Question

at the beginning of the year, rajani had $50 in savings and saved an additional $15 each week thereafter. charlotte started the year with $95 and saved $10 every week. let ( r ) represent the amount of money rajani has saved ( t ) weeks after the beginning of the year and let ( c ) represent the amount of money charlotte has saved ( t ) weeks after the beginning of the year. write an equation for each situation, in terms of ( t ), and determine the interval of time when rajani has more in savings than charlotte.
answer
attempt 1 out of 2
( r = )
( c = )
rajani has more in savings than charlotte when ( t )

Explanation:

Step1: Define Rajani's savings function

Rajani starts with $50 and saves $15 per week. So the function for Rajani's savings \( R \) in terms of weeks \( t \) is \( R = 50 + 15t \).

Step2: Define Charlotte's savings function

Charlotte starts with $95 and saves $10 per week. So the function for Charlotte's savings \( C \) in terms of weeks \( t \) is \( C = 95 + 10t \).

Step3: Find when \( R > C \)

Set up the inequality \( 50 + 15t > 95 + 10t \).
Subtract \( 10t \) from both sides: \( 50 + 5t > 95 \).
Subtract 50 from both sides: \( 5t > 45 \).
Divide both sides by 5: \( t > 9 \).

Answer:

\( R = 50 + 15t \)
\( C = 95 + 10t \)
Rajani has more savings than Charlotte when \( t > 9 \) (for \( t \) in weeks, \( t \) is a positive integer and \( t > 9 \))