Consider the quadrilateral above. It undergoes the following sequence of transformations:

The exact ratio of of the volume of the cone that is occupied by the sphere is 4/(1+√2)³.

What is a cone?

A cone is a three-dimensional geometric shape that narrows smoothly from a flat base (typically circular base) to a point called the apex or vertex (which creates an axis to the centre of base).

Given that the height and radius of the cone is the same.

Assume that the radius of the cone is x.

Thus the slant height of the cone is √(x² + x²) = x√2.

Assume the radius of the sphere is r.

△OEC and △BDC are similar triangles according to AAA rule.

Thus, BD/OE =BC/OC

x/r = x√2/(x -r)

Cross multiply:

x(x - r) = xr√2

x² - rx = xr√2

x² = xr√2 + rx

rx( 1+√2) = x²

r = x²/[x( 1+√2)]

r = x/ (1+√2)

The volume of the sphere is 4/3 ∏ r³ = 4/3 ∏ [ x/ (1+√2)]³

The volume of the cone is 1/3 ∏r²h =  1/3 ∏x³

The ratio of the volume of the sphere to the cone is

4/3 ∏ [ x/ (1+√2)]³  : 1/3 ∏r³

= 4 : (1+√2)³

= 4/(1+√2)³

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Answer:

Step-by-step explanation: Yes, they can match in 2, 5, and/or 10 equal rows. This is because the number of items in the set can be divided evenly by 2, 5, and 10. For example, if there are 20 items in the set, it can be divided into 2 rows of 10, 5 rows of 4, or 10 rows of 2.

1. A city wants to reduce its carbon footprint by switching to renewable energy sources. The city's current energy consumption is measured in gigawatt hours (GWh) and they want to reduce it by 20% in the next 5 years. The city's current energy consumption is 2,000 GWh per year. What is the target energy consumption in GWh per year that the city should aim for to achieve their goal?

2. A company wants to reduce its water consumption by 50% in the next 10 years. The company currently consumes 500,000 gallons of water per day. How many gallons of water per day should the company aim to consume in 10 years to achieve their goal?

Algebraic expression explaining real world problem is - A store is having a sale where all items are marked down 12% and an additional $2.50 is taken off the final price.

If a customer wants to find out the original price of an item before the sale, they can use the equation 12(x+2.50)=78, where x is the original price of the item.

Algebraic expressions are known as  expressions made up of :

A mathematical statement with variables, constants, coefficients, and algebraic operations is known as an algebraic expression. A good example of an algebraic expression is 5x2+6xyc. Algebraic expressions do not use equal signs, in contrast to algebraic equations. In contrast, since they are not derived from integer constants and algebraic operations, transcendental numbers like and e are not algebraic.

A store is having a sale where all items are marked down 12% and an additional $2.50 is taken off the final price. If a customer wants to find out the original price of an item before the sale, they can use the equation 12(x+2.50)=78, where x is the original price of the item.

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On solving the provided question we can say that the inequality equation can solved as  = [tex]\sqrt[4]{xyzw}[/tex]

An inequality in mathematics is a relationship between two expressions or values that is not equal. Thus, imbalance leads to inequality. An inequality creates the link between two values that are not equal in mathematics. Egality is distinct from inequality. When two values are not equal, most commonly use the not equal sign (). Different inequalities are used to contrast values, no matter how little or large. Many simple inequalities may be resolved by modifying the two sides until the variables are all that remain. But a number of things contribute to inequality: Negative values on both sides are divided or added. Trade off the left and right.

here,

the inequality equation can solved as

[tex]x^m+y^m+z^m+w^m/x^m+y^-m+z^-m+w^-m= (xyzw)^{m/2}[/tex]

[tex]u = x+y/2 \geq \sqrt{xy}[/tex]

v = z+w /2 [tex]\geq \sqrt{zw}[/tex]

u+v/2 = x+y+z+w+/4 [tex]\geq \sqrt{uw} \geq \sqrt{\sqrt{xy} \sqrt{zw} }[/tex]

= [tex]\sqrt[4]{xyzw}[/tex]

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Answer:

