In Calculus, We Define A Function F With Domain R To Be Strictly Increasing Provided That For All Real (2024)

The length of the diagonal of the rectangle is 11.70 in.

What is a diagonal?

A diagonal of a polygon is a line segment that joins two vertices of the polygon which are not already joined by an edge of the polygon.

Since, there is no information about which polygon's diagonal length we are asked to find, so let us consider that the given side lengths are of a rectangle,

Now, please refer to the figure attached in the solution,

We know that, the diagonal of a rectangle divides it into two right triangles,

Here our rectangle is ABCD and the diagonal AC divides it into rt. Δ ABC and rt. Δ ADC,

Considering the rt. Δ ABC,

The diagonal AC is acting as the hypotenuse of rt. Δ ABC,

So, using Pythagoras theorem,

AC² = AB² + BC²

AC = √4²+11²

AC = √121+16

AC = 11.70

Since, the diagonals of a rectangle are congruent so, AC = BD = 11.70

Hence, the length of the diagonals of the rectangle is 11.70 in.

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The expression of the polynomial is 0.1(x - 1)^2(x - 5)

How to determine the polynomial

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

A root of multiplicity 2 at x = 1 and A root of multiplicity 1 at x = -5y-intercept = -0.5

A polynomial is represented as

P(x) = a * (x - zero)^multiplicity

Using the above as a guide, we have the following:

P(x) = a * (x - 1)^2(x - 5)

The y-intercept is -0.5

So, we have

a * (0 - 1)^2(0 - 5) = -0.5

This gives

a =0.1

So, we have

P(x) = 0.1(x - 1)^2(x - 5)

Henc,e the expression is P(x) = 0.1(x - 1)^2(x - 5)

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The similar doll's quilt has a width of (1/4) yards.

What is similarity of shapes?

Two or more than two shapes are said to be similar if they have equal corresponding measures of angles and the ratio of corresponding sides should be the same.

Given, The daughter's quilt has a length of 2 yards and a width of 1 yard

and the other similar quilt ahs a length of (1/2) yards.

Now, (1/2)/(2) = 1/4.

Therefore, The width of a similar quilt is (1/4)×1 = 1/4 yards.

Q. A woman sews similar quilts for her daughter's doll. If the daughter's quilt has a length of 2 yards and a width of 1 yard, and the doll's quilt has a length of 1/2 yard, what is the width of the doll's quilt?

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Answer:

Veronica's attempt at ordering the hikes from least to greatest by the number of animals seen per kilometer is not correct.

To order the hikes by the number of animals seen per kilometer, we need to calculate the number of animals seen per kilometer for each hike. We can do this by dividing the number of animals seen by the length of the hike in kilometers.

Using this method, we can calculate the number of animals seen per kilometer for each hike as follows:

Rivers Edge: 8/3 = 2.67 animals/km

Wooded Marsh: 24/8 = 3 animals/km

Canyon Creek: 35/15 = 2.33 animals/km

Based on these calculations, the correct order of the hikes from least to greatest by the number of animals seen per kilometer is Rivers Edge, Canyon Creek, Wooded Marsh.

Veronica's mistake appears to be that she simply ordered the number of animals seen in each hike from least to greatest, rather than calculating the number of animals seen per kilometer.

The answer:

F^-1 (x) = x+3/2

The Fraction 9/18 should be multiplied by 1/4 to get the value Equivalent to 9/18 ÷ 2.

What is Fraction?

The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.

The number of parts into which the whole has been divided is shown by the denominator. It is positioned in the fraction's lower portion, below the fractional bar.How many sections of the fraction are displayed or chosen is shown in the numerator. It is positioned above the fractional bar in the upper portion of the fraction.

We have to solve,

9/18 ÷ 2

As, Fraction does not support the division as usually number does.

So, we have to perform the multiplication by reciprocating the number,

9/18 x 1/2

= 9/ 36

= 1/4

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The characteristics of each structural level in the corresponding boxes are hydrogen bonds, amino acids, and disulfide bonds

Protein structure is a key concept in biology, and it refers to the organization of amino acid sequences into three-dimensional structures. The different levels of protein structure are primary, secondary, tertiary, and quaternary.

On line 1, we have the primary structure, which refers to the linear sequence of amino acids in a protein. It is determined by the DNA sequence of the gene that encodes the protein. Each amino acid is joined to the next one by a peptide bond, forming a polypeptide chain.

