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Naily [24]
3 years ago
8

The distance s that an object falls is directly proportional to the square of the time t of the fall. If an object falls 16 feet

in 1​ second, how far will it fall in 2 ​seconds? How long will it take an object to fall 400 ​feet?
Mathematics
1 answer:
Lemur [1.5K]3 years ago
7 0

Answer: if S is the distance, the fact that the distance is proportional to the square of the time means that S(t) = a*t^{2} where a is a constant of proportionality, with units of \frac{feet}{s^{2} }.

We know that S(1) = 16 feet = a*1, then a = 16\frac{feet}{s^{2} }

then S(2) = 16*4 feet = 64.

and if we want to know in how much time te objet will fall 400feet, then S(t₀) = 400 feet = 16*t₀^{2}

so t₀ = \sqrt{400/16} = 5s.

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Answer:

Point A(9, 3)

General Formulas and Concepts:

<u>Pre-Algebra</u>  

Order of Operations: BPEMDAS  

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

Equality Properties  

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

<u>Algebra I</u>  

  • Coordinates (x, y)
  • Functions
  • Function Notation
  • Terms/Coefficients
  • Anything to the 0th power is 1
  • Exponential Rule [Rewrite]:                                                                              \displaystyle b^{-m} = \frac{1}{b^m}
  • Exponential Rule [Root Rewrite]:                                                                     \displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}  

<u>Calculus</u>  

Derivatives  

Derivative Notation  

Derivative of a constant is 0  

Basic Power Rule:  

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Derivative Rule [Chain Rule]:                                                                                    \displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

<em />\displaystyle y = \sqrt{x}<em />

<em />\displaystyle y' = \frac{1}{6}<em />

<em />

<u>Step 2: Differentiate</u>

  1. [Function] Rewrite [Exponential Rule - Root Rewrite]:                                   \displaystyle y = x^{\frac{1}{2}}
  2. Basic Power Rule:                                                                                             \displaystyle y' = \frac{1}{2}x^{\frac{1}{2} - 1}
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  4. [Derivative] Rewrite [Exponential Rule - Rewrite]:                                          \displaystyle y' = \frac{1}{2x^{\frac{1}{2}}}
  5. [Derivative] Rewrite [Exponential Rule - Root Rewrite]:                                 \displaystyle y' = \frac{1}{2\sqrt{x}}

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<em>Find coordinates of A.</em>

<em />

<em>x-coordinate</em>

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  2. [Multiplication Property of Equality] Multiply 2 on both sides:                      \displaystyle \frac{1}{3} = \frac{1}{\sqrt{x}}
  3. [Multiplication Property of Equality] Cross-multiply:                                      \displaystyle \sqrt{x} = 3
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<em>y-coordinate</em>

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∴ Coordinates of A is (9, 3).

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives

Book: College Calculus 10e

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