5 & -4

4 & -3

3 & -2

2 & -1

Step-by-step explanation:

5 + (-4) = 1

4 + (-3) = 1

3 + (-2) = 1

2 + (-1) = 1

The coordinates of the linear equation for (m, s) are; (0, 4), (2, 3), (-2, 5), (8, 0)

This is a one to one mapping and example of a function

We are given the equation as;

2m + 4s = 16

Thus;

When m = 0, we have;

2(0) + 4s = 16

4s = 16

divide both sides by 4 to get;

s = 4

When s = 3;

2m + 4(3) = 16

2m + 12 = 16

2m = 16 - 12

Divide both sides by 2 to get;

m = 4/2

m = 2

When m = -2

we have;

2(-2) + 4s = 16

-4 + 4s = 16

4s = 20

divide both sides by 4 to get;

s = 5

When s = 0, we have;

2m + 4(0) = 16

2m + 0 = 16

2m = 16

Divide both sides by 2 to get;

m = 16/2

m = 8

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The range of the given function y = 3x - 2 for the domain (-1, 2, 4) is

A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.

Domain of the function is the values of x for which the function is defined.

Range of the function is the value of the function when we put the values in the domain.

Given function,

y = 3x - 2

Given the domain (-1, 2, 4)

x = -1, then y = (3 × -1) - 2 = -5

x = 2, then y = (3 × 2) - 2 = 4

x = 4, then y = (3 × 4) - 2 = 10

Range is (-5, 4, 10).

Hence the range of the function is (-5, 4, 10).

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The events Grade 5 and Science"  are independent events,  P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) .Option B is the correct option.

What is frequency?

The frequency (f) of a specific value is the number of occurrences of the value in the data. The frequency distribution of a variable is the collection of all possible values and the frequencies associated with these values. Frequency distributions are represented graphically as frequency tables or charts.

Given table is

                   English      Science   Total

Grade 5           24              36           60

Grade 8           16               24           40

Total                40              60           100

Total students is 100.

Total student in grade 5 is 60.

The number of students who prefer science is 60.

The number of students who are in grade 5 and prefer science is 36.

The probability of grade 5 is P(grade 5) = 60/100 =0.6

The probability of science is P(science) = 60/100 = 0.6

The probability of Grade 5 ∩ Science is P(Grade 5 ∩ Science) = 36/100 =0.36.

Thus P(Grade 5 ∩ Science) =P(Grade 5) ⋅ P(Science).

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Distance traveled after toll payment is 1.2 miles on highway 40.

The distance may be calculated using a curved route. Displacement measurements can only be made along straight lines. Distance is path-dependent, meaning it varies depending on the direction followed. Displacement simply depends on the body's beginning and ending positions; it is independent of the route.

Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.

The size or extent of the displacement between two points is referred to as distance. Keep in mind that the distance between two points and the distance traveled between them are not the same. The entire length of the journey taken between two points is known as the distance traveled. Travel distance is not a vector.

Distance traveled before toll payment =3/5 miles on highway 40

Distance traveled after toll payment =2*3/5  = 6/5 =

1.2 miles on highway 40.

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Answer:

4,4

Step-by-step explanation:

44x1/10

The triangle has length of the side h is equal to 97.1 inches approximately to the nearest tenth, using the sine rule.

The sine rule is a relationship between the size of an angle in a triangle and the opposing side.

For the triangle ΔGHI, by sine rule;

g/sinG = h/sinH = i/sinI

We first derive the angle G as follows:

151° + 7° + G = 180° {sum of interior angles of a triangle}

G = 180° - 158

G = 22°

Hence by sine rule;

75/sin22° = h/sin151°

h = (75 × sin151°)/sin22° {cross multiplication}

h = 97.1

Therefore, by application of the sine rule we have the length of the side h = 97.1 inches approximately to the nearest tenth

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Answer:

Copy the table into your notebook and include columns to find the endpoint (upper value) for each interval. Figure out the cumulative frequency and

Step-by-step explanation:

Therefore , the solution to the given problem of derivative comes out to be 2cos(mx+ny) for both the cases of partial derivative.