On line 2, we have the secondary structure, which refers to the local folding of the polypeptide chain. The two structures depicted on the lines under line 2 are the alpha-helix and the beta-sheet. The alpha-helix is a coiled structure in which the polypeptide chain is twisted into a spiral shape, held together by hydrogen bonds. The beta-sheet is a flat structure in which the polypeptide chain is folded into a zigzag pattern, also held together by hydrogen bonds.

On line 3, we have the tertiary structure, which refers to the overall three-dimensional shape of a single polypeptide chain. It is determined by the interactions between amino acid side chains, such as hydrophobic interactions, hydrogen bonds, disulfide bonds, and van der Waals forces. The tertiary structure can be globular or fibrous, depending on the protein's function.

On line 4, we have the quaternary structure, which refers to the arrangement of multiple polypeptide chains in a protein complex. Some proteins consist of a single polypeptide chain, while others consist of two or more chains that interact with each other to form a functional unit. The quaternary structure is also determined by the same types of interactions as the tertiary structure.

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The part of the expression x(5 + y) that goes to the parentheses is 5 + y.

The correct option is 4.

What is an expression?

One mathematical expression makes up a term. It might be a single variable (a letter), a single number (positive or negative), or a number of variables multiplied but never added or subtracted. Variables in certain words have a number in front of them. A coefficient is a number used before a phrase.

Given:
A phrase: “x times the quantity 5 plus y.”

5 plus y 5 + y

x times the quantity 5 plus y = x(5 + y)

The complete expression is,

x(5 + y).

Therefore, 5 + y is the required expression.

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The equation of the plane that passes through the point [tex]$(2,-1,3)$[/tex] and is perpendicular to the line [tex]$\mathbf{v}$[/tex] can be expressed in terms of a normal vector [tex]$\mathbf{n}$[/tex] and a point [tex]$\mathbf{p}$[/tex] on the plane as [tex]$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$[/tex].

To find the normal vector [tex]$\mathbf{n}$[/tex], we must first find a vector parallel to the line [tex]$\mathbf{v}$[/tex]. We can do this by taking the cross product of two non-parallel vectors [tex]$\mathbf{u_1}$ and $\mathbf{u_2}$[/tex] that lie on the line. Let [tex]$\mathbf{u_1} = (1,2,1)$ and $\mathbf{u_2} = (3,4,3)$[/tex]. Then, [tex]$\mathbf{n}=\mathbf{u_1}\times\mathbf{u_2}=(-4,8,-4)$[/tex].The equation of the plane that passes through the point [tex]$(2,-1,3)$[/tex] and is perpendicular to the line [tex]$\mathbf{v}$[/tex] can be expressed in terms of a normal vector [tex]$\mathbf{n}$[/tex] and a point [tex]$\mathbf{p}$[/tex] on the plane as [tex]$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$[/tex].

We can now calculate the equation of the plane as [tex]$\mathbf{n}\cdot(\mathbf{x}-\mathbf{p})=0$[/tex], where [tex]$\mathbf{x}=(x,y,z)$ and $\mathbf{p}=(2,-1,3)$[/tex]. Thus, the equation of the plane is [tex]$-4x+8y-4z+17=0$.[/tex]

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The height of the wall is 14 feet

What are similar triangles?

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

Therefore ;

TH/ JS = TV/VS

4/JS = 6/21

21× 4 = 6 JS

JS = 21× 4/6

JS = 84/6

JS = 14 feet

therefore the height of the wall is 14 feet

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In a case whereby you have 1,000,000 number cubes, each measuring one inch on a side. If you arranged the cubes on the floor to make a square, then the square would fit in your classroom.

How can the square fitbe determined?

You possess one inch on each side, one million number cubes. How tall would the cubes be if they were placed one on top of the other to form a massive tower?

The fact that the high of one cube= 1 inch

The high of 1,000,000 cubes can be calculated as :

=1*1,000,000 = 1,000,000= 1 * 10^6 inch.

The area of a square A= b^2

b= length side of the square

1,000,000 = b^2

b = 1000 inch

Therefore, the dimensions of the square can be expressed as ( 1,000 in x 1,000 in)

But 1 in = 2.54 cm

1,000 inch = (1,000*2.54)

=2,540 cm-

=25.40 meters

Considering this in metres, the dimensions of the square can be expressed as: ( 25.40 m x 25.40 m)

We can conclude that it possible that the square will fit in the classroom.

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The probability of failure on the trial 9 for the given binomial experiment is 0.15.

