Prove usin's Mathematical induction 2^n ≤(n+1)!, for n≥0​

The two curves intersect when [tex]y=0[/tex] and [tex]y=1[/tex]. Over the range [tex]0\le y\le1[/tex], the curve [tex]x=12y^2[/tex] lies above [tex]x=12y^3[/tex]. Hence the area of the shaded region is

[tex]\displaystyle \int_0^1 (12y^2 - 12y^3) \, dy = 4y^3 - 3y^4\bigg|_0^1 = (4-3)-(0-0) = \boxed{1}[/tex]

The solutions to the equation 1 / (x - 1) + 2/x = x / (x - 1) are two solution which are x = 2; and x = 1

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables.

An independent variable is a variable that does not depend on any other variable for its value while a dependent variable is a variable that depends on other variable.

Given the equation:

1 / (x - 1) + 2/x = x / (x - 1)

Multiplying through the equation by x(x - 1):

x + 2(x - 1) = x(x)

x² - 3x + 2 = 0

x² - 2x - x + 2 = 0

(x - 2)(x - 1) = 0

x = 2; and x = 1

The solutions to the equation 1 / (x - 1) + 2/x = x / (x - 1) are two solution which are x = 2; and x = 1

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Answer:

Step-by-step explanation:

[tex]\frac{1}{2} x=a=\frac{2}{3} y\\\frac{1}{2}x=a\\x=2a \\\frac{2}{3} y=a\\y=\frac{3}{2} a\\x+y=2a+\frac{3}{2} a=\frac{4a+3a}{2} =\frac{7}{2} a=na\\n=\frac{7}{2}[/tex]

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

I don't know what you mean by g. and h., so I'll just skip that part.

I'm assuming you're asking about some arbitrary finite set [tex]A[/tex].

a. If [tex]A[/tex] has 64 subsets, then [tex]A[/tex] has [tex]\log_2(64) = \boxed{6}[/tex] elements. This is because a set of [tex]n[/tex] elements has [tex]2^n[/tex] subsets/elements in its power set.

b. A proper subset is a subset that doesn't contain all the elements of the parent set. This means we exclude the set [tex]A[/tex] from its power set. The power set itself would have 32 elements, so [tex]A[/tex] would have [tex]\log_2(32) = \boxed{5}[/tex] elements.

c. The empty set is a proper subset of any non-empty set. However, if [tex]A=\emptyset[/tex], then it has no proper subsets. So [tex]A[/tex] must be the empty set and have [tex]\boxed{0}[/tex] elements.

d. By the same reasoning as in part (b), if [tex]A[/tex] has 255 proper subsets, then it has a total of 256 subsets, and [tex]\log_2(256) = \boxed{8}[/tex].

The best estimate for the correlation coefficient of the data is: C. 0.5.

Correlation can be defined as a quantitative measure that describes the relationship between variables. Correlation coefficient, r, is a number that ranges from -1 to 1.

The farther the correlation coefficient from 0, the stronger the relationship between two variables, and also, the more closer the data points are on a plotted graph. Also, a negative correlation value will show a trendline on a plot that slopes downward, while positive correlation value will show a trendline that slopes upwards.

In the graph given, the points are moderately closer to each other. Also, the trendline slopes upward, which suggest a positive correlation.

Since the points are moderately spaced from each other, the correlation coefficient would positive and be approximately 0.5.

Therefore, the best estimate for the correlation coefficient of the data is: C. 0.5.

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Answer:

Step-by-step explanation:

Option C.

Step-by-step explanation:

Initial number of clients = 18

Number of Increase per week = 4

We need to use an arithmetic sequence to write a function.

So, the required arithmetic sequence for four weeks is

18, 22, 26, 30

Here, first term is 18 and common difference is 4. The range is  [18,30].

The explicit formula of n arithmetic sequence is

where, a is first term and d is common difference.

Substitute a=18 and d=4 in the above formula.

The function notation is

The function is f(x)=4x+14 and the range is  18 ≤ y ≤ 30.

Therefore, the correct option is C.

Answer:

Step-by-step explanation:

Equivalent polar coordinates will have the same modulus and some multiple of 360° added to the argument, or the opposite modulus and some odd multiple of 180° added to the argument.

  a∠b° = a∠(b+360n)° = -a∠(b +180 +360n)°

Equivalents will be ...

