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Minchanka [31]
3 years ago
7

Which of the following cannot be the lengths of the three sides of a triangle? A 3.4.5 B 6.7.10 C 8.38 D 8.7.15

Mathematics
1 answer:
lesantik [10]3 years ago
3 0

Answer:

D

Step-by-step explanation:

For three line segments to be able to form any triangle you must be able to take any two sides, add their length and this sum be greater than the remaining side.

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Two fewer than a number doubled is the same as the number decreased by 38. if n is "the number," which equation could be used to
Alona [7]
   
\displaystyle \\ 
\text{We use the following equation:} \\  \\ 
 \frac{n}{2} = n - 38 \\  \\ 
We solve for n: \\  \\ 
 \frac{n}{2} = n - 38 ~~\Big| \times 2 \\  \\  
n = 2 \times  n - 2 \times 38 \\  \\ 
n = 2n - 76 \\  \\ 
2n - n = 76 \\  \\ 
\boxed{n = 76} \\  \\ 
\text{Verification: } \\  \\ 
\frac{n}{2} = n - 38 \\  \\ 
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\boxed{38 = 38  ~~~ Correct }



8 0
3 years ago
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How do I evaluate this using trigonometric substitution?<br><br>∫dx/(81x^2+4)^2
Daniel [21]

Answer:

\displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C

General Formulas and Concepts:

<u>Alg I</u>

  • Terms/Coefficients
  • Factor
  • Exponential Rule [Dividing]: \displaystyle \frac{b^m}{b^n} = b^{m - n}

<u>Pre-Calc</u>

[Right Triangle Only] Pythagorean Theorem: a² + b² = c²

  • a is a leg
  • b is a leg
  • c is hypotenuse

Trigonometric Ratio: \displaystyle sec(\theta) = \frac{1}{cos(\theta)}

Trigonometric Identity: \displaystyle tan^2\theta + 1 = sec^2\theta

TI: \displaystyle sin(2x) = 2sin(x)cos(x)

TI: \displaystyle cos^2(\theta) = \frac{cos(2x) + 1}{2}

<u>Calc</u>

Integration Rule [Reverse Power Rule]:                                                                \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

Integration Property [Multiplied Constant]:                                                         \displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx

IP [Addition/Subtraction]:                                                             \displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx

U-Substitution

U-Trig Substitution: x² + a² → x = atanθ

Step-by-step explanation:

<u>Step 1: Define</u>

\displaystyle \int {\frac{dx}{(81x^2 + 4)^2}}

<u>Step 2: Identify Sub Variables Pt.1</u>

Rewrite integral [factor expression]:

\displaystyle \int {\frac{dx}{[(9x)^2 + 4]^2}}

Identify u-trig sub:

\displaystyle x = atan\theta\\9x = 2tan\theta \rightarrow x = \frac{2}{9}tan\theta\\dx = \frac{2}{9}sec^2\theta d\theta

Later, back-sub θ (integrate w/ respect to <em>x</em>):

\displaystyle tan\theta = \frac{9x}{2}  \rightarrow \theta = arctan(\frac{9x}{2})

<u>Step 3: Integrate Pt.1</u>

  1. [Int] Sub u-trig variables:                                                                                 \displaystyle \int {\frac{\frac{2}{9}sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta
  2. [Int] Rewrite [Int Prop - MC]:                                                                           \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta
  3. [Int] Evaluate exponents:                                                                                \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4tan^2\theta + 4]^2}} \ d\theta
  4. [Int] Factor:                                                                                                      \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4(tan^2\theta + 1)]^2}} \ d\theta
  5. [Int] Rewrite [TI]:                                                                                              \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4sec^2\theta]^2}} \ d\theta
  6. [Int] Evaluate exponents:                                                                                \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{16sec^4\theta} \ d\theta
  7. [Int] Rewrite [Int Prop - MC]:                                                                          \displaystyle \frac{1}{72} \int {\frac{sec^2\theta}{sec^4\theta} \ d\theta
  8. [Int] Divide [ER - D]:                                                                                         \displaystyle \frac{1}{72} \int {\frac{1}{sec^2\theta} \ d\theta
  9. [Int] Rewrite [TR]:                                                                                            \displaystyle \frac{1}{72} \int {cos^2\theta} \ d\theta
  10. [Int] Rewrite [TI]:                                                                                              \displaystyle \frac{1}{72} \int {\frac{cos(2\theta) + 1}{2}} \ d\theta
  11. [Int] Rewrite [Int Prop - MC]:                                                                          \displaystyle \frac{1}{144} \int {cos(2\theta) + 1} \ d\theta
  12. [Int] Rewrite [Int Prop - A/S]:                                                                          \displaystyle \frac{1}{144} [\int {cos(2\theta) \ d\theta + \int {1} \ d\theta]  