What is binomial experiment?

An experiment with a set number of independent trials and just two results is referred to as a binomial experiment. The results of these experiments can be classified as either successes or failures. Due to the nature of the experiment's subject, results may only be categorised as successes or failures.

Due to the set number of outcomes that might occur in each experiment's trial, binomial experiments are unique from other types of studies. Binomial experiments, in particular, can only ever produce one of two outcomes.

The probability of success and failure of a trial is same for every number of trials.

Given that,

n = 12 trials

Success = 0.85

The failure is given as:

failure = 1 - success of trial

failure = 1 - 0.85

failure = 0.15

Hence, the probability of failure on the trial 9 for the given binomial experiment is 0.15.

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Answer:

Step-by-step explanation:

The answer is TRUE, If you want to compare the value 3 to the value 5 to see if 3 is greater than 5, you would enter the following formula: =3>5.

The two numbers of the form a+bi and c+di where a,b,c, and d are real numbers, and i is the imaginary unit, always resulting in the correct mathematical result in the same form.

This is because complex numbers are a mathematical construct and the operations defined on them, such as addition, subtraction, multiplication, and division, follow the rules of arithmetic. These operations preserve the form of the numbers, so that the result of any two complex numbers, when operated upon, will always be in the form of a complex number.

For example, when adding two complex numbers (a + bi) and (c + di), the real parts add up to form a new real part, and the imaginary parts add up to form a new imaginary part, resulting in a new complex number in the form (a + c) + (b + d)i.

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Answer:(A) 1.314.08

Step-by-step explanation:

Inflation

Given by the question:
Based on the values given, the stemplot titled "speed limit" could be constructed as follows:

1 | 0

4 | 5 5 5 5

5 | 0 5 5 5 5 5

6 | 0 0 0 5 5 5 5

7 | 0 0

The stem represents the tens digit of each value, and the leaves represent the ones digit. For example, the first value of 10 has a stem of 1 and a leaf of 0.

Based on the stemplot, we can make the following observations:

The speed limits range from 10 to 70.There are no speed limits between 11 and 44.The most common speed limit is 55, which appears 5 times.There are four speed limits of 45 and two speed limits of 50 and 60.There are a total of 23 speed limits.

Therefore, the statement that is true about the data shown in the stemplot is that the most common speed limit is 55.

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The answers to each part is given below.

What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

Given is that Greenville Fine Jewelry charges $12 per charm, plus $14 for the bracelet. Hanson Jewelers, in contrast, charges $11 per charm and $21 for the bracelet.

For the Greenville Fine Jewelry, we can write -

A{x} = 14 + 12x

For the Hanson Jewelers, we can write -

B(x) = 21 + 11x

To add a certain number of charms to her bracelet, the cost will not be same.

Therefore, the answers to each part is given above.

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The required height of the kite to the nearest tenth of a meter is 25.2 m.

What is the sine ratio in a triangle?

In a right-angled triangle, the ratio of the perpendicular (with respect to an angle) to the hypotenuse gives the sine value of that angle. This relationship is used to solve problems regarding height and distance chapter.

Draw the figure as per the given instructions.

The position of the kite is at A and the string is AC which is 37 m long. The angle of elevation is ∠ACB, which is 43°. The height of the kite is AB.

So, apply the formula of sine ratio (perpendicular/hypotenuse) = sinα, where α be the opposite angle of that perpendicular.

AB/AC = sin∠ACB

AB/37 = sin43°

AB = 37sin43°

≈37×0.68

= 25.16

≈ 25.2

Therefore, the obtained answer is 25.2 m.

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The solution to the system of equation y = -2x + 3 and y = 5x - 4 is (1,1).

What is the solution to the system of equation?

Given the system of equation in the question;

y = -2x + 3

y = 5x - 4

To solve the system of equation, plug equation one to into equation two and solve for x.

y = 5x - 4

Plug in y = -2x + 3

-2x + 3 = 5x - 4

Solve for x

-2x - 5x = - 4 - 3

-7x = -7

x = -7 / -7

x = 1

Now, plug x = 1 into equation one and solve for x.

y = -2x + 3

Plug in x = 1

y = -2( 1 ) + 3

y = -2 + 3

y = 1

Therefore, the value of x is 1 and y is also 1.

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[tex]D(x)=\frac{f(x)}{g(x)}[/tex]

[tex]D'(x)=\frac{f'(x)g(x)-g'(x)f(x)}{(g(x))^{2} }[/tex]

What is quotient formula of differentiation?

