5. The amount of time, t, to finish a race and
the number of laps, I, around a track

We can use the integral  x=4secθ  for each of the indefinite integrals below.

In the given question we can use either 4tanθ,4sinθ or 4secθ to substitute and simplify the given indefinite integrals,

the 1st integral is,

∫ √(x²-16) dx

we can use the trigonometric substitution x=4secθ in this integral, due to which after simplification it will become as,

∫16 secθtan²θ dθ

which then can easily be solved.

the 2nd integral is,

∫ x²/√(x²-16) dx

we can use the trigonometric substitution x=4sinθ in this integral, due to which after simplification it will become as,

∫16 sin²θ dθ

which then can be solved by using a reduction formula.

the 3rd integral is,

∫ dx /(16+x²)³

we can use the trigonometric substitution x=4tanθ in this integral, due to which after simplification it will become as,

∫(1/16⁴) cos⁴θ dθ

the 4th integral is,

∫ (x²-16)^(5/2)dx

we can use the trigonometric substitution x=4secθ in this integral, due to which after simplification it will become as,

∫4096 secθtan⁶θ dθ

the 5th integral is,

∫  dx /(16-x²)^(3/2)

we can use the trigonometric substitution x=4sinθ in this integral, due to which after simplification it will become as,

∫(1/16² cos²θ) dθ

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A logarithm is an operation used to determine the exponent of a given base number.

What is logarithm?

A logarithm is a mathematical function that describes the relationship between the power to which a number (called the base) must be raised, and the resulting value. The logarithm of a number is the power to which the base must be raised to produce that number.

Start by moving the logarithm to the left-hand side of the equation: log(x²-15) = log(2x)

Next, use the property of logarithms that log(a/b) = log(a) - log(b) to get x²-15 = 2x

then you can simplify the equation by subtracting 2x from both sides x²-2x-15=0

then factor the equation to get (x - 5)(x + 3) = 0

Finally, use the Zero Product Property to solve for x. The solutions will be x = 5 or x = -3

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The required word problem of expression  2(5+w)=42, can be given as twice the sum of a number, and 5 is 42,

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

Here,
The question seems to be incomplete, from sources complete expression is given as 2(5+w)=42,

So word problem of expression 2(5+w)=42 can be given as,
Twice the sum of a number and 5 is 42, when we sum a number with 5 and doubled that number gives 42 as a product.

Thus, the required word problem of expression  2(5+w)=42, can be given as twice the sum of a number, and 5 is 42,

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Answer:

Step-by-step explanation:

Answer:

a)  No inverse function

[tex]\textsf{b)} \quad c^{-1}(x)=5+\sqrt{x-2},\;\; c\geq2[/tex]

[tex]\textsf{c)} \quad f^{-1}(x)=-\dfrac{2x}{3x-5}[/tex]

Step-by-step explanation:

Given function:

[tex]a(x)=-4x^2+3[/tex]

To find the inverse of the function, swap x and y:

[tex]\implies x=-4y^2+3[/tex]

Rearrange the equation to isolate y:

[tex]\implies x-3=-4y^2[/tex]

[tex]\implies y^2=-\dfrac{x-3}{4}[/tex]

[tex]\implies y^2=\dfrac{3-x}{4}[/tex]

As the domain and range of function a(x) are unrestricted, the inverse relation is a “sideways” parabola and therefore it is not a function, since it fails the vertical line test.

Given function:

[tex]c(x)=(x-5)^2+2,\;\;x\geq5[/tex]

As the domain of function c(x) is restricted to x ≥ 5, the range is also restricted.

To find the inverse of the function, swap x and y:

[tex]\implies x=(y-5)^2+2[/tex]

Rearrange the equation to isolate y:

[tex]\implies x-2=(y-5)^2[/tex]

[tex]\implies \pm\sqrt{x-2}=y-5[/tex]

[tex]\implies y=5\pm\sqrt{x-2}[/tex]

As the domain of function c(x) is restricted to x ≥ 5 then:

[tex]\implies y=5+\sqrt{x-2}[/tex]

Replace y with c⁻¹(x):

[tex]\implies c^{-1}(x)=5+\sqrt{x-2}[/tex]

The domain of the inverse of a function is the same as the range of the original function.  Therefore, the domain of the inverse function is restricted to x ≥ 2.