The definition of a derivative is the simultaneous change that occurs at a certain place. Implicit & explicit functions are typically distinguished from one another. Explicit functions are those where the value of the known individual entity "x" directly determines the value of the known dependent variable "y."

Here,

Given : f(x,y)= sin^(2) (mx+ny)"

Partial derivative with respect to x

=> dx/dy = 2sin (mx+ ny) *m

=> d²x/dy = 2cos(mx+ny)

Partial derivative with respect to y

=>  dx/dy = 2sin (mx+ ny) *n

=> d²x/dy = 2cos(mx+ny)

Therefore , the solution to the given problem of derivative comes out to be 2cos(mx+ny) for both the cases of partial derivative.

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Answer:

----------------------------------------

Given sequence from zeros to 6th terms:

It is a GP with the common ratio of 1/2 as each term is half the previous one.

Use the nth term equation, considering the first term is t(1) = 8, and common ratio r = 1/2:

Answer:

a)  Geometric sequence

[tex]\textsf{b)} \quad t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

Step-by-step explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

From inspection of the given table, t(n) halves each time n increases by 1.

Therefore, the sequence is geometric with a common ratio of 1/2.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the initial value (when n = 1) and the common ratio into the formula to create an equation for the nth term:

[tex]\implies t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

Answer:

-3

Step-by-step explanation:

rise over run method

3 up and 1 to the left which makes the slope negative

Considering the coordinates of the preimage of line AB represented by A (6, -4), B (2, -8) and coordinates of the image of line A'B' represented by A' (9, -6), B'(3, -12) the dilation factor is 3/2

The transformation rule for dilation is as follows

(x, y) for a scale factor of r → (rx, ry)

following similar procedure for the given problem we have

A (6, -4) transformed to A' (9, -6)

B (2, -8) transformed to B' (3, -12)

hence 6r = 9, and -4r = -6

r = 9/6 = -6/-4

r = 3/2

The dilation factor of the transformation is 3/2

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Answer:

Step-by-step explanation:

The length of the two diagonals of a rectangle are said to be equivalent.

From the Pythagorean theorem, the diagonal (a) will be the hypotenuse in the equation and the length (b) and width (c) will be its opposite and adjacent.

From the figure given, the length of half of two diagonal of rectangles are given.

We can simply solve for the value of x

TO = RO

3x + 1 = 2x + 4 (combine like terms by first transposing it to the other side of the equation and changing its sign from (+) to (-) or vice versa).

3x - 2x = 4 - 1

x = 3

End Balance: $3,839.25 + $2,579.20 - $2,190.05 = $5,735.15, so the reconciled balance of $5,735.15.

This theory is used to compare the bank statements to the check register in order to properly reconcile the two balances.

The theory used in this question is the bank reconciliation theory.

Step 1: Begin by looking at the two balances - the check register balance and the bank statement balance.

In this case, the check register balance is $3,839.25 and the bank statement balance is $3,450.10.

Step 2: Add up all of the deposits that are not reported on the bank statement. In this case, the deposits are $2,000, $135.67, $254.77, and $188.76.

Step 3: Subtract all of the checks that are not reported on the bank statement. In this case, the checks are $567.89, $23.83, $598.33, and $1,000.

Step 4: Add the deposits and subtract the checks from the check register balance.

In this case, the check register balance ($3,839.25) plus the deposits ($2,579.20) minus the checks ($2,190.05) gives a reconciled balance of $5,735.15.

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Answer: If the model kit for the Dornier Do17 airplane has a wingspan of 10 inches and the scale of the kit is 1 inch to 72 inches. This means that for every inch on the model, the equivalent measurement on the actual Dornier Do17 airplane is 72 inches.

We know that the wingspan of the model is 10 inches and we want to find the actual wingspan of the airplane so we'll use the scale factor of 1:72

To find the actual wingspan, we'll multiply the wingspan of the model by the scale factor.

10 inches * 72 inches/1 inch = 720 inches

Therefore, the actual wingspan of the Dornier Do17 airplane is 720 inches or 60 feet.

Step-by-step explanation:

(A) f(-1) = 1

(B) The range of the function f(x) is [-4, ∞).