Quotient rule in calculus is method finding the derivative of the differentiable functions which are in the division form

There are different methods to prove the quotient rule formula, given as,

Using derivative and limit propertiesUsing implicit differentiationUsing chain rule

Here, we are using implicit differentiation method to solve this quotient rule,

Let us take a differentiable function ,

[tex]D(x)=\frac{f(x)}{g(x)}[/tex]--------(1)

So, [tex]f(x)={D(x)}*{g(x)}[/tex]

Using the product rule we get,

[tex]f'(x)= D'(x).g(x)+g'(x).D(x)[/tex] solving for [tex]D'(x)[/tex] we get,

[tex]\frac{f'(x)-g'(x).D(x)}{g(x)} = D'(x)[/tex]------(2)

substitute for D(x) sub (1) in (2)

[tex]D'(x) =\frac{f'(x)-g'(x).\frac{f(x)}{g(x)} }{g(x)}[/tex]

⇒[tex]D'(x) =\frac{f'(x)g(x)-g'(x){f(x)} }{(g(x))^{2} }[/tex]

Hence,[tex]D'(x)=\frac{f'(x)g(x)-g'(x)f(x)}{(g(x))^{2} }[/tex] proved.

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we can purchase a maximum of 23 hats if we buy them for $3.50 each, or a maximum of 27 hats if we buy them for $2.97 each.

What is Equation?

Two or more expressions with an Equal sign is called as Equation.

Function to represent the total cost of the order in terms of the number of hats ordered:

C(n) = {3.50n, if n < 25;

2.97n, if n >= 25}

where n is the number of hats ordered, and C(n) is the cost of the order.

If the budget for hats is $80, we can set C(n) equal to 80 and solve for n:

3.50n = 80, if n < 25;

2.97n = 80, if n >= 25

For n < 25, we get:

n = 80/3.50 ≈ 22.86 =23

if the budget is $80 and the hats cost $3.50 each, we can buy a maximum of 23 hats.

For n >= 25, we get:

n = 80/2.97 ≈ 26.96

So if the budget is $80 and the hats cost $2.97 each, we can buy a maximum of 27 hats.

Therefore, we can purchase a maximum of 23 hats if we buy them for $3.50 each, or a maximum of 27 hats if we buy them for $2.97 each.

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The trigonometry ratios have their values to be

sin(x) = √3/2cos(x) = 0.5tan(x) = √3

How to determine the trigonometry ratios

The complete question is added as an attachment

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

Hypotenuse = 10

Adjacent = 5

Opposite = 5√3

Using the above as a guide, we have the following:

sin(x) = Opposite/Hypotenuse

cos(x) = Adjacent/Hypotenuse

tan(x) = Opposite/Adjacent

Using the above as a guide, we have the following:

sin(x) = 5√3/10 = √3/2

cos(x) = 5/10 = 0.5

tan(x) = 5√3/5 = √3

The above are the values of the trigonometry ratios based on the attached figure

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The graph of the given inequality of x > 23 is attached.

What is an Inequality?

The relationship between two expressions or values that are not equal to each other is called inequality.

A number line can be used to represent numbers placed on regular intervals. A number line can be used to represent an inequality.

Given that the inequality of x > 23

We are asked to plot the given inequality on a number line.

x > 23

The above inequality says that, the value of x is equal to or greater than 23.

Hence, the graph of the given inequality is attached.

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The number of calories burned per minute be 23.33333.

What is meant by system of equations?

Equations simultaneously, or a system of equations Several equations in algebra must be solved concurrently (i.e., the solution must satisfy all the equations in the system). There must be an equal number of equations and unknowns for a system to have a singular solution.

In order to locate the point where the lines intersect when the equations are graphed, systems of equations must be solved. The (x,y) ordered pair of this intersection point is regarded as the system's solution.

Let j be the number of calories burned by Shelly while running and c be the number of calories she burns while cycling.

The system of equations be

45j + 30c = 1350

30j + 45c = 1350

45j + 30(10)=1350

j = 23.33333

Therefore, the value j be 23.33333.

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The equations must intersect at a single point then it have the unique solution and the value of h and k are h = -1 and k = 6

A system has no solution if the equations are inconsistent, meaning there is no way to satisfy both equations simultaneously. It has a unique solution if the equations intersect at a single point, meaning there is only one set of values that satisfies both equations.

