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2 years ago

Which results only in a horizontal compression of Y = by a factor of 6?

Mathematics

1 answer:

elena55 [62]2 years ago

3 0

Answer:

Y = 6*y

Step-by-step explanation:

We have Y = b * y by a factor of 6.

That is, b = 6.

now, to find what results only in a horizontal compression of y = b * y by a factor of 6.

By transformation rule, the function would be a horizontal compression f (a * x) if a> 1.

Therefore, knowing the above, the answer would be:

Y = 6 * y

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Factor 8a^2+8ab+8ac+3a+3b+3c

nalin [4]

Answer:

(8a + 3)(a + b + c)

Step-by-step explanation:

8a^2 + 8ab + 8ac + 3a + 3b + 3c =

= 8a(a + b + c) + 3(a + b + c)

= (8a + 3)(a + b + c)

5 0

3 years ago

The lengths of the sides of a triangle are x,x+1 and 7 centimeters and the perimeter of the triangle is 56 centimeters. The equa

____ [38]

Answer:

the longest side is 25

Step-by-step explanation:

the equation would be x+ x+1 + 7 for the 3 sides of the triangle

x + (x+1) + 7=56

combine like terms

2x+8 = 56

subtract 8 from each side

2x = 48

divide by 2

x = 24

the sides are

x=24

x+1 = 25

the longest side is 25

7 0

3 years ago

How do I evaluate this using trigonometric substitution? ∫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:

Alg I

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

Pre-Calc

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

Calc

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:

Step 1: Define

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

Step 2: Identify Sub Variables Pt.1

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 x):

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

Step 3: Integrate Pt.1

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

Step 4: Identify Sub Variables Pt.2

Determine u-sub for trig int:

u = 2θ

du = 2dθ

Step 5: Integrate Pt.2

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

Step 6: Triangle

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 a = 2

Leg b = 9x

Leg c = ?

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

Substitute into trig ratios:

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

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

Step 7: Integrate Pt.3

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

3 0

2 years ago

Melody used the ratio table at the right to find the number of people served with 15 pounds of ground turkey find her error and

Solnce55 [7]

The first column is the ratio you start with. The next column in the table should be the reduced fraction. To properly reduce a fraction, you divide the same number on the top and bottom of the fraction. Melody instead subtracted 1 in the top and bottom of the fraction to reduce it. She should have divided the top and bottom by 2 to get the second column of 1 pound of turkey and 3 people served.
The final column is designed to be the answer to the initial question of how many people are fed by 15 pounds of ground turkey. Starting with column 2 (the reduced ratio of 1/3), multiply the top by 15 to get 15 pounds of ground turkey. Then do the same to the bottom of the fraction. 3 x 15 = 45 people fed.
So the table should have been
Pounds of turkey 2 1 15
People served 1 3 45
Melody's overriding error was in believing that adding or subtracting numbers to both the top and bottom of a ratio/fraction keep its value the same. Instead, only multiplying or dividing the top and bottom by the same number creates an equivalent fraction.

7 0

3 years ago

Your friend takes a job cleaning up a neighbor's yard and mowing the grass,and asks you and two other friends to help. Your frie

daser333 [38]

Answer: $50

Step-by-step explanation: Multiply $25 times 2

5 0

2 years ago