Given function:

[tex]f(x)=\dfrac{5x}{3x+2}[/tex]

To find the inverse of the function, swap x and y:

[tex]x=\dfrac{5y}{3y+2}[/tex]

Rearrange the equation to isolate y:

[tex]\implies x(3y+2)=5y[/tex]

[tex]\implies 3xy+2x=5y[/tex]

[tex]\implies 3xy-5y=-2x[/tex]

[tex]\implies y(3x-5)=-2x[/tex]

[tex]\implies y=-\dfrac{2x}{3x-5}[/tex]

Replace y with f⁻¹(x):

[tex]\implies f^{-1}(x)=-\dfrac{2x}{3x-5}[/tex]

The resulting vector has the same dimensions as the original, 1 x 3.

A rectangular array of numbers, symbols, or expressions set up in rows and columns is known as a matrix. They are employed in a wide variety of mathematical disciplines and contexts, including calculus, statistics, physics, and engineering, as well as linear algebra. Matrices are an essential tool for solving systems of linear equations and representing linear transformations since they may be added, subtracted from, and multiplied by other matrices.

The matrix multiplication of [2 1 0] and [1 -1 2 -1 -2 1 0 1 1] is:

[2 1 0] x [1 -1 2 -1 -2 1 0 1 1] = [2(1) + 1(-1) + 0(2), 2(-1) + 1(-2) + 0(1), 2(0) + 1(1) + 0(1)]

= [2 -3 1]

Therefore, the product of the two matrices is a row vector [2 -3 1], which has the dimensions 1 x 3.

If we subtract 1 from each element of this row vector, we get:

[2 -3 1] - 1 = [1 -4 0]

So, the resulting vector has the same dimensions as the original, 1 x 3.

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From the given option , the completed sentence is : An odd function is symmetric with respect to the origin .

The Odd Functions are functions which return the negative inverse when x is replaced with "-x" . which means that function f(x) is an odd function when f(-x) = -f(x) .

The Odd Functions are symmetrical about origin because ; the Odd function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.

Therefore , the odd functions are symmetric about the origin .

The given question is incomplete , the complete question is

Select the correct choice that completes the sentence below.

An odd function is symmetric with respect to the _____  .

(a) y-axis

(b) origin

(c) line y = x

(d ) x-axis.

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IfIf Ryan claims that the cost of Plan A and Plan B differ a lot. The statement that best explains why Ryan was misled by the graph is: b.) The scale of intervals on the horizontal axis of both graphs is inconsistent.

Based on the graph we can see that the graph is inconsistent based on the fact that the scale of intervals on the horizontal axis of Plan A  graph and Plan B graph are inconsistent.

The reasons why the both graphs are inconsistent was  because the lines did not intersect or are not on the same line which means that they are parallel lines.

Therefore the correct option is B.

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Answer:

Step-by-step explanation:

The two graphs use different vertical scales.

By the method of logical reasoning and the seating positions of seven people it is obtained that F sits third to the left of C.

What is logical reasoning?

The technique of applying systematic, logical steps based on mathematical logic to solve a problem is known as logical reasoning. Based on the information provided and mathematical principles, inferences can be drawn.

This seating positing problem can be solved by logical reasoning.

There are seven people - A, B, C, D, E, F, G.

Firstly, A sits third from one of the extreme ends.

Then two people sit between A and B.

The positions can be -

__ __ A __ __ B __ or __ B __ __ A __ __

Now, F sits fourth to the left of B.

So, second seating position gets eliminated.

Also, C sits at the immediate left of B.

__ F A __ C B __

Now, G sits third to the right of E.

E does not sit at any of the extreme ends.

D F A E C B G

So, the seating arrangement is obtained.

Therefore, F sits third to the left of C.

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Seven people named A, B, C, D, E, F, and G are sitting in a row facing north but not necessarily in the same order. A sits third from one of the extreme ends. Two people sit between A and B. C sits at the immediate left of B. F sits fourth to the left of B. G sits third to the right of E. E does not sit any of the extreme ends.

What is the position of F with respect to C?