What is the range of the function?

In mathematics, a function's range can refer to one of two closely related ideas: Its codomain is a function. The picture of the action Given two sets X and Y, a binary relation f between them is a function if and only if there is precisely one y in Y such that f connects x to y for every x in X.

For the given function we have to find the value of f(-1) and to find the range of the function.

From graph we can see that,

f(-1) = 1

The range of the function is the interval in which function cover the x - axis.

The range of the function is [-4, ∞).

Hence,

(A) f(-1) = 1

(B) The range of the function f(x) is [-4, ∞).

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Here, on solving the provide question to us, we got to know that - equation  210 feet represents  = 825/4

Since equations are essentially questions, efforts to systematically find answers to those questions have been the driving force behind the development of mathematics. From straightforward algebraic equations that just require addition or multiplication to differential equations, exponential equations that use exponential expressions, and integral equations, there are many different types of equations that range in complexity.

Here,

15 feet represents = 1/8  of an inch

1 feet represents = 1/8 X 15

The growth of mathematics has been fueled by efforts to methodically find answers to equations, which are basically questions. There are many distinct forms of equations that range in complexity, from simple algebraic equations that just need addition or multiplication to differential equations, exponential equations that employ exponential expressions, and integral equations 210 feet represents = 1/8 X 15 X 210 = 825/4

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The surface area of the capsule is calculated by adding the surface area of the cylinder and the surface area of the hemispherical bowl.

A hemispherical bowl is a type of bowl with a rounded bottom that is shaped like half of a sphere. It is often used in laboratories, kitchens, and dining rooms. The bowl’s curved surface prevents the contents from spilling, which makes it an ideal choice for mixing, swirling, and scooping. Additionally, the bowl’s shape makes it look aesthetically pleasing, so it can be used as a decorative piece as well.

The surface area of the cylinder = 2πr2 + 2πrh

Where r is the radius of the cylinder and h is the height of the cylinder.

The radius of the cylinder = 12 cm

The height of the cylinder = 16 cm

Therefore, the surface area of the cylinder = 2π(12)2 + 2π(12)(16) = 1536π cm2

The surface area of the hemispherical bowl = 2πr2

Where r is the radius of the hemispherical bowl.

The radius of the hemispherical bowl = 12 cm

Therefore, the surface area of the hemispherical bowl = 2π(12)2 = 144π cm2

The total surface area of the capsule = 1536π cm2 + 144π cm2 = 1680π cm2

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The number of red counters in the bag is given as follows:

8.

A probability is obtained by the division of the number of desired outcomes by the number of total outcomes.

The probabilities for this problem are given as follows:

The sum of the probabilities is of one, hence the value of x is obtained as follows:

2x + x + 0.45 + 0.25 = 1

3x = 0.3

x = 0.1.

Hence the proportion of red is given as follows:

0.2.

There are 18 blue counters in the bag, which is of 0.45 of the total amount, hence the total amount is obtained as follows:

0.45n = 18

n = 18/0.45

n = 40.

Meaning that the number of red counters is given as follows:

0.2 x 40 = 8.

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The set of 4 integers are is given by the equation A = { 1 , 2 , 3 , 4 }

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the set of 4 integers be represented as set A

Now , the equation will be

Let the first number be 1

The absolute value of 1 is | 1 | = 1

Let the second number be 2

The absolute value of 2 is | 2 | = 2

Let the third number be 3

The absolute value of 3 is | 3 | = 3

Let the fourth number be 4

The absolute value of 4 is | 4 | = 4

And , | 1 | < | 2 | < | 3 | < | 4 |

Hence , the set of integers are { 1 , 2 , 3 , 4 }

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The domain and range of the piecewise function is;

B) Domain: x < 2 or x ≥ 3.; range: y > -3

The piecewise function is given as;

f(x) = {-¹/₂x - 2, if x < 2

        {2x - 4, if x ≥ 3

The domain of a function is the set of all possible input values of the function and as such in this case the domain is x < 2 or x ≥ 3.

