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Sagot :
Answer:
[tex]\displaystyle \frac{dA}{dt} = 102 \ m^2/day[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
- Left to Right
Geometry
Area of a Rectangle: A = lw
- l is length
- w is width
Calculus
Derivatives
Derivative Notation
Implicit Differentiation
Differentiation with respect to time
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle l = 10 \ meters[/tex]
[tex]\displaystyle \frac{dl}{dt} = 4 \ m/day[/tex]
[tex]\displaystyle w = 23 \ meters[/tex]
[tex]\displaystyle \frac{dw}{dt} = 1 \ m/day[/tex]
Step 2: Differentiate
- [Area of Rectangle] Product Rule: [tex]\displaystyle \frac{dA}{dt} = l\frac{dw}{dt} + w\frac{dl}{dt}[/tex]
Step 3: Solve
- [Rate] Substitute in variables [Derivative]: [tex]\displaystyle \frac{dA}{dt} = (10 \ m)(1 \ m/day) + (23 \ m)(4 \ m/day)[/tex]
- [Rate] Multiply: [tex]\displaystyle \frac{dA}{dt} = 10 \ m^2/day + 92 \ m^2/day[/tex]
- [Rate] Add: [tex]\displaystyle \frac{dA}{dt} = 102 \ m^2/day[/tex]
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Implicit Differentiation
Book: College Calculus 10e
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