Let's break down the question step by step:
### Part (a): Writing a Rule to Describe the Function
We will create a rule that describes the distance [tex]\( d \)[/tex] a snail travels given a certain time [tex]\( t \)[/tex] in minutes, considering the snail travels at a constant rate of [tex]\( 2.58 \)[/tex] feet per minute.
The general rule or function to describe this relationship would be:
[tex]\[ d(t) = 2.58 \times t \][/tex]
Here, [tex]\( t \)[/tex] represents the time in minutes, and [tex]\( d(t) \)[/tex] represents the distance traveled in feet. This rule expresses a linear relationship between time and distance, where the rate of travel (slope) is [tex]\( 2.58 \)[/tex] feet per minute.
### Part (b): Calculating the Distance Traveled in 9 Minutes
Now we will calculate how far the snail will travel in 9 minutes using the rule we just established.
Using the function [tex]\( d(t) = 2.58 \times t \)[/tex]:
1. Substitute [tex]\( t = 9 \)[/tex] minutes into the function:
[tex]\[ d(9) = 2.58 \times 9 \][/tex]
2. Perform the multiplication:
[tex]\[ d(9) = 23.22 \, \text{feet} \][/tex]
So, the snail will travel 23.22 feet in 9 minutes.
### Comparing the Options
Given these calculations:
- [tex]\( d(t) = t + 2.58 ; 11.58 \, \text{ft} \)[/tex] is incorrect both in form and result.
- [tex]\( d(t) = \frac{t}{2.58} ; 3.49 \, \text{ft} \)[/tex] is also incorrect in form and result.
- [tex]\( d(t) = 2.58 t ; 23.22 \, \text{ft} \)[/tex] is correct in both form and result.
- [tex]\( d(t) = 9 t ; 23.22 \, \text{ft} \)[/tex] is incorrect in form even though the result matches.
Thus, the correct rule to describe the function is:
[tex]\[ d(t) = 2.58 t \][/tex]
And the snail will travel [tex]\( 23.22 \)[/tex] feet in 9 minutes.
