9. Nathan's bank balance has increased in a linear manner since he started his part-time job. If after 20 weeks of work his bank balance was at $$ \$ 560 $$ and after 21 weeks of work it was at $$ \$ 585 $$, determine: a. the rule that relates the size of his bank balance, $$ A $$, and the time (in weeks) worked, $$ t $$ b. the amount in his account after 200 weeks c. the initial amount.

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UpStudy AI · Accepted Answer

1. Since Nathan's bank balance increases linearly, we can express it as  
   $$
   A = mt + b,
   $$  
   where $$ m $$ is the weekly increase and $$ b $$ is the initial amount (when $$ t=0 $$).

2. We are given two points:
   - At $$ t = 20 $$:  
     $$
     560 = 20m + b.
     $$
   - At $$ t = 21 $$:  
     $$
     585 = 21m + b.
     $$

3. Subtract the first equation from the second to find the slope $$ m $$:
   $$
   585 - 560 = (21m + b) - (20m + b) \Rightarrow 25 = m.
   $$

4. Substitute $$ m = 25 $$ into one of the equations to solve for $$ b $$. Using $$ t = 20 $$:
   $$
   560 = 25(20) + b \Rightarrow 560 = 500 + b \Rightarrow b = 560 - 500 = 60.
   $$

5. Therefore, the linear rule that relates Nathan's bank balance $$ A $$ and the time $$ t $$ is:
   $$
   A = 25t + 60.
   $$

6. After 200 weeks, substitute $$ t = 200 $$ into the expression:
   $$
   A = 25(200) + 60 = 5000 + 60 = 5060.
   $$

7. The initial amount in his bank account (when $$ t = 0 $$ weeks) is:
   $$
   A(0) = 25(0) + 60 = 60.
   $$

Thus, the answers are:  
a. $$ A = 25t + 60 $$  
b. The amount after 200 weeks is $$ \$5060 $$  
c. The initial amount is $$ \$60 $$
a. The bank balance $$ A $$ after $$ t $$ weeks is given by $$ A = 25t + 60 $$. b. After 200 weeks, Nathan has $$ \$5060 $$ in his account. c. The initial amount in his account was $$ \$60 $$.

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