<u>Step 4: Identify Sub Variables Pt.2</u>

Determine u-sub for trig int:

u = 2θ

du = 2dθ

<u>Step 5: Integrate Pt.2</u>

  1. [Ints] Rewrite [Int Prop - MC]:                                                                       \displaystyle \frac{1}{144} [\frac{1}{2} \int {2cos(2\theta) \ d\theta + \int {1 \theta ^0} \ d\theta]
  2. [Int] U-Sub:                                                                                                     \displaystyle \frac{1}{144} [\frac{1}{2} \int {cos(u) \ du + \int {1 \theta ^0} \ d\theta]
  3. [Ints] Integrate [Trig/Int Rule - RPR]:                                                             \displaystyle \frac{1}{144} [\frac{1}{2} sin(u) + \theta + C]
  4. [Expression] Back Sub:                                                                                 \displaystyle \frac{1}{144} [\frac{1}{2} sin(2 \theta) + arctan(\frac{9x}{2}) + C]
  5. [Exp] Rewrite [TI]:                                                                                           \displaystyle \frac{1}{144} [\frac{1}{2}(2sin(\theta)cos(\theta)) + arctan(\frac{9x}{2}) + C]
  6. [Exp] Multiply:                                                                                                 \displaystyle \frac{1}{144} [sin(\theta)cos(\theta) + arctan(\frac{9x}{2}) + C]
  7. [Exp] Back Sub:                                                                                             \displaystyle \frac{1}{144} [sin(arctan(\frac{9x}{2}))cos(arctan(\frac{9x}{2})) + arctan(\frac{9x}{2}) + C]

<u>Step 6: Triangle</u>

Find trig values:

\displaystyle tan\theta = \frac{9x}{2}

\displaystyle \theta = arctan(\frac{9x}{2})

tanθ = opposite / adjacent; solve hypotenuse of right triangle, determine trig ratios:

sinθ = opposite / hypotenuse

cosθ = adjacent / hypotenuse

Leg <em>a</em> = 2

Leg <em>b</em> = 9x

Leg <em>c</em> = ?

  1. Sub variables [PT]:                                                                                         \displaystyle 2^2 + (9x)^2 = c^2
  2. Evaluate exponents:                                                                                      \displaystyle 4 + 81x^2 = c^2
  3. [Equality Property] Square root both sides:                                                  \displaystyle \sqrt{4 + 81x^2} = c
  4. Rewrite:                                                                                                           c = \sqrt{81x^2 + 4}

Substitute into trig ratios:

\displaystyle sin\theta = \frac{9x}{\sqrt{81x^2 + 4}}

\displaystyle cos\theta = \frac{2}{\sqrt{81x^2 + 4}}

<u>Step 7: Integrate Pt.3</u>

  1. [Exp] Sub variables [TR]:                                                                               \displaystyle \frac{1}{144} [\frac{9x}{\sqrt{81x^2 + 4}} \cdot \frac{2}{\sqrt{81x^2 + 4}} + arctan(\frac{9x}{2}) + C]
  2. [Exp] Multiply:                                                                                                 \displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C]
  3. [Exp] Distribute:                                                                                             \displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C
3 0
2 years ago
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