  (2, (-60 +360n)°) = (2, 300°)

  (-2, (-60 +180 +360n)°) = (-2, -240°) = (-2, 120°)

Equivalents will be ...

  (-2, (240 +360n)°) = (-2, -120°)

  (2, (240 +180 +360n)°) = (2, -300°) = (2, 60°)

Expanding the desired form, we have

[tex]A \sin(\omega t + \phi) = A \bigg(\sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi)\bigg)[/tex]

and matching it up with the given expression, we see that

[tex]\begin{cases} A \sin(\omega t) \cos(\phi) = 2 \sin(4\pi t) \\ A \cos(\omega t) \sin(\phi) = 5 \cos(4\pi t) \end{cases}[/tex]

A natural choice for one of the symbols is [tex]\omega = 4\pi[/tex]. Then

[tex]\begin{cases} A \cos(\phi) = 2 \\ A \sin(\phi) = 5 \end{cases}[/tex]

Use the Pythagorean identity to eliminate [tex]\phi[/tex].

[tex](A\cos(\phi))^2 + (A\sin(\phi))^2 = A^2 \cos^2(\phi) + A^2 \sin^2(\phi) = A^2 (\cos^2(\phi) + \sin^2(\phi)) = A^2[/tex]

so that

[tex]A^2 = 2^2 + 5^2 = 29 \implies A = \pm\sqrt{29}[/tex]

Use the definition of tangent to eliminate [tex]A[/tex].

[tex]\tan(\phi) = \dfrac{\sin(\phi)}{\cos(\phi)} = \dfrac{A\sin(\phi)}{\cos(\phi)}[/tex]

so that

[tex]\tan(\phi) = \dfrac52 \implies \phi = \tan^{-1}\left(\dfrac52\right)[/tex]

We end up with

[tex]y(t) = 2 \sin(4\pi t) + 5 \cos(4\pi t) = \boxed{\pm\sqrt{29} \sin\left(4\pi t + \tan^{-1}\left(\dfrac52\right)\right)}[/tex]

where

• amplitude:

[tex]|A| = \boxed{\sqrt{29}}[/tex]

• angular frequency:

[tex]\boxed{4\pi}[/tex]

• phase shift:

[tex]4\pi t + \tan^{-1}\left(\dfrac 52\right) = 4\pi \left(t + \boxed{\frac1{4\pi} \tan^{-1}\left(\frac52\right)}\,\right)[/tex]

Answer:

The area of windowpane is answer.

Answer:

The area of the window pane

Step-by-step explanation:

The area of an object is the width x length of said object.

Ex: A  4 x 4 sized square has an area of 16 ( 4 x 4 = 4 by 4)

P.S. The area of Andrea´s windowpane is 648.

From the 2 selected employees, the probability that they will both be females is; C: 4/15

We are given;

Number of male employees = 7

Number of female employees = 8

Total number of employees = 8 + 7 = 15

Now, 2 random employees are chosen and so the number of ways of selecting this is; 15C2

Now, out of the 2 selected employees, the probability that they will be female = 8C2/15C2 = 28/105 = 4/15

Thus, we can conclude that from the 2 selected employees, the probability that they will both be females is; C: 4/15

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Answer:

0

Step-by-step explanation:

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

because2^3=8

Answer:

Step-by-step explanation:

why is it so hard omg

the area of the minor segment is 0. 29 cm^2

From the information given, we have the following parameters;

It is important to note the formula for area of a sector is given as;

Area = πr² + θ/360° - 1/ 2 r² sin θ

The value for π = 3.142

θ = 30°

Now, let's substitute the values

Area = 3. 142 × 5² × 30/ 360 - 1/ 2 × 5² × sin 30

Find the difference

Area = 3. 142 × 25 × 1/ 12 - 1/ 2 × 25 × 1/2

Multiply through

Area = 6. 54 - 6. 25

Area = 0. 29 cm^2

The area of the minor segment is given as 0. 29 cm^2

Thus, the area of the minor segment is 0. 29 cm^2

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The dimensions of the pool would give a quadratic equation x² + 28x - 770 = 0

For the pool, we have that;

Let the width = x feet

Length = (x+8) feet

The pool alongside the walkway gives;

