Answer:

General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<u>Algebra I</u>
- Terms/Coefficients
- Expand by FOIL (First Outside Inside Last)
- Factoring
<u>Calculus</u>
Differentiation
Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integration Property:

- Fundamental Theorem of Calculus:

- Arc Length Formula:
![\displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20AL%20%3D%20%5Cint%5Climits%5Ea_b%20%7B%5Csqrt%7B1%2B%20%5Bf%27%28x%29%5D%5E2%7D%7D%20%5C%2C%20dx)
- Surface Area Formula:
![\displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20SA%20%3D%202%5Cpi%20%5Cint%5Climits%5Ea_b%20%7Bf%28x%29%20%5Csqrt%7B1%2B%20%5Bf%27%28x%29%5D%5E2%7D%7D%20%5C%2C%20dx)
Step-by-step explanation:
<u>Step 1: Define</u>
![\displaystyle y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}\\Interval: [0, \frac{1}{3}]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B1%7D%7B3%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20-%20x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5C%5CInterval%3A%20%5B0%2C%20%5Cfrac%7B1%7D%7B3%7D%5D)
<u>Step 2: Differentiate</u>
- Basic Power Rule:

- [Derivative] Simplify:

- [Derivative] Simplify:

<u>Step 3: Integrate Pt. 1</u>
- Substitute [Surface Area]:
![\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{1+ [\frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}]^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20SA%20%3D%202%5Cpi%20%5Cint%5Climits%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D_0%20%7B%28%5Cfrac%7B1%7D%7B3%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20-%20x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%29%20%5Csqrt%7B1%2B%20%5B%5Cfrac%7B1%7D%7B6%5Csqrt%7Bx%7D%7D%20-%20%5Cfrac%7B3%5Csqrt%7Bx%7D%7D%7B2%7D%7D%5D%5E2%7D%7D%20%5C%2C%20dx)
- [Integral - √Radical] Expand/Add:

- [Integral - √Radical] Factor:

- [Integral - Simplify]:

- [Integral] Integration Property:

<u>Step 4: Integrate Pt. 2</u>
- [Integral] Define:

- [Integral] Assumption of Positive/Correction Factors:

- [Integral] Expand - FOIL:

- [Integral] Integrate - Basic Power Rule:

- [Expression] Multiply:

<u>Step 5: Integrate Pt. 3</u>
- [Integral] Substitute/Integral - FTC:

- [Integrate] Evaluate FTC:

- [Expression] Multiply:

<em>It is in ft² because it is given that our axis are in ft.</em>
<u>Step 6: Find Amount of Glass</u>
<em>Convert ft² to in² and multiply by 0.015 in (given) to find amount of glass.</em>
- Convert ft² to in²:

- Multiply:

And we have our final answer! Hope this helped on your Calc BC journey!
Given:
Population of a colony in 2000 = 250
Population of a colony in 2004 = 400
To find:
The average rate of change.
Solution:
Let y be the population of the colony at x year.
The population of a colony grows from 250 in the year 2000 to 400 in the year 2004. So, the two points are (2000,250) and (2004,400).
Formula for slope or average rate of change is




Therefore, the average rate of change of population is 37.5.
Yes it is true if you need me to explain lmk
Answer:
$7.50
Step-by-step explanation:
Since there is a 50% chance that the card is odd, the expected payoffs would be $15 * 50%=0.5*15=7.5. Hope this helps!