sin^4x rewrite the power as a product of two squared terms

Considering the definition of combination, if no meal is repeated, 210 different meal arrangements are possible.

Combinations of m elements taken from n to n (m≥n) are called all the possible groupings that can be made with the m elements in such a way that not all the elements enter; the order does not matter and the elements are not repeated.

To calculate the number of combinations, the following formula is applied:

[tex]C=\frac{m!}{n!(m-n)!}[/tex]

The term "n!" is called the "factorial of n" and is the multiplication of all numbers from "n" to 1.

Joseph is planning dinners for the next 4 nights. There are 10 meals to choose from and no meal is repeated.

So, you know that:

Replacing in the definition of combination:

[tex]C=\frac{10!}{4!(10-4)!}[/tex]

Solving:

[tex]C=\frac{10!}{4!6!}[/tex]

[tex]C=\frac{3,628,800}{24x720}[/tex]

[tex]C=\frac{3,628,800}{17,280}[/tex]

C= 210

Finally, if no meal is repeated, 210 different meal arrangements are possible.

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The positive square roots of the number 151,321 according to the task content can be determined by means of division as; 389.

It follows from.the task content above that the number given is; 151,321 whose positive square roots is to be determined.

Upon testing different integers as divisor on the number 151,321; it is concluded that the only positive integer by which 151,321 can be divided to result in a whole is; 389.

Hence, the positive square root of the number 151,321 is; 389.

Consequently, it can be concluded that the positive square root of the number, 151,321 as in the task content is; 389 which is itself a prime number as it is only divisible by 1 and itself.

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Answer:

Y=mx+b

Step-by-step explanation:

The answer is y = -4/5x + 4.

We know the general form for slope intercept form is :

y = mx + c

The given equation is in standard form.

4x + 5y = 20

Bring 4x to the other side by subtracting 4x on both sides.

4x + 5y - 4x = 20 - 4x

5y = -4x + 20

Divide 5 on both sides to isolate the variable y.

5y x 1/5 = 1/5 x (-4x + 20)

y = -4/5x + 4

The standard deviations above the mean that a student have to score to be publicly recognized will be 0.674.

From the information given, it was stated that the students who score in the top 16 percent are recognized publicly for their achievement by the Department of the Treasury.

Based on the information given, it should be noted that the appropriate thing to do is to find the z score for the 75th percentile.

This will be looked up in the distribution table. In this case, the value is 0.674. Therefore, standard deviations above the mean that a student have to score to be publicly recognized will be 0.674.

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The missing value in the logarithm are as follows:

Using logarithm rule,

logₐ b - logₐ c = logₐ (b / c)

logₐ b + logₐ c = logₐ (b × c)

Therefore,

log₃ 7 - log₃ 2 = log₃ (7 / 2)

log₉ 7 + log₉ 4 = log₉ (7 × 4) =  log₉ 28

log₆ 1 / 81 = log₆ 81⁻¹ = log₆ 3⁻⁴ =  - 4 log₆ 3

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Answer:

D.

Step-by-step explanation:

We can solve by simply isolating x:

[tex]-122 < -3(-2-8x)-8x\\-122 < 6+24x-8x\\-122 < 6+16x\\-128 < 16x\\-8 < x\\OR\\x > -8[/tex]

You can check by plugging in any value greater than -8

-122<-3(-2-8(-7))-8*-7

-122<-3(-2+56)+56

-122<-3(54)+56

-122<-162+56

-122<-106

[tex]number \: of \: movies = 3 \\ permutations = 2 \\ \\ c( \gamma ) = 2! \times 1 = 2 \: ways[/tex]

[tex]g(1) \: \: \: \: g(2) \: \: \: \: nc(17) \\ g(2) \: \: \: \: g(1) \: \: \: \: nc(17)[/tex]

Answer:

  x = 8

Step-by-step explanation:

Diagonals of a kite cross at right angles. That gives us a relation that can be solved for x.

The measure shown is equal to the angle measure of 90°.

  14x -22 = 90

We can solve this 2-step linear equation in the usual way.

  14x = 112 . . . . . . step 1, add the opposite of the constant to get x alone

  x = 112/14 = 8 . . . step 2, divide by the coefficient of x

The value of x is 8.

Answer: 1308m

Step-by-step explanation:

Top and Bottom: 19 x 16 x 2 = 608

Sides: 16 x 10 x 2 = 320

Front and Back: 19 x 10 x 2 = 380

608 + 320 + 380 = 1308

Answer:   x = 2

Work Shown:

x+5 = 7

x+5-5 = 7-5

x = 2

Subtract 5 from both sides to isolate x. This is to undo the plus 5.

