Question

at the beginning of the summer, there are 9 rabbits living in a field. each week for the rest of the summer, another rabbit moves in. define a unit for the amount of time since summer started. enter a variable for the amount of time since summer started and use this variable to write an expression for the number of rabbits living in the field. 1 how many rabbits will be living in the field 35 days after summer starts? 2. when there are 16 rabbits living in the field, how many weeks has it been? after completing the worksheet, graph your model. quantity name unit expression question 1 question 2 time 7 rabbits rabbits 14 plot point plot point rabbits time 0 1 2 3 4 5 6 7 8 9 10 11 0 3 6 9 12 15 18 21 -1 0

Explanation:

Step1: Define the model

Let \( t \) be the number of weeks since summer started. The initial number of rabbits is 9, and each week 1 more rabbit moves in. So the number of rabbits \( R \) is given by the linear equation \( R = 9 + t \).

Step2: Solve for \( t \) when \( R = 35 \)

We set up the equation \( 35 = 9 + t \). Subtract 9 from both sides: \( t = 35 - 9 = 26 \) weeks. But the question asks for days. Since 1 week = 7 days, the number of days \( d = 26\times7 = 182 \) days? Wait, no, wait. Wait, maybe I misread. Wait, the first part: "How many rabbits will be living in the field 35 days after summer starts?" Wait, 35 days is \( \frac{35}{7}=5 \) weeks. So using \( R = 9 + t \), where \( t \) is weeks. So \( t = 5 \), so \( R = 9 + 5 = 14 \). Wait, the table has 14 for Question 1. Then Question 2: When there are 16 rabbits, \( 16 = 9 + t \), so \( t = 7 \) weeks, which is \( 7\times7 = 49 \) days? Wait, the table has 7 for Question 2 (time in weeks? Or days?). Wait, the problem says "Define a unit for the amount of time since summer started. Enter a variable for the amount of time since summer started and use this variable to write an expression for the number of rabbits living in the field."

Let's re-express: Let \( t \) be the number of weeks (unit: weeks). Then number of rabbits \( R = 9 + t \).

1. For 35 days: 35 days \( = \frac{35}{7} = 5 \) weeks. So \( R = 9 + 5 = 14 \), which matches the table.

1. For \( R = 16 \): \( 16 = 9 + t \implies t = 7 \) weeks, which is 7 weeks, or \( 7\times7 = 49 \) days. But the table has 7 for Question 2 (time). So the unit for time here is weeks? Wait, the table's "Time" row has 7 for Question 2. So maybe the first question is "How many rabbits after 35 days (5 weeks)?" So \( R = 9 + 5 = 14 \). The second question: "When there are 16 rabbits, how many weeks? \( t = 16 - 9 = 7 \) weeks, so 7 weeks, which is 49 days, but the table has 7 (weeks) for Question 2.

But the first part of the problem (the first question) is "How many rabbits will be living in the field 35 days after summer starts?" So 35 days is 5 weeks. So \( R = 9 + 5 = 14 \), which is what the table shows. Then the second question: "When there are 16 rabbits, how many weeks has it been?" \( 16 = 9 + t \implies t = 7 \) weeks, so 7 weeks, which is 49 days, but the table has 7 (weeks) for Question 2.

Wait, maybe the initial model is \( R = 9 + t \), where \( t \) is weeks. So:

1. 35 days is 5 weeks, so \( R = 9 + 5 = 14 \) rabbits (matches the table's 14 for Rabbits, Question 1).

1. For 16 rabbits, \( t = 16 - 9 = 7 \) weeks (matches the table's 7 for Time, Question 2).

So the first question: 35 days is 5 weeks, so number of rabbits is 14. The second question: 7 weeks (or 49 days) when there are 16 rabbits.

But the user's question (the first part) is "How many rabbits will be living in the field 35 days after summer starts?" So using the model \( R = 9 + t \), \( t \) in weeks. \( t = 35/7 = 5 \). So \( R = 9 + 5 = 14 \).

Answer:

For the first question (35 days after summer starts), the number of rabbits is \(\boldsymbol{14}\). For the second question (when there are 16 rabbits), the time is \(\boldsymbol{7}\) weeks (or 49 days, but the table uses weeks as the unit for time in Question 2).