that Sarah charges each year, we can use the rate of change (m) and the initial value (b).
The equation would be:
Fee(year) = Rate of change * Year + Initial value
Substituting the values we found:
Fee(year) = $100 * Year + $350
Therefore, the equation that represents the fees Sarah charges each year is:
Fee(year) = $100 * Year + $350
If the yearly increase is x, then
(750-350)/(2014-2010) = 400/4 = 100
Now you have a point and a slope, so
y-350 = 100(x-2010)