Answer:

Step-by-step explanation:

Answer:

-4

Step-by-step explanation:

7x^3 + 5x^2 - 2
= 7*-1 + 5*1 - 2

= -7 + 5 - 2

= -4

Answer:

3.2

Step-by-step explanation:

4 x 8 / 8 + 2

32/10

=3.2

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Answer:

80 in.²

Step-by-step explanation:

The total surface area of the pyramid is the sum of the area of the base and the areas of the 4 triangular sides.

Square: area = s²

Square: side = 4 in.

Triangular side: area = bh/2

Triangular side: base = 4 in.; height = 8 in.

Area of the base: s² = (4 in.)² = 16 in.²

Total area of the 4 triangular sides: 4 × bh/2 = 2bh = 2 × 4 in. × 8 in. = 64 in.²

Surface area = 16 in.² + 64 in.² = 80 in.²

The possible rational roots of the given equation are 1 and -3

From the question, we are to determine all the possible rational roots of the given equation

The given equation is

x⁴ -2x³ -6x² +22x -15 = 0

To determine the rational roots, we will test for values that make the equation equal to zero

(-1)⁴ -2(-1)³ -6(-1)² +22(-1) -15

1 + 2 - 6 - 22 -15

= -40

∴ -1 is not a root of the equation

(1)⁴ -2(1)³ -6(1)² +22(1) -15

1 - 2 - 6 + 22 -15

= 0

∴ 1 is one of the roots of the equation

(-2)⁴ -2(-2)³ -6(-2)² +22(-2) -15

16 + 16 - 24 - 44 -15

= -51

∴ -2 is not a root of the equation

(2)⁴ -2(2)³ -6(2)² +22(2) -15

16 - 16 - 24 + 44 -15

= 5

∴ 2 is not a root of the equation

(-3)⁴ -2(-3)³ -6(-3)² +22(-3) -15

81 + 54 - 54 -66 -15

= 0

∴ -3 is one of the roots of the equation

(3)⁴ -2(3)³ -6(3)² +22(3) -15

81 - 54 - 54 + 66 -15

= 24

∴ 3 is not a root of the equation

The other roots of the equation are irrational roots.

Hence, the possible rational roots of the given equation are 1 and -3

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[tex] \sqrt[3]{b {}^{2} } = b {}^{ \frac{2}{3} } [/tex]

[tex]f(4) = 2(3) {}^{4} = 2 \times 81 = 162[/tex]

[tex]f( \frac{1}{2} ) = \frac{1}{3} (9) {}^{ \frac{1}{2} } + 5 = \frac{ \sqrt{9} }{ 3} + 5 = 6[/tex]

[tex]f(5) = - 4(2) {}^{ - 5 + 1} = - 4(2) {}^{ -4} = \frac{2 {}^{2} }{2 {}^{4} } = \frac{1}{2 {}^{2} } = \frac{1}{4} [/tex]

[tex]f(10) = 2e {}^{0.15 \times 10} = 2e {}^{1.5} = 8.96[/tex]

[tex]y = 10(b) {}^{x} \: \: \\ 10 = 10(b) {}^{0} \: duh \\ 2 = 10b {}^{1} \\ b = \frac{2}{10} = \frac{1}{5} [/tex]

[tex]y = a(b) {}^{x} \\ 3 = a(b) {}^{0} \\ a = 3 \\ \\75 = 3(b) {}^{2} \\ b {}^{2} = 25 \\ b = + 5 \: \: \: or \: \: \: \: b = - 5[/tex]

[tex]s = 1000(1 + \frac{0.045}{4} ) {}^{4 \times 10} = 1564.38[/tex]

[tex]s = 2500.e {}^{0.07 \times 20} = 10138[/tex]

Answer:

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

The average of the five consecutive numbers ending with b in discuss when expressed in terms of a is; Choice D; a+3.

First, since it was given in the task content that the average of six positive consecutive odd integers starting with a is equal to b, it therefore follows that;

(a+a+2+a+4+a+6+a+8+a+10)/6 = b

6b=6a+30

b=a+5

Also, let the average of the consecutive intergers ending with b be denoted by; x.

(b+b-1+b-2+b-3+b-4)/5 = x

=(5b-10)/5

=b–2

The average, x=b – 2 (where b = a-5)

Ultimately, the value of the required average is; = a+5-2 = a+3.

