Answer:
i guess the answer is c= 14 or 20 .
Answer:
c.5
Step-by-step explanation:
first do the S.R of 9=3
6 times three is 18
18 divided by 3 is 6
6-1 is 5
Answer:
73.46yd²
Step-by-step explanation:
Area of Triangle = ½ * b * h
= ½ * 16 * 19
= 152yd²
Area of Circle= pi * r² = 5² pi = 25 pi = 78.54yd²
area of shaded part= 152 - 78.54
= 73.46yd²
The <em>linear</em> function y = 3 · w - 1 represents the number of sea shells found in each week.
The speed of the <em>driven</em> gear is 180 rounds per minute.
<h3>How to use direct and inverse relationships to analyze situations</h3>
In the first problem we have an example of <em>linear</em> progression, in which the number of sea shells is increased linearly every week. After a quick analysis, we conclude that the <em>linear</em> function y = 3 · w - 1, a kind of <em>direct</em> relationship.
In the second problem, we must an <em>inverse</em> relationship to determine the speed of the <em>driven</em> gear. Please notice that the speed of the gear is inversely proportional to the number of teeths. Then, we proceed to calculate the speed:
![\frac{v_{1}}{v_{2}} = \frac{N_{2}}{N_{1}}](https://tex.z-dn.net/?f=%5Cfrac%7Bv_%7B1%7D%7D%7Bv_%7B2%7D%7D%20%3D%20%20%5Cfrac%7BN_%7B2%7D%7D%7BN_%7B1%7D%7D)
If we know that
,
and
, then the speed of the driven gear is:
![v_{1} = v_{2}\times \frac{N_{2}}{N_{1}}](https://tex.z-dn.net/?f=v_%7B1%7D%20%3D%20v_%7B2%7D%5Ctimes%20%5Cfrac%7BN_%7B2%7D%7D%7BN_%7B1%7D%7D)
![v_{1} = 60\,rpm \times \frac{60}{20}](https://tex.z-dn.net/?f=v_%7B1%7D%20%3D%2060%5C%2Crpm%20%5Ctimes%20%5Cfrac%7B60%7D%7B20%7D)
![v_{1} = 180\,rpm](https://tex.z-dn.net/?f=v_%7B1%7D%20%3D%20180%5C%2Crpm)
The speed of the <em>driven</em> gear is 180 rounds per minute.
To learn more on inverse relationships: brainly.com/question/4147411
#SPJ1
Answer:
![\displaystyle y = \sqrt{2}e^{\frac{1}{2} ln|2 + x^2|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20%5Csqrt%7B2%7De%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%20%2B%20x%5E2%7C%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality<u>
</u>
<u>Algebra I</u>
- |Absolute Value|
- Functions
- Function Notation
- Exponential Rule [Multiplying]:
![\displaystyle b^m \cdot b^n = b^{m + n}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20b%5Em%20%5Ccdot%20b%5En%20%3D%20b%5E%7Bm%20%2B%20n%7D)
<u>Algebra II</u>
- Logarithms and Natural Logs
- Euler's number e
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative of a constant is 0
Differential Equations
Antiderivatives - Integrals
Integration Constant C
Integration Property [Multiplied Constant]: ![\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bcf%28x%29%7D%20%5C%2C%20dx%20%3D%20c%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx)
U-Substitution
Logarithmic Integration
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<em />
<em />
<em />
<em />
<em />
<u>Step 2: Rewrite</u>
<em>Separation of Variables</em>
- Rewrite Derivative Notation:
![\displaystyle \frac{dy}{dx} = \frac{xy}{2 + x^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7Bxy%7D%7B2%20%2B%20x%5E2%7D)
- [Division Property of Equality] Isolate <em>y</em>'s:
![\displaystyle \frac{1}{y} \frac{dy}{dx} = \frac{x}{2 + x^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7By%7D%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7Bx%7D%7B2%20%2B%20x%5E2%7D)
- [Multiplication Property of Equality] Rewrite Derivative Notation:
![\displaystyle \frac{1}{y} dy = \frac{x}{2 + x^2} dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7By%7D%20dy%20%3D%20%5Cfrac%7Bx%7D%7B2%20%2B%20x%5E2%7D%20dx)
<u>Step 3: Find General Solution Pt. 1</u>
<em>Integration</em>
- [Equality Property] Integrate both sides:
![\displaystyle \int {\frac{1}{y}} \, dy = \int {\frac{x}{2 + x^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Cfrac%7B1%7D%7By%7D%7D%20%5C%2C%20dy%20%3D%20%5Cint%20%7B%5Cfrac%7Bx%7D%7B2%20%2B%20x%5E2%7D%7D%20%5C%2C%20dx)
