Math Problem Statement

Jackie's grandfather gave her $250 for her birthday. Jackie asked her mother to save it for her and only give her $10 each week. The amount of money remaining each week can be found using the function A(t) = 250 - 10t, where t is the number of weeks since her birthday. What are the domain and range of the function for this situation?

Solution

To determine the correct answer for this problem, let's break down the function and analyze its domain and range.

Given Information

  • Function: A(t)=25010tA(t) = 250 - 10t
  • Context: tt represents the number of weeks since Jackie’s birthday, and A(t)A(t) is the amount of money remaining.
  • Initial amount: $250
  • Spending: $10 each week

Analyzing the Domain

  1. Starting Point: At t=0t = 0, Jackie has $250 (initial amount).

  2. End Condition: Jackie spends $10 each week. When she runs out of money, A(t)A(t) will be 0.

    Setting A(t)=0A(t) = 0: 25010t=0250 - 10t = 0 10t=25010t = 250 t=25t = 25

    So, Jackie will have money from t=0t = 0 to t=25t = 25 weeks.

    Therefore, the domain for this situation is 0t250 \leq t \leq 25, where tt is an integer (since we are counting weeks).

Analyzing the Range

  1. Maximum Value: At t=0t = 0, A(0)=250A(0) = 250.

  2. Minimum Value: At t=25t = 25, A(25)=0A(25) = 0.

    Therefore, the range of A(t)A(t) is 0A(t)2500 \leq A(t) \leq 250.

Answer Choice

The correct answer matches this domain and range.

Looking at the options:

  • Option C matches our analysis:**
    • Domain: 0t250 \leq t \leq 25, tZt \in \mathbb{Z}
    • Range: 0A(t)2500 \leq A(t) \leq 250

Thus, the correct answer is Option C.


Would you like a detailed explanation of each step?

Here are 5 related questions that could deepen your understanding:

  1. How would the domain and range change if Jackie spent $5 per week instead of $10?
  2. What would happen to the function if the initial amount was increased to $300?
  3. How would you determine the domain and range if Jackie spent a different amount each week?
  4. Can you create a similar function if Jackie deposited $10 each week instead of spending it?
  5. What does the graph of A(t)=25010tA(t) = 250 - 10t look like over the domain 0t250 \leq t \leq 25?

Tip: Always identify the initial condition and rate of change to help determine the domain and range in word problems like these.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Domain and Range

Formulas

A(t) = 250 - 10t

Theorems

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Suitable Grade Level

Grades 8-10