3. chicagos average monthly rainfall, $r = f(t)$ inches, is given as a function of the month, $t$, in the table below (january is $t = 1$).
| t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| r | 1.8 | 1.8 | 2.7 | 3.1 | 3.5 | 3.7 | 3.5 | 3.4 |
(a) solve and interpret.
i. $f(t) = 3.5$
ii. $f(t) = f(2)$
(b) assume there is more data stored in another table that is not on this worksheet. determine weather the following has, one, no, or possibly multiple solutions. explain your reasoning.
i. $f(15) = $
ii. $f(t) = 3.6$

Question

3. chicagos average monthly rainfall, $r = f(t)$ inches, is given as a function of the month, $t$, in the table below (january is $t = 1$).
| t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| r | 1.8 | 1.8 | 2.7 | 3.1 | 3.5 | 3.7 | 3.5 | 3.4 |
(a) solve and interpret.
 i. $f(t) = 3.5$
 ii. $f(t) = f(2)$
(b) assume there is more data stored in another table that is not on this worksheet. determine weather the following has, one, no, or possibly multiple solutions. explain your reasoning.
 i. $f(15) = $
 ii. $f(t) = 3.6$

Accepted Answer

### Part (a) i: Solve $ f(t) = 3.5 $
# Explanation:
## Step 1: Identify $ t $ with $ R = 3.5 $
From the table, find $ t $ where $ R = 3.5 $. The table shows $ R = 3.5 $ when $ t = 5 $ and $ t = 7 $.
## Step 2: Interpret the solution
$ t = 5 $ (May) and $ t = 7 $ (July) have an average monthly rainfall of 3.5 inches.
# Answer: $ t = 5 $ and $ t = 7 $; Interpretation: May (t=5) and July (t=7) have an average monthly rainfall of 3.5 inches.

### Part (a) ii: Solve $ f(t) = f(2) $
# Explanation:
## Step 1: Find $ f(2) $
From the table, when $ t = 2 $, $ R = 1.8 $, so $ f(2) = 1.8 $.
## Step 2: Find $ t $ with $ R = 1.8 $
From the table, $ R = 1.8 $ when $ t = 1 $ (January) and $ t = 2 $ (February).
## Step 3: Interpret the solution
$ t = 1 $ (January) and $ t = 2 $ (February) have the same average monthly rainfall as February (t=2).
# Answer: $ t = 1 $ and $ t = 2 $; Interpretation: January (t=1) and February (t=2) have the same average monthly rainfall as February (t=2).

### Part (b) i: Determine solutions for $ f(15) $
# Explanation:
## Step 1: Analyze the function
$ f(t) $ is a function of the month $ t $, where $ t = 1 $ is January. $ t = 15 $ corresponds to a specific month (15 - 12 = 3, so March of the next year, but the function is defined for months, so $ t = 15 $ is a valid input (a specific month). A function has exactly one output for each input, so $ f(15) $ has exactly one solution.
# Answer: One solution. Reason: A function assigns exactly one output to each input, so $ f(15) $ (where $ t = 15 $ is a valid month input) has exactly one value.

### Part (b) ii: Determine solutions for $ f(t) = 3.6 $
# Explanation:
## Step 1: Analyze the data range
From the given table, the rainfall values are 1.8, 1.8, 2.7, 3.1, 3.5, 3.7, 3.5, 3.4. 3.6 is between 3.5 and 3.7 (e.g., between $ t = 6 $ (3.7) and $ t = 5,7 $ (3.5)). Since there is more data (not shown), the rainfall could equal 3.6 in one, none, or multiple months (e.g., if the function is continuous or has multiple points with 3.6). So it could have multiple solutions, one, or none, but "possibly multiple" is the most reasonable (as it's between existing values, so maybe one or more months have 3.6).
## Step 2: Conclusion
Since 3.6 is within the range of observed rainfall (3.5 to 3.7) and there's more data, $ f(t) = 3.6 $ could have multiple solutions (or one, or none, but "possibly multiple" is appropriate as we don't know the full data).
# Answer: Possibly multiple solutions. Reason: 3.6 is between existing rainfall values (3.5 and 3.7) in the table, and with more data, there may be one or more months (t) where rainfall is 3.6.

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QUESTION IMAGE

Question

Explanation:

Part (a) i: Solve \( f(t) = 3.5 \)

Step 1: Identify \( t \) with \( R = 3.5 \)

Step 2: Interpret the solution

Step 1: Find \( f(2) \)

Step 2: Find \( t \) with \( R = 1.8 \)

Step 3: Interpret the solution

Step 1: Analyze the function

Answer:

Part (a) ii: Solve \( f(t) = f(2) \)