- [1st Integral] Integrate [Logarithmic Integration]:
![\displaystyle ln|y| = \int {\frac{x}{2 + x^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20ln%7Cy%7C%20%3D%20%5Cint%20%7B%5Cfrac%7Bx%7D%7B2%20%2B%20x%5E2%7D%7D%20%5C%2C%20dx)
<u>Step 4: Identify Variables</u>
<em>Identify variables for u-substitution for 2nd integral.</em>
u = 2 + x²
du = 2xdx
<u>Step 5: Find General Solution Pt. 2</u>
- [2nd Integral] Rewrite [Integration Property - Multiplied Constant]:
![\displaystyle ln|y| = \frac{1}{2}\int {\frac{2x}{2 + x^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20ln%7Cy%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cint%20%7B%5Cfrac%7B2x%7D%7B2%20%2B%20x%5E2%7D%7D%20%5C%2C%20dx)
- [2nd Integral] U-Substitution:
![\displaystyle ln|y| = \frac{1}{2}\int {\frac{1}{u}} \, du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20ln%7Cy%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cint%20%7B%5Cfrac%7B1%7D%7Bu%7D%7D%20%5C%2C%20du)
- [2nd Integral] Integrate [Logarithmic Integration]:
![\displaystyle ln|y| = \frac{1}{2} ln|u| + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20ln%7Cy%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20ln%7Cu%7C%20%2B%20C)
- [Equality Property] e both sides:
![\displaystyle e^{ln|y|} = e^{\frac{1}{2} ln|u| + C}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20e%5E%7Bln%7Cy%7C%7D%20%3D%20e%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7Cu%7C%20%2B%20C%7D)
- Simplify:
![\displaystyle |y| = e^{\frac{1}{2} ln|u| + C}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%7Cy%7C%20%3D%20e%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7Cu%7C%20%2B%20C%7D)
- Rewrite [Exponential Rule - Multiplying]:
![\displaystyle |y| = e^{\frac{1}{2} ln|u|} \cdot e^C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%7Cy%7C%20%3D%20e%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7Cu%7C%7D%20%5Ccdot%20e%5EC)
- Simplify:
![\displaystyle |y| = Ce^{\frac{1}{2} ln|u|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%7Cy%7C%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7Cu%7C%7D)
- Back-Substitute:
![\displaystyle |y| = Ce^{\frac{1}{2} ln|2 + x^2|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%7Cy%7C%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%20%2B%20x%5E2%7C%7D)
Our general solution is
.
<u>Step 6: Find Particular Solution</u>
- Substitute in point:
![\displaystyle |2| = Ce^{\frac{1}{2} ln|2 + 0^2|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%7C2%7C%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%20%2B%200%5E2%7C%7D)
- Evaluate |Absolute Value|:
![\displaystyle 2 = Ce^{\frac{1}{2} ln|2 + 0^2|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%20%2B%200%5E2%7C%7D)
- |Absolute Value| Evaluate exponents:
![\displaystyle 2 = Ce^{\frac{1}{2} ln|2 + 0|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%20%2B%200%7C%7D)
- |Absolute Value| Add:
![\displaystyle 2 = Ce^{\frac{1}{2} ln|2|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%7C%7D)
- |Absolute Value| Evaluate:
![\displaystyle 2 = Ce^{\frac{1}{2} ln(2)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%20%3D%20Ce%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%282%29%7D)
- [Division Property of Equality] Isolate <em>C</em>:
![\displaystyle \sqrt{2} = C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%7B2%7D%20%3D%20C)
- Rewrite:
![\displaystyle C = \sqrt{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20C%20%3D%20%5Csqrt%7B2%7D)
- Substitute in <em>C</em> [General Solution]:
![\displaystyle y = \sqrt{2}e^{\frac{1}{2} ln|2 + x^2|}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20%5Csqrt%7B2%7De%5E%7B%5Cfrac%7B1%7D%7B2%7D%20ln%7C2%20%2B%20x%5E2%7C%7D)
∴ Our particular solution is
.
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Differential Equations
Book: College Calculus 10e