Let's take a look at the first system of equations, 19.

Equation 1: x + hx = 2

Equation 2: 4x + 8x = k

For the system to have no solution, the equations must be inconsistent. This means they must be parallel and never intersect. To achieve this, we can set the slopes of the lines to be equal but the y-intercepts different. In other words, we need to choose h and k such that:

h = -2 and k = 5

Substituting these values into the equations, we get:

Equation 1: x - 2x = 2

Equation 2: 4x + 8x = 5

You can see that these two equations do not have a common solution. They are inconsistent, meaning there is no point that satisfies both equations.

For the system to have a unique solution, the equations must intersect at a single point. This means the slopes and y-intercepts must be different. To achieve this, we can choose h and k such that:

h = -1 and k = 6

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Answer:

To find the answer, we need to subtract the amount of bags Ben used from the amount he bought:

6 and 3/5 - 4 and 1/5

To subtract fractions, we need to have a common denominator. The smallest common denominator for 5 and 20 is 20, so we'll convert both fractions:

6 and 3/5 = 6 * 5/5 + 3/5 = 30/5 + 3/5 = 33/5

4 and 1/5 = 4 * 20/20 + 1/5 = 80/20 + 1/5 = 81/20

Now we can subtract:

33/5 - 81/20

We need to convert the mixed number to an improper fraction and find a common denominator:

33/5 - 81/20 = 132/20 - 81/20

Now we can subtract:

132/20 - 81/20 = 51/20

So Ben has 51/20 bags of rocks left. This can be simplified to 2 and 11/20 bags.

Answer:

2 2/5

Step-by-step explanation:

6 3/5 - 4 1/5 = 2 2/5

Answer:

Step-by-step explanation:

250 per week base

15%of 1800 he made is = 270

270+250 =520

He made 520$ in the week he made 1800$ in sales

Final answer:

Jaxson's salary is a combination of a fixed $250 plus 15% commission on his weekly sales. For $1800 in sales, he will earn $520. If the sales amount is x dollars, then he will earn $250 + 0.15*x dollars in a week.

Explanation:

Jaxson's income consists of a guaranteed base salary of $250 per week and a commission of 15% on his total weekly sales. If Jaxson made $1800 in sales in a week, his total income for the week would be his base salary plus his commission i.e., $250 + 0.15*$1800 = $520.

Similarly, if Jaxson made x dollars in sales in a week, his total income for that week would be his base salary plus his commission on the sales i.e., $250 + 0.15*x dollars.

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Answer:

13.3 miles

Step-by-step explanation:

Picture a right triangle with legs of 3 and 13. The hypotenuse is the straight line distance back to the starting point. Use the Pythagorean theorem to find the hypotenuse (c):

c² = 13² + 3² = 178

c = √178 = 13.3 miles

The product of the two fractions Tami multiplied will be a negative number between 0 and -1.

How to find the product of the multiplication

Since the first fraction is negative greater than -1 but less than 0

and the second fraction is positive, greater than 0 but less than 1, their product will be negative.

This is because when a negative number multiplies a positive number the product is a negative number

Moreover, since both numbers are in fraction and less than whole their multiplication will also be a reduction tending towards zero

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The kinetic energy (in joule) of the asteroid is, 33 million

What is kinetic energy ?

The kinetic energy is the energy associated with the particle having mass m and moving with the velocity v.

To find the kinetic energy of the asteroid, we need to use the formula:

KE = (1/2)mv²

where KE is the kinetic energy, m is the mass of the asteroid, and v is its velocity.

The mass of the asteroid can be calculated using its density and volume:

V = (4/3)πr³

where V is the volume of the asteroid, r is its radius, and π is the mathematical constant pi.

Since the asteroid is spherical, its radius is half its diameter, which is 0.54 km.

So,

V = (4/3)π(0.54 km)³

= 0.65 km³

The mass of the asteroid is then:

m = density × volume

= 3000 kg/m³ × 10⁹ m³/km³ × 0.65 km³

= 1.95 × 10¹² kg

Now we can calculate the kinetic energy:

KE = (1/2)mv²

= (1/2) × 1.95 × 10¹² kg × (23.0 km/sec)²

= 1.32 × 10²³ joule

To convert this to megatons of TNT, we divide by the energy of one megaton:

= 1.32 × 10³ joule / (4 × 10¹⁵ joule/megaton)

= 3.3 × 10⁷ megatons

Hence, the KE of asteroid is 33 million

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