Sure, let's solve this step by step.
### Part A: Writing a Rule to Describe the Function
First, we need to write a rule to describe the function of how far the snail travels based on the given rate. We're given that the snail travels at a rate of 2.37 feet per minute.
To describe the distance [tex]\( d \)[/tex] the snail travels in [tex]\( t \)[/tex] minutes, we can use the formula for distance:
[tex]\[ d(t) = \text{rate} \times \text{time} \][/tex]
Given the rate is 2.37 feet per minute, our function [tex]\( d(t) \)[/tex] would be:
[tex]\[ d(t) = 2.37 \times t \][/tex]
So, option (b) [tex]\( d(t) = 2.37 \times t \)[/tex] is the correct rule to describe the function.
### Part B: Finding the Distance Traveled in 6 Minutes
Next, we need to find out how far the snail will travel in 6 minutes using the rule we just formulated. We substitute [tex]\( t = 6 \)[/tex] into our function:
[tex]\[ d(6) = 2.37 \times 6 \][/tex]
Using the given result, the distance traveled is calculated to be:
[tex]\[ d(6) = 14.22 \, \text{feet} \][/tex]
Therefore, the correct option is [tex]\( d(t) = 2.37 t ; 14.22 \, \text{ft} \)[/tex].
