What is the range of S(t)?

Steps to Solve

1. Understanding Shelby's Savings Function

Shelby's savings function $S(t)$ starts at $100 and increases by $25 for each deposit she makes. The function can be expressed as: $$ S(t) = 100 + 25t $$ where $t$ is the number of deposits she makes.

2. Finding Maximum Savings

Shelby wants to buy a tablet that costs $250. Therefore, we need to set the savings function equal to this amount to find the maximum deposits: $$ 100 + 25t \leq 250 $$

3. Solving for $t$

Rearranging the inequality: $$ 25t \leq 150 $$

Dividing both sides by 25: $$ t \leq 6 $$

This means Shelby can make at most 6 deposits.

4. Calculating S(t) for t = 0 to 6

Now we can find the range of $S(t)$ by calculating the savings for $t$ values from 0 to 6:

• For $t = 0$: $S(0) = 100 + 25(0) = 100$
• For $t = 1$: $S(1) = 100 + 25(1) = 125$
• For $t = 2$: $S(2) = 100 + 25(2) = 150$
• For $t = 3$: $S(3) = 100 + 25(3) = 175$
• For $t = 4$: $S(4) = 100 + 25(4) = 200$
• For $t = 5$: $S(5) = 100 + 25(5) = 225$
• For $t = 6$: $S(6) = 100 + 25(6) = 250$

5. Determining the Range of S(t)

The set of values ( S(t) ) can take, as calculated, is: $$ {100, 125, 150, 175, 200, 225, 250} $$

The range of $S(t)$ consists of the whole numbers from 100 to 250, in steps of $25$.