The range is defined as the set of all possible output values of the function and in this case, we have;

For -¹/₂x - 2; If x < 2

At x = 2; -¹/₂(2) - 2 = -3

Range is; y > -3

Similarly, for 2x - 4, if x ≥ 3

At x = 3; 2(3) - 4 = 2

Range is; y ≥ 2

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The derivate of function f(x)=(5x⁶+4x³)⁴is F'(x) = 4(30x³+12x²)(5x⁶+4x³)³

The derivative of a function f(x) represents its rate of change and is denoted by f'(x) or df/dx.

The derivative Formula is given as:

f1 (x)=lim△x→0 f(x+△x) − f(x)△x.

Take the derivative: F'(x) = [tex]\frac{d}{dx} (5x^6+4x^3)^5[/tex]

Substitute the inner function by (5x⁶+4x³)

Apply the chain rule:   [tex]\frac{d}{dx} (g)^4*\frac{d}{dx} (5x^6+4x^3)^4[/tex]

Calculate the derivative: [tex]4g^3*\frac{d}{dx} (5x^6+4x^3)^4[/tex]

Substitute back: [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]

Result after applying the chain rule: F'(x) = [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]

Calculate the derivative of the sum or difference:

F'(x) =[tex]4(5x^6+4x^3)^3*(\frac{d}{dx} (5x^6))+\frac{d}{dx} (4x^3))[/tex]

Take out the coefficients F'(x) = [tex]4(5x^6+4x^3)^3*(5*\frac{d}{dx} (x^6)+4\frac{d}{dx}(x^3))[/tex]

Apply the power rule: F'(x) = 4(5x⁶+4x³)³*(5*6x³+4*3x²)

Multiply the monomials: F'(x) =  4(5x⁶+4x³)³*(30x³+12x²)

Rewrite the expression: F'(x) = 4(30x³+12x²)*(5x⁶+4x³)³

So, the answer is F'(x) =4(30x³+12x²)*(5x⁶+4x³)³

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Answer:

a)  (-∞, -1) ∪ (-1, ∞)

b)  (-∞, -4) ∪ (-4, ∞)

c)  Vertical asymptote: x = -1

    Horizontal asymptote:  y = -4

Step-by-step explanation:

Given function:

[tex]f(x)=\dfrac{1}{x+1}-4[/tex]

The domain of a function is the set of all possible input values (x-values).

When the denominator of a rational function is zero, the function is undefined.  

Rewrite the function as one fraction:

[tex]\implies f(x)=\dfrac{1}{x+1}-\dfrac{4(x+1)}{x+1}[/tex]

[tex]\implies f(x)=\dfrac{1-4(x+1)}{x+1}[/tex]

[tex]\implies f(x)=\dfrac{-4x-3}{x+1}[/tex]

Set the denominator to zero and solve for x:

[tex]\implies x+1=0[/tex]

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

Therefore, the given function is undefined when x = -1, so its domain in interval notation is:

The range of a function is the set of all possible output values (y-values).

As the domain is restricted to (-∞, -1) ∪ (-1, ∞), the range is also restricted.

To find the range of a rational function, first solve the equation for x:

[tex]\implies y=\dfrac{1}{x+1}-4[/tex]

[tex]\implies y+4=\dfrac{1}{x+1}[/tex]

[tex]\implies (y+4)(x+1)=1[/tex]

[tex]\implies x+1=\dfrac{1}{y+4}[/tex]

[tex]\implies x=\dfrac{1}{y+4}-1[/tex]

[tex]\implies x=\dfrac{1-(y+4)}{y+4}[/tex]

Set the denominator of the resultant equation ≠ 0 and solve for y:

[tex]\implies y+4 \neq 0[/tex]

[tex]\implies y \neq -4[/tex]

Therefore, the range is the set of all real numbers other than y = -4:

A vertical asymptote occurs at the x-value(s) that make the denominator of a rational function zero.

Therefore, there is a vertical asymptote at x = -1.

As the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the result of dividing the highest degree term of the numerator by the highest degree term of the denominator.

[tex]\implies f(x)=\dfrac{-4x-3}{x+1}[/tex]

Therefore, there is a horizontal asymptote at:

There are no slant asymptotes as there is a horizontal asymptote.