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andreyandreev [35.5K]
3 years ago
7

An inner city revitalization zone is a rectangle that is twice as long as it is wide. The width of the region is growing at a ra

te of 24 m per year at a time when the region is 300 m wide. How fast is the area changing at that point in time
Physics
2 answers:
lukranit [14]3 years ago
3 0

Answer:

\frac{dA}{dt} = 28800 \ m^2/year

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

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

Equality Properties

<u>Geometry</u>

  • Area of a Rectangle: A = lw

<u>Algebra I</u>

  • Exponential Property: w^n \cdot w^m = w^{n + m}

<u>Calculus</u>

Derivatives

Differentiating with respect to time

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Explanation:

<u>Step 1: Define</u>

Area is A = lw

2w = l

w = 300 m

\frac{dw}{dt} = 24 \ m/year

<u>Step 2: Rewrite Equation</u>

  1. Substitute in <em>l</em>:                    A = (2w)w
  2. Multiply:                              A = 2w²

<u>Step 3: Differentiate</u>

<em>Differentiate the new area formula with respect to time.</em>

  1. Differentiate [Basic Power Rule]:                                                                   \frac{dA}{dt} = 2 \cdot 2w^{2-1}\frac{dw}{dt}
  2. Simplify:                                                                                                           \frac{dA}{dt} = 4w\frac{dw}{dt}

<u>Step 4: Find Rate</u>

<em>Use defined variables</em>

  1. Substitute:                    \frac{dA}{dt} = 4(300 \ m)(24 \ m/year)
  2. Multiply:                        \frac{dA}{dt} = (1200 \ m)(24 \ m/year)
  3. Multiply:                        \frac{dA}{dt} = 28800 \ m^2/year
eduard3 years ago
3 0

Answer:

28,800 m²/yr

Explanation:

This rectangle has dimensions such that:

  • width = w
  • length = 2w  

We are given \displaystyle \frac{dw}{dt} = \frac{24 \ m}{yr} and want to find \displaystyle \frac{dA}{dt} \Biggr | _{w \ = \ 300 \ m} = \ ? when w = 300 m.

The area of a rectangle is denoted by Area = length * width.

Let's multiply the width and length (with respect to w) together to have an area equation in terms of w:

  • A=2w^2

Differentiate this equation with respect to time t.  

  • \displaystyle \frac{dA}{dt} =4w \cdot \frac{dw}{dt}

Let's plug known values into the equation:

  • \displaystyle \frac{dA}{dt} =4(300) \cdot (24)

Simplify this equation.

  • \displaystyle \frac{dA}{dt} =1200 \cdot 24
  • \displaystyle \frac{dA}{dt} =28800

The area is changing at a rate of 28,800 m²/yr at this point in time.

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