Make a word problem piecewise function using
y=−0.001 (x−1000)² + 3000 {450≤x≤1550}
y=−0.001(x−3000)² +3000 {2450≤x≤3550}

Here’s a word problem that incorporates the given piecewise function:

---

**Word Problem:**

A company is planning two advertising campaigns, each with different target audiences. The success of each campaign is measured by the number of potential customers who engage with the advertisement, denoted by $ y $, based on the amount of money, $ x $, invested in thousands of dollars.

1. **Campaign 1** targets a younger audience and is most effective when investments are between $450,000 and $1,550,000. The effectiveness $ y $ of this campaign is modeled by the function:
   \[
   y = -0.001(x - 1000)^2 + 3000 \quad \text{for} \quad 450 \leq x \leq 1550
   \]
   where $ x $ is the investment in thousands of dollars.

2. **Campaign 2** targets a more mature audience and is most effective when investments are between $2,450,000 and $3,550,000. The effectiveness $ y $ of this campaign is modeled by the function:
   \[
   y = -0.001(x - 3000)^2 + 3000 \quad \text{for} \quad 2450 \leq x \leq 3550
   \]
   where $ x $ is again the investment in thousands of dollars.

**Questions:**

- How much money should be invested in each campaign to achieve the maximum engagement?
- What is the maximum number of potential customers each campaign can engage?
- If the company invests $500,000 in the first campaign, how many customers are expected to engage?
- How many customers are expected to engage if $3,000,000 is invested in the second campaign?
- If the company invests $1,550,000 in the first campaign, what will the engagement be?
- Compare the effectiveness of investing $1,000,000 in the first campaign versus $3,000,000 in the second campaign.
- Determine the intervals in which the investment leads to an engagement of more than 2,000 customers for both campaigns.
- For each campaign, how does the effectiveness change as the investment increases from the minimum to the maximum limit?

**Tip:** When dealing with quadratic functions, the vertex gives the maximum or minimum value depending on whether the parabola opens upwards or downwards.

Would you like to explore any of these questions in more detail or have any other questions related to this problem?

Explore a word problem involving piecewise functions in quadratic form, modeling the engagement of advertising campaigns based on investment levels. Includes calculations for maximum engagement, investment thresholds, and comparisons between campaign effectiveness. Suitable for high school students studying quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation