Find the domain of the function f(x)=\sqrt{\sqrt{x^2-16}-3}$

Answer:

5x + 2x + 5x + (3/5x) = 180

12x + 0.6x = 180

12.06x = 180

x = 14

Step-by-step explanation:

Answer:

this very easy to factorize this because (-+)is- (+-)-

(++)+ (--)+ but the sign which is bigger we have to keep that

Answer:

  62 square units

Step-by-step explanation:

You want the surface area of a rectangular prism 3 units high, 2 units wide, and 5 units long.

The surface area of a rectangular prism is given by the formula ...

  SA = 2(LW +H(L +W))

Using the given dimensions, L=5, W=2, H=3, the area is ...

  SA = 2(5·2 +3(5 +2)) = 2(10 +3(7)) = 2(31)

  SA = 62 . . . square units

The surface area of the prism is 62 square units.

Step-by-step explanation:

it is really a simple concept : instead of the isolated variable x you grab the whole function of g(x) and put that into the place of any x appearance in f(x) : g(x) becomes the input argument for f(x).

so,

f(g(x)) = sqrt(36/(x + 3)) = 6×sqrt(1/(x + 3))

What the value of k means in the proportional relationship y =kx is; the change in depth per second of the ascent.

We are told that;

James Cameron descended to the bottom of Challenger Deep in the Marianas Trench; the deepest point in the ocean.

The vessel he rode in called DeepSea Challenger was 35,814 feet deep at its lowest point.

Now, if the time spent is denoted as x in seconds and the depth is denoted as y, the the relationship will be;

y = -kx

where -k is the constant of proportionality which is the change in depth per second of descent.

Now, if the ascent can be modeled by a different proportional relationship y = kx, then it means that k here will be the change in depth per second of the ascent.

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Answer:

D

Step-by-step explanation:

m^ (-3 · -4) = m^12

n ^ (-2 · -4) = n^8

m^12 n^8

a complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations then the position of a 60 gg oscillating mass is given by x(t) -20 sin 10t

this information is needed to solve oscillation problems of this type

the position of a 60 gg oscillating mass is given by x(t)

x = 2 cos wt = 2 cos 10t ; w = 10

velocity = dx/dt = -2 x 10 sin 10 t.=- 20 sin 10t

t = .4

velocity = -20 sin 10 x .4 = -20 sin 4 = -20 x -0.7568 = 15.136 cm /s

w = √ k / m = 10 = √ k / .05

k = 15.136  N/m

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The system does not have a periodic solution in each of 10 and 11.

A solution of the period depends on the independent variable t is called Periodic Solution. For a periodic solution x(t)( in the case of a system, x is a vector), there is a number T≠0 such that

x(t+T)=x(t)  for t∈R.

The given equations are:

      [tex]\frac{dx}{dt} =-2x-3y-xy^2\\\\\frac{dx}{dt} =y+x^3+x^2y[/tex]

F is the derivative of dx/dt and Gis the derivative of dy/dt:

[tex]F(x,y)=-2x-3y-xy^2\\G(x,y)=y+x^3+x^2y[/tex]

When Fx+Gx has the same sign on the entire domain D, then there are no periodic solutions on the domain D.

Fx+Gx=-2-y^2+1-x^2

          =-1-y^2-x^2

             <for all real numbers x,y[tex]\neq[/tex]0

Fx+Gx have the same sign  for all pairs of real numbers with x[tex]\neq[/tex]0 and y[tex]\neq[/tex]0

Thus, there are no periodic solutions in the system(excluding contact solutions)

Hence proved.

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The the probability of being dealt no hearts is 0.75 and probability of being dealt at least one heart is 0.25 .

Total number of cards = 52

A : the probability of being dealt no hearts

B: the probability of being dealt with hearts

P( A ) = 1 - P( B)

= 1 - 13/52

= 1 -0.25

=0.75

C : probability of being dealt at least one heart

P(C) = 13C1 / 52C1

= 13 / 52

=0.25

Probability is a way to tell or calculate the possibility of an event to occur. Using it, we can make predictions about the likelihood of an event happening, or how likely it is , it helps to make predictions more clear .

The probability can be between 0 and 1 here , P being 0 means a failure and P being 1 means a success.

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a. The number of students who were in math only is 7 students.

b. The number of students who were not majoring in computer science is 15.

c. The number of students who were not math or computer science is 8

From the information given, in a class of 25 students, there were 13 math majors, 10 were computer science majors, and 6 were dual majors in both.

It should be noted that the number of math majors only will be:

= 13 - 6

= 7

The number of students not majoring in computer science will be:

= 25 - 10

= 15

Those not math or computer science majors will be:

= 25 - 6 - 7 - (10 - 6)

= 8

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The set of events that the sum of the numbers showing face up is at least 9 is E = {36, 45, 46, 54, 55, 56, 63, 64, 65, 66} and the probability is 5/18.

In the roster form, the elements (or members) of a set are listed in a row between curly brackets. For the event that the sum of the numbers showing face up is at least 9 and is written as a set, look at the sum of values of blue and gray dies that has values greater than equal to 9.

Here, the total outcome S = 36

By doing that, we get,

E = {36, 45, 46, 54, 55, 56, 63, 64, 65, 66} = 10

Then, the probability of getting a sum of at least 9 is calculated as,

[tex]\begin{aligned}P(X\geq 9)&=\frac{\text{Number of favorable outcomes}}{\text{total number of outcomes}}\\&=\frac{n}{S}\\&=\frac{10}{36}\\&=\frac{5}{18}\;\text{or}\;0.28\end{aligned}[/tex]

The answer is 5/18.

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Answer:

x = 0

Step-by-step explanation:

x + 5/4 = 20/16

x + 5/4 = 5/4

x = 0

Answer:

True

Step-by-step explanation:

This is true by definition.

The height of the rocket in given time using regression equation is 586 feet.

The regression equation is, y =  7.70961x² + 176.30826x - 12.44515

Given,

A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet

That is,

x ; 0.6, 1.3, 1.8, 2.5, 3.1

y ; 108, 218, 285, 368, 436

We have to write a quadratic regression equation for the given set;

The quadratic regression equation is;

y =  7.70961x² + 176.30826x - 12.44515

Now,

We have to find the height at a time of 3 seconds using the equation;

So,

y =  7.70961 x 3² + 176.30826 x 3 - 12.44515

y =  69.38649 + 528.92478 - 12.44515

y = 585.86612

y = 586

Height of the rocket= 586 feet

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Answer:

  see the attachment

Step-by-step explanation:

You want to match several expressions with fractional exponents to their equivalent radical forms.

The relationship between the two forms is ...

  [tex]a^{\frac{m}{n}}=\sqrt[n]{a^m}[/tex]

If n=2, or m=1, these are omitted from the radical form.

The attachment shows the matching values.

From the first two rows we can see that:

Fill in the missing values.

The long-term probability that a given marked rat is in room 4 is 1/3

What is meant by probability?

In statistics, probability is defined as the measure of the likelihood of an event to happen and it measures the certainty of the event.

Here we have given that Forty rats are placed at random in a house having 4 rooms. There is one door between rooms 1 and 2, one door between rooms 1 and 4, one door between rooms 2 and 4, one door between rooms 2 and 3, and two doors between rooms 3 and 4 see the book for a picture). There are no doors between rooms 1 and 3. After each minute, a rat may change rooms, or stay still.

And we need to find the  the long-term probability that a given marked rat is in room 4.

By using the eigenvalues method, we know that

Room 4 has the 3 exits that is from Room 1, Room 2 and Room 3.

So, here we have 3 possibilities for the exit so, the rat can choose any one these three ways,

So, the probability can be written as,

=> 1/3

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The DF value we get after is 143 and the integration value we get is 1.76

What is a value ?

Value in ethics and social sciences refers to the degree of significance of something or an activity, with the goal of deciding the best courses of action to take or the best ways to live, or to define the significance of certain activities.