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Answer:

W = (P -2L)/2

Step-by-step explanation:

P = 2(L + W)  Distribute the 2

P = 2L + 2W   Subtract 2L from both sides

P - 2L = 2W  Divide both sides by 2

(P-2L)/2 = W  

Step-by-step explanation:

I= 74810

II=66733

III=71392

iv=239904

Answer:

c)  3 units

d)  g(x) - f(x) = x² + 2x

e)  (-∞, -2] ∪ [0, ∞)

Step-by-step explanation:

To calculate the length of FC, first find the coordinates of point C.

The y-value of point C is zero since this is where the function f(x) intercepts the x-axis.  Therefore, set f(x) to zero and solve for x:

[tex]\implies 1-x^2=0[/tex]

[tex]\implies x^2=1[/tex]

[tex]\implies \sqrt{x^2}=\sqrt{1}[/tex]

[tex]\implies x= \pm 1[/tex]

As point C has a positive x-value,  C = (1, 0).

To find point F, substitute the x-value of point C into g(x):

[tex]\implies g(1)=2(1)+1=3[/tex]

⇒ F = (1, 3).

Length FC is the difference in the y-value of points C and F:

[tex]\begin{aligned} \implies \sf FC& = \sf y_F-y_C\\ & = \sf 3-0\\ & =\sf 3\:units \end{aligned}[/tex]

Given functions:

[tex]\begin{cases}f(x)=1-x^2\\ g(x)=2x+1 \end{cases}[/tex]

Therefore:

[tex]\begin{aligned}\implies g(x)-f(x) & = (2x+1) - (1-x^2)\\& = 2x+1-1+x^2\\& = x^2+2x\end{aligned}[/tex]

The values of x for which g(x) ≥ f(x) are where the line of g(x) is above the curve of f(x):

Point A is the y-intercept of both functions, therefore the x-value of point A is 0.

To find the x-value of point E, equate the two functions and solve for x:

[tex]\begin{aligned}g(x) & = f(x)\\\implies 2x+1 & = 1-x^2\\x^2+2x & = 0\\x(x+2) & = 0\\\implies x & = 0, -2\end{aligned}[/tex]

As the x-value of point E is negative ⇒ x = -2.

Therefore, the values of x for which g(x) ≥ f(x) are:

Answer:

a)

A = (0, 1)

B = (-1, 0)

C = (1, 0)

D = (-0.5, 0)

b) E = (-2, -3)

c) FC = 3 units

d) x² + 2x

e) x ≤ -2 and x ≥ 0

Explanation:

This question displays one equation of a linear function g(x) = 2x + 1 and a parabolic function f(x) = 1 - x².

a)

A point is where the linear function cuts the y axis.

y = 1 - (0)²

y = 1

A = (0, 1)

B and C point is where the parabolic function cuts the x axis.

1 - x² = 0

-x² = -1

x² = 1

x = ±√1

x = -1, 1

B = (-1, 0), C = (1, 0)

D point is where the linear function cuts x axis.

2x + 1 = 0

2x = -1

x = -1/2 or -0.5

D = (-0.5, 0)

b)

E point is where both equations intersect each other.

y = y

2x + 1 = 1 - x²

x² + 2x = 0

x(x + 2) = 0

x = 0, x = -2

y = 1, y = -3

E = (-2, -3)

c)

C : (1, 0)

To find F point

y = 2(1) + 1

y = 3

F : (1, 3)

[tex]\sf Distance \ between \ two \ points = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]

[tex]\sf d = \sqrt{(1 - 1)^2 + (3 - 0)^2}[/tex]

[tex]\sf d = \sqrt{0 + 3^2}[/tex]

[tex]\sf d = 3[/tex]

FC length = 3 units

d)

g(x) - f(x)

(2x + 1) - (1 - x²)

2x + 1 - 1 + x²

x² + 2x

e)

g(x) ≥ f(x)

2x + 1 ≥ 1 - x²

x² + 2x ≥ 0

x(x + 2) ≥ 0

[tex]\boxed{If \ x \ \geq \ \pm \ a \ then \ -a \ \leq x \ \ and \ x \ \geq \ a }[/tex]

x ≤ -2 and x ≥ 0

Answer:

1. f(x) is reflected across the x-axis

2. f(x) is translated 1 unit up

3. f(x) is vertically scaled by a factor of 2

4. f(x) is reflected across the x-axis AND is vertically scaled by a factor of 2

5. f(x) is vertically scaled by a factor of 3 AND is translated 1 unit down

6. f(x) is vertically scaled by a factor of 1/6 AND is translated 1 unit up

Solving question:

(1)  [tex]g(x) = -f(x)[/tex]

This graph has been reflected in the x axis. Equation:  [tex]\sf g(x) = -\dfrac{2}{x}[/tex]

(2) [tex]g(x) = f(x) + 1[/tex]

Graph has been translated 1 units up vertically. Equation: [tex]\sf g(x) = \dfrac{2}{x} +1[/tex]

(3) [tex]g(x) = 2f(x)[/tex]

This graph has been stretched vertically by a factor of 2. Equation: [tex]\sf g(x) = \dfrac{4}{x}[/tex]

(4) [tex]g(x) = -2f(x)[/tex]

This graph has been reflected in the x axis and stretched vertically by a factor of 2. Equation: [tex]\sf g(x) = -\dfrac{4}{x}[/tex]

(5)  [tex]g(x) = 3f(x) - 1[/tex]

This graph has been stretched vertically by a factor of 3 and translated 1 units down. Equation: [tex]\sf g(x) = \dfrac{6}{x} -1[/tex]

(6)   [tex]g(x) = \frac{1}{6} f(x) + 1[/tex]

This graph has been stretched vertically by a factor of 1/6 and translated 1 units up. Equation: [tex]\sf g(x) = \dfrac{1}{3x} +1[/tex]

Answer:

Most you can do is factor out the x and turn it into   x (3x + 7)

roots would be x = 0 and x = -7/3

Answer:  x(3x+7)

Step-by-step explanation:

You would factor the x out of both of your values and put on the outside of the parenthesis. And you put the two numbers that you have left inside of the parenthesis. And that is as far down as this function can be factored.

a. The arc length is given by the integral

[tex]L(r) = \displaystyle \int_3^\infty \sqrt{x'(t)^2 + y'(t)^2} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \sqrt{\left(\frac{t\cos(t) - r\sin(t)}{t^{r+1}}\right)^2 + \left(-\frac{t\sin(t) + r\cos(t)}{t^{r+1}}\right)^2} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \sqrt{\frac{(t^2+r^2)\cos^2(t) + (t^2+r^2)\sin^2(t)}{\left(t^{r+1}\right)^2}} \, dt \\\\ ~~~~~~~~ = \boxed{\int_3^\infty \frac{\sqrt{t^2+r^2}}{t^{r+1}} \, dt}[/tex]

b. The integrand roughly behaves like

[tex]\dfrac t{t^{r+1}} = \dfrac1{t^r}[/tex]

so the arc length integral will converge for [tex]\boxed{r>1}[/tex].

c. When [tex]r=3[/tex], the integral becomes

[tex]L(3) = \displaystyle \int_3^\infty \frac{\sqrt{t^2+9}}{t^4} \, dt[/tex]

Pull out a factor of [tex]t^2[/tex] from under the square root, bearing in mind that [tex]\sqrt{x^2} = |x|[/tex] for all real [tex]x[/tex].

[tex]L(3) = \displaystyle \int_3^\infty \frac{\sqrt{t^2} \sqrt{1+\frac9{t^2}}}{t^4} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \frac{|t| \sqrt{1+\frac9{t^2}}}{t^4} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \frac{t \sqrt{1+\frac9{t^2}}}{t^4} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \frac{\sqrt{1+\frac9{t^2}}}{t^3} \, dt[/tex]

since for [tex]3\le t<\infty[/tex], we have [tex]|t|=t[/tex].

Now substitute

[tex]s=1+\dfrac9{t^2} \text{ and } ds = -\dfrac{18}{t^3} \, dt[/tex]

Then the integral evaluates to

[tex]L(3) = \displaystyle -\frac1{18} \int_2^1 \sqrt{s} \, ds \\\\ ~~~~~~~~ = \frac1{18} \int_1^2 s^{1/2} \, ds \\\\ ~~~~~~~~ = \frac1{27} s^{3/2} \bigg|_1^2 \\\\ ~~~~~~~~ = \frac{2^{3/2} - 1^{3/2}}{27} = \boxed{\frac{2\sqrt2-1}{27}}[/tex]

a) The improper integral in simplified form is equal to [tex]L = \int\limits^{\infty}_{3} {\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}} } \, dt[/tex].

b) r > 1 for a spiral with finite length.

c) The length of the spiral when r = 3 is (1 - 2√2) / 9 units.

a) The arc length formula for 2-dimension parametric functions is defined below:

L = ∫ √[(dx / dt)² + (dy / dt)²] dt, for [α, β]         (1)