Width = x + 5 + 5 = (x + 10) feet

Length = x + 8 + 5 + 5 = (x + 18) feet

Total area of the pool with walkway = 950 square feet

Note that formula for area is given as

Area = length * width = 950

Equate the length and width

(x+18)  × (x + 10) = 950

Using the  FOIL method, we have;

(x × x )+ (x × 10) + (18 × x) + (18 × 10) = 950

x² + 10x + 18x + 180 -950 = 0  

collect like terms

x² + 28x - 770 = 0

Thus, the dimensions of the pool would give a quadratic equation x² + 28x - 770 = 0

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From the information given, F(x) = 9.3319419 or approximately 9. This is  problem relating to math functions

Recall that we are given x to be = 16.1

Hence,

[tex]f\left(x\right)=\ 2x^{\frac{1}{4}}\ +\ 3x^{\frac{1}{2}}[/tex] =

2 (16.1) ⁰°²⁵ + 3 (16.1)⁰°⁵

= 32.2⁰°²⁵ + 48.3⁰°⁵

= 2.3821218 +6.9498201

f(x) = 9.3319419

f(x) approximately 9

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According to the confidence interval only persons who watch at least 9.9 hours TV per day are eligible.

According to the statement

we have given that the mean of 7 hours per day watching TV, and a standard deviation of 1.8 hours.

And we have to find that the At least how many hours of daily TV watching are necessary for a person to be eligible for the interview.

So, For this purpose,

The confidence interval is is a range of estimates for an unknown parameter. A confidence interval is computed at a designated confidence level.

Let us Assume x is the amount of hours that 5% of the persons exceeds.

Then P(z<(x-7)/1.8) = 0.95

From a standard normal table, we know that:

(x-7)/1.8 = 1.6449

x-7 = 2.9607

x = 9.9607

So, According to the confidence interval only persons who watch at least 9.9 hours TV per day are eligible.

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The formula that shows the relationship between the distance, rate and time is t = d/r and t = 5.

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables.

An independent variable is a variable that does not depend on any other variable for its value while a dependent variable is a variable that depends on other variable.

Given that:

d = rt

Hence:

t = d/r

For d = 40, r = 8:

t = 40 / 8

t = 5

The formula that shows the relationship between the distance, rate and time is t = d/r and t = 5.

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The point estimate of difference of the sample his will be -20.

Based on the information given, the. following can be depicted:

n1 = 13

x1 = 141

s1 = 13

n2 = 9

x2 = 161

s2 = 12

The point estimate of difference will be:

= 141 - 161

= -20

The margin of error to be used in constructing the confidence interval will be calculated by multiplying the standard error which is 5.467 and the critical value. This will be:

= 5.467 × 2.528

= 13.822

The margin of error is 13.822.

The confidence interval will now be:

= (-20 + 13.822) and (-20 - 13.822)

= -6.178 and -33.822

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Answer:  11

Step-by-step explanation:

XY + YZ = XZ

  XY + 3 = 14

         XY = 11

The dependent variable here is task performance. The independent variable in the question is musical genres.

This is the term that is used to refer tp the variable that is of interest. It is the variable that one is trying to determine.

The independent variable is the x variable. It is the variable that helps us check the impact that it has on the dependent variable.

The task performance is what we want to know here. To do so we want to use the genre of music.

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Answer:

What is the expected value when rolling a fair 12-sided die?

132=6.5

Recall the definition of the expected value:

E[X]=∑ni=1xp(x)

Here, we have 12 possible values the die can take on, and the probability of each one is the same: 1/12. So we can just pull the term p(x), the probability of each outcome, out of the summation:

E[X]=112∑12i=1x

A useful trick here is that the sum of the numbers 1..N=N(N+1)2. So:

E[X]=11212(13)2

E[X]=12(13)12(2)

E[X]=132

Step-by-step explanation:

hope it helps you :)

Step-by-step explanation:

since the sumation of f(x) of a probability is 1

thw probability to win is o.5 and to lose is o.5 so expected value is xf(x

your expected value will b 0.5 multiply by 5 thats is 2.5 thats your expected gain

Step-by-step explanation: Given that the graph shows the side view of a water slide with dimensions in feet.

We are to find the rate of change between the points (0, ?) and (?, 40).