The value describes the value of each digit in relation to its position in the number. The place value and face value of the digit are multiplied to arrive at the answer. Value equals Place Value minus Face Value. Consider this: If we take 45 into consideration. The tens column is where the fourth digit is located in this instance.

The things that someone values are known as their values. In other terms, values are what a person or a group of people see as "important."

(a)

DF for Factor A = 6 - 1 = 5

DF for Factor B = 4 - 1 = 3

DF for Interaction of Factor A and B = 5*3 = 15

Number of replications per cell = 6

Total no of observations = 6*(N_Factor A)*(N_Factor B) = 6 * 6 * 4 = 144

DF Residuals = 144 - 5 - 3 - 15 - 1 = 120

Total DF = N - 1 = 143

(b) Critical F values

Factor A = F(5,120) = 2.30

Factor B = F(3,120) = 2.69

Interaction = F(15,120) = 1.76

(c) As F statistics for Factor A, Factor B as well as Interaction is >> Fc, Hence we can Reject the Null Hypothesis and warrant the rejection of the claim for equality of means

Hence there is a significant difference in Factor A and there is a significant difference in Factor B.

(d) As the F statistics for Interaction = 4.2 >> Fc (1.76), Hence we can conclude that there is a significant Interaction impact of Factor A and Factor B.

Hence there is a significant difference for Interaction term.

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Answer:

0.5a + 0.75p ≤ 23

Step-by-step explanation:

Cost of a apples: 0.5a

Cost of p pears: 0.75p

The total cost must be $23 or less.

0.5a + 0.75p ≤ 23

The best prediction of the number of clicks on the advertisement if 1,500 people visit the website is; C: 83

We want to find the best prediction of the number of clicks on the advertisement if 1,500 people visit the website.

Now, we see the coordinates from the table;

(1045, 60)

(1106, 63)

(1500, y)

where y  is the best prediction of the number of clicks on the advertisement if 1,500 people visit the website.

By interpolation, we can say that;

(y - 63)/(1500 - 1106) = (63 - 60)/(1106 - 1045)

(y - 63)/394 = 3/39

(y - 63)/394 = 1/13

Cross multiply to get;

y - 63 = 394/13

y = 93

Looking at the options, the closest option is option C.

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Answer:

-12

Step-by-step explanation:

If g(x) = 5x, then plugging in -60 for g(x) gives -60 = 5x. Dividing both sides of this equation by 5 gives -12 = x. Therefore, the correct answer is -12.

Taylor Series for following parts,

a) f(x) = 7 - (7/2)x² + (7/24)x⁶ +......

b)f(x) = 21 - 18(x+2) + 25(x + 2)² - 8(x+3)³ +......

c)f(x) = 2 + 2log(2)(x -1) +(log2)²(x-1)² + (1/3 (log2)³(x-1)³ + --------

d)f(x) = x² + ----

The Taylor series giving the expansion of the function f(x) in the neighborhood of the point a. If the function is continuous in the neighborhood, all its derivatives exist, and the series converges, then The form f(x) = f(a)+f′(a)/1!(x−a)+(f′′(a)/2!)(x−a)²+⋯+(fⁿ(a)/n! )(x−a)ⁿ --(1) where fⁿ(a) is the nᵗʰ derivative of f(x) evaluated at a.

a) f ( x) = 7 cos(-x) = 7 cos(x) , a = 0

f(a ) = 7 cos(0°) = 7

f'(x) = - 7 sin(x) , f'(a) = 7 sin(0°) = 0

f"(x) = - 7cos(x) , f"(a) = -7 cos(0°) = -7

f"'(x) = 7 sin(x) , f"'(a) = 7 sin(0° ) = 0

....................................

put all the values in above equation (1) we get,

f(x) = 7 + 0/1! (x -0) - (7/2!)(x-0)² )+ 0 +(7/4!) (x-0)⁴-0 + ----------

f(x) = 7 - (7/2)x² + (7/24)x⁶ +......