The values of m and b on the linear equation:

2 + 3x + 5 - 2x = y

are

m = 1

b = 7

A general linear equation can be written as in the slope-intercept form as:

y = m*x + b

Where m is the slope and b is the y-intercept.

Here we have the linear equation

2 + 3x + 5 - 2x = y

Now we want to write this in the slope-intercept form, to do so, we just need to simplify the left side, then we will get:

2 + 3x + 5 - 2x = y

(2 + 5) + (3x - 2x)  =y

7 + x = y

Then the linear equation is:

y = 1*x + 7

So the values are:

a = 1

b = 7

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The volume of the toy is 35cm^3. The solution has been obtained using the formula of triangular pyramid.

What is a triangular pyramid?

A triangular pyramid has a triangle-shaped base, and its apex is where the three triangular faces converge. There is a unique type of triangular pyramid called a tetrahedron, which features equilateral triangles on each face. The formula for a triangular pyramid consists of the volume and surface area of the pyramid, which are used to determine the height and slant height as well as the three triangular-shaped sides.

In the given figure, x represents the base(b), y represents the base height(h) and z represents the height(H).

Let's first find the pyramid's base area(B) which is given by;

⇒B = 1/2 * b * h

⇒B = 1/2 * 3 * 7

⇒B = 10.5cm^2

The volume of the toy is given by:

⇒V = 1/3 * B * H

⇒V = 1/3 * 10.5 * 10

⇒V = 35cm^3

Hence, the volume of the toy is 35cm^3.

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The height of the pole used in the question is; 12.15 ft

The 3 main trigonometric ratios are;

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

Now, since the guy wire makes an angle of 13 degrees with the ground, it means the guy wire will be the hypotenuse of the right angle triangle and the height of the pole which we will label as h will be the opposite side of the triangle. Thus;

height/54 = sin 13

height = 54 * sin 13

height = 54 * 0.225

Height = 12.15 ft

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Since the Fourier transform of a real value signal is the product of the even cosine transform and the odd sine transform, it is conjugate symmetric and consequently symmetric about the frequency 0.

A real-valued signal's Fourier transform is a conjugate symmetric function, which means it is symmetric about frequency 0. This is demonstrated by the fact that a real-valued signal's Fourier transform may be expressed as the product of its cosine and sine transforms:

F(w) = Fc(w) plus Fs (w)

Where the cosine and sine transforms, respectively, are denoted by Fc(w) and Fs(w). Both the sine and cosine transforms are symmetric about the frequency zero because the sine transform is an odd function and the cosine is an even function. As a result, a real-valued signal's Fourier transform also exhibits symmetry near frequency 0.

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The complete question is

g suppose that w(t) is a real-valued signal. prove that its fourier transform magnitude is symmetric around frequency 0.

The elements of the set A'(B∩C)is  { r, t , v, x, y, z}

It is important to note that the universal set is the set containing all the elements for a given event.

From the information given, we have that;

The signs used to represent sets are;

To determine the set,

A'(B∩C)

A' = { r, t, v, x, z}

B∩C = { y, z}

Now, the set A'(B∩C)

A'(B∩C) = { r, t , v, x, y, z}

Hence, the set is { r, t , v, x, y, z}

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Answer:
Maria is able to spend less than $12.
C < 12

Step-by-step explanation:

If Maria has already spent 17 dollars, that means she has 12 dollars left. C cannot be greater than 12. It has to be less.

So think of this C < $12
So if we do the total on one side and the $17 and C on the other side of inequality we get
C + 17 < 29

It cannot be equal to 29 so that's why we don't use a less than or equal to symbol.

Simplify this to get C < 12

I hope this is able to help you :D

Mrs. Barton should use 2 cups of peanut butter in her huge batch with 1 cup of oil.

What is fraction?

Fractions are a way of representing part of a whole. They are written as two numbers separated by a slash. The top number is called the numerator, and the bottom number is called the denominator. For example, in the fraction 1/2, 1 is the numerator and 2 is the denominator. This fraction represents one part of a total of two parts.

Mrs. Carter's famous peanut butter cookies call for 1 cup of peanut butter for every 1/2 of a cup of oil. Today, she wants to make a huge batch with 1 cup of oil.

From the above question:

1/2 cup of oil = 1 cup of peanut butter

1 cup of oil = x

Cross Multiply

1/2x = 1 × 1

x = 1 ÷ 1/2

x = 1 × 2/1

x = 2 cups of peanut butter.

Therefore, for huge batch with 1 cup of oil, she should use 2 cups of peanut butter.

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The expression to determine the distance between the lowest and the highest point is -

{x} = 14505 + |- 282|

Given is that the highest elevation in California is 14,505 feet, and the lowest elevation is − 282 feet.

The expression to determine the distance between the lowest and the highest point can be written as -

{x} = 14505 + |- 282|

where -

|  |  represents modulus function.

Therefore, the expression to determine the distance between the lowest and the highest point is -

{x} = 14505 + |- 282|

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Yes, the piecewise functions that coincide with the given function are [(x + 3)-1, when x is greater than or equal to 3, [(x-3)/2] -1, when x is less than 3, [(x+3)/2] -1, when x is greater than -3, and -(x+3)/2] -1, when x is greater than or equal to -3.

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The ratio may be written if it is a component of a percentage.

6 is to 5 is 12 is to 10 .

Ratios can be expressed as a notation, like 1/2, or as a ratio, like 6:5. A ratio's value is determined by dividing the first and second numbers. Consider the ratio 12:10 as an example. "6 to 5" should be read as the ratio. Any fraction, like 6/5, can also be expressed as a ratio, which is written as 6:5.

A colon or a fraction bar are two typical symbols representing ratios. You can also use words.

6 : 5 = 6/5 = 6 to 5

The ratio may be written if it is a component of a percentage.

6 is to 5

as in ...

12 is to 10 what 6 is to 5, and vice versa.

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Answer:

I believe it's 350 R(remainder) 7

The order of the functions' average rates is h(x), g(x) and f(x)

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

The functions

The interval is given as

4 ≤ x ≤ 7

And the average rate is calculated as

Rate = (f(x₂) - f(x₁))/(x₂ - x₁)

So, we have

f(x) rate = (10 + 10)/(7 - 4) = 6.67

g(x) rate = (19 - 7)/(7 - 4) = 4

For h(x), we have

h(7) = 7^2 - 9 * 7 + 5 = -9

h(4) = 4^2 - 9 * 4 + 5 = -15

So, we have

h(x) rate = (-15 + 9)/(7 - 4) = -2

When compared, we have

-2 less than 4 less than 6.67

So, the order is h(x), g(x) and f(x)

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The amount of water placed in the fountain is given as follows:

D. 9,600π in³

The volume of a cylinder of radius r and height h is given by the equation presented as follows:

V = πr²h.

The parameters for the fountain in this problem are given as follows:

r = 30 in, h = 16 in.

Then the volume of the entire fountain is given as follows:

V = π x 30² x 16

V = 14,400π in³.

Two thirds of this entire volume is given as follows:

2/3 x  14,400π in³ = 9,600π in³

Meaning that the correct option is given by option D.

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The values of IJ and JK in parallelogram HIJK are 52 and 80 respectively

Recall that in every parallelogram, the opposite sides are equal and parallel.

This implies that IJ ║ HK and HI ║ JK

⇒ 5s - 88 = 3s - 32

Collecting the like terms we have

5s-3s = -32 +88

2s = 56

This implies that s = 28

By substitution we have s as 3(28) -32

= 84-32 = 52 and and 4(28)-32 = 80

The sides of the parallelogram are 80 and 52 respectively

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The angular speed of 14 revolutions per second is equal to 87.96 radians per second.

Angular speed is a measure of how fast an object is rotating or revolving around an axis. It is typically measured in units of angular displacement per unit of time, such as degrees per second, revolutions per minute (RPM), or radians per second.

To convert angular speed from revolutions per second (RPS) to radians per second (RAD/s), we can use the following formula:

Angular speed (RAD/s) = 2π x Revolutions per second (RPS)

Given that the angular speed is 14 revolutions per second, we can substitute this value into the formula:

Angular speed (RAD/s) = 2π x 14 RPS

Angular speed (RAD/s) = 87.96 RAD/s

So, 14 revolutions per second is equivalent to 87.96 radians per second.

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Answer:

20 sides

Step-by-step explanation:

The number of sides of a regular polygon, if each of its interior angles is 162° is 20.