From the graph, we note that

the y co-ordinate for the x co-ordinate 0 is 80 and the x co-ordinate for the y co-ordinate 40 is 5.

So, the given two points are (0, 80) and (5, 40).

The rate of change for a function f(x) between the points (a, b) and (c, d) is given by

Therefore, the rate of change for the given function between the points (0, 80) and (5, 40) is

Thus, the required rate of change is -8.

Based on the percentage of the first, second, and third acids, the milliliters of each acid that should be used to make 100 ml of 18% are:

Assuming the concentration of the 40% acid is denoted as x, the other acid concentrations would be:

10% acid = 4x

20% = 100 - 5x

The target solution is 100ml of 18%.

Solving gives:

(10% × 4x) + (20% × (100 - 5x)) + (40% × x) = 18% x 100

(80% × x) + 20 - x = 18

(80x - 100x) / 100 = 18 - 20

-20x / 100 = -2

20x = 200

x = 200 / 20

x = 10 ml

The 10% solution:

= 10 x 4

= 40 ml

20% acid:

= 100 - (5 x 10)

= 50 ml

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Answer:

The function is odd because f(-x) = -f(x)

Step-by-step explanation:

[tex]f(x)=8x^3[/tex]
[tex]f(-x)=8(-x)^3[/tex]
[tex]f(-x)=-8x^3[/tex]
[tex]-f(x)=-(8x^3)[/tex]
[tex]-f(x)=-8x^3=f(-x)[/tex]
[tex]f(-x)=-f(x)[/tex], therefore the function is odd

The function is odd because f(-x)=-f(x).

A relation is a function if it has only One y-value for each x-value.

The given function is f(x)=8x³.

We need to find which statement would be true in the given options.

Let us find this by taking x as -x.

f(-x)=8(-x)³

We know that when a negative sign is multiplied three times we get negative sign.

f(-x)=-8x³

-f(x)=-(8x³)

-f(x)=-8x³=f(-x)

f(-x)=-f(x)

By the definition of odd function f(-x)=-f(x).

Hence, the function is odd because f(-x)=-f(x).

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Answer:

$580

Step-by-step explanation:

{371x(100-34)/100}+371=580

The original price (rounded to the nearest dollar) is $497.

In this problem, we need to identify what's the original price of the new mattress.

Since we know that the sale price is $371 and is 34% lesser than the original price, we need to add 371 and 34%.

Grab a calculator and add 371 and 34%

we get:

371 + 34% = 497.14

Now round it to the nearest whole number, we get: 497

This is how the answer gets 497.

Hope this helps :)

The factors of the equation of power 4, [tex]y=x^{3}-52x^{2} +15x^{4} +16+20x[/tex]  are x=1, -2, 4/3, -2/5.

Given equation is [tex]y=x^{3}-52x^{2} +15x^{4} +16+20x[/tex],

By performing L-division method, we get

[tex]y = (x-1)(15x^{3} +16x^{2} -36x-16)[/tex]

Then again doing the same L-division method to the cubic equation [tex]15x^{3} +16x^{2} -36x-16[/tex], we get

[tex]15x^{3} +16x^{2} -36x-16 = (x+2)(15x^{2} -14x-8)[/tex]

Therefore, [tex]y = (x-1)(x+2)(15x^{2}-14x-8)[/tex]

Then finally the roots of the quadratic equation [tex]15x^{2}-14x-8[/tex] are (x-(4/3)) and (x+(2/5))

Hence, [tex]y=x^{3}-52x^{2} +15x^{4} +16+20x = (x-1)(x+2)(x-\frac{4}{3} )(x+\frac{2}{5} )[/tex]

Therefore, the roots of the equation of power 4, [tex]y=x^{3}-52x^{2} +15x^{4} +16+20x[/tex] are x=1, -2, 4/3, -2/5.

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Answer:

A and D

Step-by-step explanation:

Let's find the slope of the functions in all tables.

A. -1/2

B. -1/2

C. -1/2

D. 1

E. -1/2

Option D. is out since it has a different slope.

The answer has to be two tables out of A, B, C, and E.

Start with Table A.

As x goes from 8 to 6, y goes up by 1.

We can create points for x = 0 and x = 2

x = 2, y = 9

x = 8, y = 6

This is exactly table D.

Answer: Tables A and D