b) f(x) = x⁴ + x² + 1, a = -2

f(a= -2) = (-2)⁴ +(-2)² +1 = 21

f'(x) = 4x³ + 2x

f'(a= -2) = 4(-2)³ + 2×(-2) = -36

f"(x) = 12x² + 2

f"( a = -2) = 12(-2)² + 2 = 50

.....

put all the values in above equation (1) we get,

f(x) = 21 - (36/2)(x+2) + (50/2!)(x + 2)² - ( 48/3!)(x+3)³ +......

f(x) = 21 - 18(x+2) + 25(x + 2)² - 8(x+3)³ +......

c) f(x) = 2ˣ ; a = 1

f( a= 1) = 2¹ = 2

f'(x) = 2ˣ log(2)

f'(a= 1) = 2 log(2)

f"(x) = 2ˣ ( log(2) )²

f"( a = 1) = 2 ( log(2) )²

................

put all the values in above formula we get,

f(x) = 2 + 2log(2) +( 2/2!)(log2)² + (2/3!)(log2)³

+--------

f(x) = 2 + 2log(2)(x -1) +(log2)²(x-1)² + (1/3(log2)³(x-1)³ + --------

d) f(x) = x tan⁻¹(x²) ; a= 0

f(a=0) = 0

f'(x) = x (1/(1+x⁴) + tan⁻¹(x²)

f'(a = 0) = 0

f"(x) = ((1+x⁴) - x 4x³)/(1+x⁴)² + (1/(1+x⁴)

= (2/(1+x⁴) - 4x⁴/(1+x⁴)²

f"( a = 0) = 2 - 0 = 2

f"'(x) = - 8x³/(1+x⁴)² - 12x³/ (1+x⁴)² + 16x⁷/(1+x⁴)⁴

f"'(a = 0) = 0

..............

f(x) = 0 +0 + 2/2! (x-0)² + 0 +-------

f(x) = x² + ----

Hence, we get all required Taylor Series for

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The equation of the linear regression is y = 1.49x- 2.76

From the question, we have the following parameters that can be used in our computation:

X 1 2 3 4 5 6 7 8

Y 0,5 0,8 1,2 1,9 3 4,8 7,5 11,9

Rewrite as

X 1 2 3 4 5 6 7 8

Y 0.5  0.8  1.2  1.9  3  4.8 7.5 11.9

When the points are entered in a graphing calculator, we have

The regression equation is represented as

Regression Equation

ŷ = bX + a

Substitute the known values in the above equation, so, we have the following representation

b = SP/SSX = 62.6/42 = 1.49048

a = MY - bMX = 3.95 - (1.49*4.5) = -2.75714

So, we have

y = 1.49048X - 2.75714

Approximate

y = 1.49x- 2.76

Hence, the regression equation is y = 1.49x- 2.76

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Complete question

Given that

X 1 2 3 4 5 6 7 8

Y 0,5 0,8 1,2 1,9 3 4,8 7,5 11,9

Calculate the regression linear equation

Using the Pythagorean triplet, the value of AB is 2.

In the given question we have to find the length of AB.

From the given diagram we can see that;

ED = 2, DC = 8 and BC = 6.

To solve this question we use the Pythagorean Triplet.

Pythagorean triples consist of the three positive numbers a, b, and c, where a^2+b^2 = c^2. The symbols for these triples are (a,b,c). Here, a represents the right-angled triangle's hypotenuse, b its base, and c its perpendicular.

Since we have to find the value of AB.

So we can write it as

EC^2 = AE^2+AC^2

(ED+DC)^2 = AE^2+(AB+BC)^2

Now putting the value

(2+8)^2 = AE^2+(6+AB)^2

10^2 = AE^2+(6+AB)^2

As we know that triplet (6,8,10).

The value of EC is 10, AE = 6, the value of AC can't be 6 because one part of value of AC is already given 6. So the value of AC is 8. AS BC=6, so AB=2.

Hence, the value of AB is 2.

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The growth constant is 1.0986 and the trout population will increase to 14600 after 2.1 years. The result is obtained by using the logistic equation.

