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:
\(\[ \text{distance} = \text{rate} \times \text{time} \]\)
So, if the rate is 2 miles per hour:
\(\[ \text{distance} = 2 \times \text{time} \]\)
Let's define this function \(\( D(t) \)\):
\(\[ D(t) = 2t \]\)
where \(\( t \)\) 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:
\(\[ \text{calories} = \text{calories\_per\_mile} \times \text{distance} \]\)
So, if 110 calories are burnt per mile:
\(\[ \text{calories} = 110 \times \text{distance} \]\)
Let's define this function \(\( C(d) \)\):
\(\[ C(d) = 110d \]\)
where \(\( d \)\) 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:
\(\[ D(t) = 2t \]\)
\(\[ C(d) = 110d \]\)
The composite function \(\( C(D(t)) \)\) means we first calculate the distance walked using time \(\( t \)\), and then we use that distance to calculate the calories burned.
\(\[ C(D(t)) = 110 \times (2t) \]\)
Simplifying, we get:
\(\[ C(D(t)) = 220t \]\)
So the composite function \(\( C_D(t) \)\) that gives calories burned given time walked is:
\(\[ C_D(t) = 220t \]\)
d. Using the composite function \(\( C_D(t) \)\), we can find out how many calories are burned in 3 hours of walking.
\(\[ t = 3 \]\)
\(\[ C_D(3) = 220 \times 3 \]\)
\(\[ C_D(3) = 660 \]\)
Answer: In 3 hours of walking, you burn 660 calories.