If we know that [tex]\dot x (t) = \frac{t \cdot \cos t - r \cdot \sin t}{t^{r+1}}[/tex], [tex]\dot y(t) = \frac{t\cdot \sin t + r\cdot \cos t}{t^{r + 1}}[/tex], α = 0 and β → + ∞ then their arc length formula is:

[tex]L = \int\limits^{\infty}_{3} {\sqrt{\left(\frac{t\cdot \cos t - r\cdot \sin t}{t^{r + 1}}\right)^{2}+\left(\frac{t\cdot \sin t + r\cdot \cos t}{t^{r+1}}\right)^{2}} } \, dt[/tex]

By algebraic handling and trigonometric formulae (cos ² t + sin² t = 1):

[tex]L = \int\limits^{\infty}_{3} {\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}} } \, dt[/tex]      (2)

The improper integral in simplified form is equal to [tex]L = \int\limits^{\infty}_{3} {\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}} } \, dt[/tex].

b) By ratio comparison criterion, we notice that √(t² + r²) is similar to √t² = t and [tex]\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}}[/tex] is similar to [tex]\frac{t}{t^{r +1}} = \frac{1}{t^{r}}[/tex].

The integral found in part a) has a finite length if and only the governing grade of the denominator is greater that the governing grade of the numerator. and according to the ratio comparson criterion, the absolute value of the ratio is greater than 0 and less than 1. Therefore, r > 1 for a spiral with finite length.

c) Now we proceed to integrate the function:

L = ∫ [√(t² + 9) / t⁴] dt, for [3, + ∞].

L = ∫ [t · √(1 + 9 / t²) / t⁴] dt, for [3, + ∞].

By using the algebraic substitutions: u = 1 + 9 / t², du = - (18 / t³) dt → - (1 / 18) du.

L = ∫ √u du, for [3, + ∞].

L = - (1 / 9) · √(u³), for [3, + ∞].

L = - (1 / 9) · [√(1 + 9 / t²)³], for [3, + ∞].

L = - (1 / 9) · [√(2³) - √(1³)]

L = - (1 / 9) · (2√2 - 1)

L = (1 - 2√2) / 9

The length of the spiral when r = 3 is (1 - 2√2) / 9 units.

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Answer:

Arithmetic => each new term differs from the previous term by a fixed amount

an = a1 + d (n − 1)

Geometric => each element after the first is obtained by multiplying the previous number by a constant factor

an = a1 (r)^(n − 1)

4,8,16,32 the difference is not fixed so it is a geometric so it is ratio

the ratio is 2 and n is 5 so 4*(2)^4 =4*16=64

To generate the terms of a series given in sigma notation, replace the index of summation with consecutive integers from the first value to the last value of the index.

if you also want the sum of them

arithmetic -> (n/2)(a1+an)

geometric -> (a1*(1-r^n))/(1-r) or

      when the sequence is infinite you can use a1/(1-r)

Step-by-step explanation:

Arithmetic => 1,3,5,7,9,11,13,15....

Geometric => 1,2,4,8,16,32,64....

Answer:

0

Step-by-step explanation:

because 1-0=1*2^3=8

because2^3=8

Answer:

2x + 8

Step-by-step explanation:

(4x + 5) - (2x - 3)

4x + 5 - 2x + 3

2x + 8

Answer: 2x+8

Step-by-step explanation:

4x+5-2x+3

=2x+8

Answer:

[tex]\dfrac{8}{\sin(6^\circ)} \approx 76.53[/tex]

Step-by-step explanation:

[tex]UX \sin(6^\circ) = 8 \Rightarrow \boxed{UX = \dfrac{8}{\sin(6^\circ)} \approx 76.53}[/tex]

The best sampling strategy would be a stratified sample.

Samples may be classified as:

For this problem, the 4 different brands of the recorders must be considered, hence the buyers should be divided into groups, and a proportion of each group should be sampled, hence a stratified sample should be used.

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Using translation concepts, it is found that:

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.

The rule applied for each vertex of the rectangle is given as follows:

(x,y) -> (x + 7, y - 2).

The new coordinates of A' are given as follows:

(-2 + 7, 2 - 2) = (5,0).

The new coordinates of B' are given as follows:

(-2 + 7, 4 - 2) = (5,2).

The new coordinates of C' are given as follows:

(2 + 7, 4 - 2) = (9,2).

The new coordinates of D' are given as follows:

(2 + 7, 2 - 2) = (9,0).

Since there are only two values for the x-coordinates and two values for the y-coordinates, if the corresponding vertices were connected with line segments, a rectangle would be formed.

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