The increase of population can be found by using the logistic equation. It is

[tex]P(t) = \frac{K}{1 + Ae^{-kt} }[/tex]

Where

Sunset Lake is stocked with the rainbow trout. We have

Find the growth constant k and time t when P(t) = 14600!

A = (K - P₀)/P₀

A = (28000 - 2800)/2800

A = 25200/2800

A = 9

After 1 year, we have 7000 rainbow trout. The growth constant is

[tex]7000 = \frac{28000}{1 + 9e^{-k(1)} }[/tex]

[tex]1 + 9e^{-k} = 4[/tex]

[tex]9e^{-k} = 3[/tex]

[tex]e^{-k} = \frac{1}{3}[/tex]

k = - ln (1/3)

k = 1.0986

Use k value to find the time when the population will increase to 14600!

[tex]14600 = \frac{28000}{1 + 9e^{-1.0986t} }[/tex]

[tex]1.9178 = 1 + 9e^{-1.0986t}[/tex]

[tex]0.9178 = 9e^{-1.0986t}[/tex]

[tex]\frac{0.9178 }{9} = e^{-1.0986t}[/tex]

[tex]t = \frac{ln \: 0.10198}{-1.0986}[/tex]

t = 2.078

t ≈ 2.1 years

It is in another 1.1 years after t = 1.

Hence, the growth constant k is 1.0986 the population will increase to 14600 when t is 2.1 years.

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the base of a solid is a circular disk with radius 3. Therefore, the volume of the solid is 27π.

The volume of a solid is determined using the formula V = πr²h, where r is the radius of the base and h is the height of the solid. In this problem, the base is a circular disk with radius 3, and the cross-sections are isosceles right triangles with hypotenuse lying along the base. Since the cross-sections are isosceles right triangles, the height of each triangle is equal to the radius of the base, which is 3. Therefore, the height of the solid is also 3. Substituting these values into the formula gives V = π(3)²(3), which simplifies to V = 27π.

Therefore, the volume of the solid is:

V = πr²h

V = π(3)²(3)

V = 27π

Therefore, the volume of the solid is 27π.

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The expression (2/15x - 5) - (-0.7x + 2) is simplified to 5/12x - 3

An algebraic expression can be defined as an expression made up of variables, terms, coefficients, factors and constants.

These expressions are also made up of arithmetic operations.

From the information given, we have;

2/15x - 5) - (-0.7x + 2)

expand the bracket

2/15x - 5 + 7/10x - 2

collect like terms

2/15x + 7/10x - 5 + 2

Add the like terms

4x + 21x /30 - 3

Add the numerators

25x/60 - 3

Divide the common terms, we have;

5/12x - 3

Hence, the expression is 5/12x - 3

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The exponential function is  y = 2400(1.78)x

Since 1.78 < 1, the function is an exponential decay function.

The function can also be written as 2400(1 - 0.04)x.

The rate of decrease is 4% (that's 0.04 expressed as a percent)

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The area of the regular hexagon value calculated using given perimeter of the regular hexagon is equal to option C. 600√3 square meters.

As given in the question,

Perimeter of the regular hexagon is equal to 'P' = 120m

Relation between the area of the regular hexagon 'A' and the perimeter of the regular hexagon 'P is given by :

A = √3 × ( P²/24 )

  = √3 × [(120)²/ 24]

  = √3 × ( 14400 / 24 )

  = √ 3 × 600

 = 600√3 square meters

Therefore, the area of the regular hexagon using the perimeter of the regular hexagon is equal to Option C.  600√3 square meters.

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The computation of the future value is $92,691.08

As we know that

Future value = Present value × (1 + interest rate)^number of years

where, the Present value is $24275

The Interest rate is 4.05%

And, the number of the year is 10 years

Now placing these values to the above formula;

So, the future value is;

Future value = Present value × (1 + interest rate)^number of years

= $24275× (1 + 0.0405)^10

= $24275× 1.448298166

= $92,691.08

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