On Saturday morning, Avery took a hike in the hills near her house. The table below shows the cumulative number of calories she burned on Saturday. Time spent hiking (min) Cumulative number of calories burned 5568 10591 15614 20637 Write an equation for a line that represents Avery’s cumulative calories burned as a function of time, in minutes. Be sure to define your variables. Answer _____________________________________________________________ Did you write your equation in point-slope form,, or slope-intercept form,푦 − 푦 1 = 푚(푥 − 푥 1 ) ? Explain why you chose the form that you did.푦 = 푚푥 + 푏 _____________________________________________________________ Identify the set of numbers that best defines the range of this relationship and explain what it means in context.
Let's define our variables:
- Time spent hiking (min) = x
- Cumulative number of calories burned = y
To find the equation of the line that represents Avery's cumulative calories burned as a function of time, we can use the slope-intercept form, y = mx + b.
Using the points (5568, 10591) and (20637, 15614), we can find the slope (m) first:
m = (y2 - y1) / (x2 - x1)
m = (15614 - 10591) / (20637 - 5568)
m = 5023 / 15069
m ≈ 0.333
Now, let's substitute the slope and one of the points into the equation:
10591 = 0.333 * 5568 + b
10591 = 1854.144 + b
b ≈ 8736.856
Therefore, the equation that represents Avery's cumulative calories burned as a function of time is:
y = 0.333x + 8736.856
For the form of the equation, I chose slope-intercept form, y = mx + b, because it allows us to directly see the slope (m) and the y-intercept (b), which have a clear interpretation in the context of the problem.
The range of this relationship (cumulative calories burned) would be the set of y-values, which in this case would be all the possible cumulative number of calories burned. Based on the data given, the range can be defined as [10591, 15614]. This means that Avery burned a minimum of 10,591 calories and a maximum of 15,614 calories during the time she spent hiking.