Great! Let's solve each part of the question step by step.
1.
a. Given that you walk at a rate of 2 miles per hour, let's define a function that calculates the distance walked based on the time.
Input: Time (hours)
Output: Distance walked (miles)
The distance walked can be calculated using the formula:
[tex]\[ \text{distance} = \text{rate} \times \text{time} \][/tex]
So, if the rate is 2 miles per hour:
[tex]\[ \text{distance} = 2 \times \text{time} \][/tex]
Let's define this function [tex]\( D(t) \)[/tex]:
[tex]\[ D(t) = 2t \][/tex]
where [tex]\( t \)[/tex] is the time in hours.
b. Given that you burn 110 calories per mile, let's define a function that calculates the calories burned based on the distance walked.
Input: Distance (miles)
Output: Calories burned
The calories burned can be calculated using the formula:
[tex]\[ \text{calories} = \text{calories\_per\_mile} \times \text{distance} \][/tex]
So, if 110 calories are burnt per mile:
[tex]\[ \text{calories} = 110 \times \text{distance} \][/tex]
Let's define this function [tex]\( C(d) \)[/tex]:
[tex]\[ C(d) = 110d \][/tex]
where [tex]\( d \)[/tex] is the distance in miles.
c. Now, let’s create a composite function that calculates the calories burned based on the time walked.
We already have:
[tex]\[ D(t) = 2t \][/tex]
[tex]\[ C(d) = 110d \][/tex]
The composite function [tex]\( C(D(t)) \)[/tex] means we first calculate the distance walked using time [tex]\( t \)[/tex], and then we use that distance to calculate the calories burned.
[tex]\[ C(D(t)) = 110 \times (2t) \][/tex]
Simplifying, we get:
[tex]\[ C(D(t)) = 220t \][/tex]
So the composite function [tex]\( C_D(t) \)[/tex] that gives calories burned given time walked is:
[tex]\[ C_D(t) = 220t \][/tex]
d. Using the composite function [tex]\( C_D(t) \)[/tex], we can find out how many calories are burned in 3 hours of walking.
[tex]\[ t = 3 \][/tex]
[tex]\[ C_D(3) = 220 \times 3 \][/tex]
[tex]\[ C_D(3) = 660 \][/tex]
Answer: In 3 hours of walking, you burn 660 calories.
