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The number y of raccoons in an area after x years can be modeled by the function y= 0.4x^2+2x+2. When were there about 45 raccoons in the area? Round your answer to the nearest year

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Answer:

A timeframe of 8 years is when there were 45 raccoons in the area.

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Algebra I

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Standard Form:
[tex]\displaystyle ax^2 + bx + c = 0[/tex]

Quadratic Formula:
[tex]\displaystyle x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \begin{aligned}y & = 0.4x^2 + 2x + 2 \\y & = 45 \ \text{raccoons} \\\end{aligned}[/tex]

Step 2: Find Specific Year

We are trying to find the year when there were 45 raccoons present in the area. From first glance, we see we probably can't factor the quadratic expression, so let's set up to use the Quadratic Formula:

  1. [Model Equation] Substitute in y:
    [tex]\displaystyle 0.4x^2 + 2x + 2 = 45[/tex]
  2. [Equality Properties] Rewrite [Standard Form]:
    [tex]\displaystyle 0.4x^2 + 2x + 2 = 45 \rightarrow 0.4x^2 + 2x - 43 = 0[/tex]
  3. [Standard Form] Identify:
    [tex]\displaystyle 0.4x^2 + 2x - 43 = 0 \\\downarrow \\\begin{aligned}a & = 0.4 \\b & = 2 \\c & = -43 \\\end{aligned}[/tex]

Now that we have our variables from Standard Form, we can use the Quadratic Formula to find which years when there were 45 raccoons present in the area:

  1. [Quadratic Formula] Substitute in variables:
    [tex]\displaystyle x=\frac{-2 \pm \sqrt{2^2 - 4(0.4)(-43)}}{2(0.4)}[/tex]
  2. [Order of Operations] Evaluate:
    [tex]\displaystyle \begin{aligned}x & = \frac{-2 \pm \sqrt{2^2 - 4(0.4)(-43)}}{2(0.4)} \\& = \frac{-2 \pm 8.53229}{0.8} \\& = -13.1654, 8.16536\end{aligned}[/tex]

Since time cannot be negative, we can isolate the other root to obtain our final answer:

[tex]\displaystyle\begin{aligned}x & = 8.16536 \ \text{years} \\& \approx \boxed{ 8 \ \text{years} } \\\end{aligned}[/tex]

∴ we have found the approximate amount of years to be 8 years when there were 45 raccoons in the area.

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Learn more about Algebra I: https://brainly.com/question/16442214

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Topic: Algebra I

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