Following example 1, to find the area of an isosceles triangle with height 4 cm and width 8 cm, we could use horizontal slices:
a slice at height ℎ has width =
so that we may find the area by integratingDu(answer) ????ℎdh, with
a=
????=

The probability of 30-39 age group is 18.83%. The probability of a given age group is the frequency of that group divided by the total frequency.

To find the probability of the age group 30-39, divide the frequency of this group (7845) by the total frequency (9831 + 7845 + 6869 + 6323 + 5410 + 5279 = 41,478). So the probability of 30-39 is 7845/41478 = 0.19.

In probability distributions, probabilities are calculated as the ratio of the number of occurrences of an event to the total number of possible events. To find the probability of an event in a specific age group, we need to calculate the ratio of the frequency of that age group to the total frequency of all age groups.

In this case, the frequency of the age group 30-39 is 7845. The total frequency of all age groups is 9831 + 7845 + 6869 + 6323 + 5410 + 5279 = 41,657.

So, the probability of the age group 30-39 is 7845/41657 = 0.1883 or 18.83%.

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Answer:

9.5

Step-by-step explanation:

[tex]1. \ \frac{AB}{PQ} =\frac{BM}{QN} ; = >[/tex]

[tex]2. \ \ \frac{11}{4.4} =\frac{BM}{3.8} ; \ = > \ BM=\frac{11*3.8}{4.4} =9.5.[/tex]

The multiplication of the expression given is 60x² - 28x - 8.

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

The given expression is (5x + 1) (12x - 8).

We have to multiply the expressions and simplify.

For any numbers or algebraic expressions, a, b, c and d,

(a + b) (c + d) = ac + ad + bc + bd

(5x + 1) (12x - 8) = (5x × 12x) + (1 × 12x) + (5x × -8) + (1 × -8)

                         = 60x² + 12x - 40x - 8

                         = 60x² - 28x - 8

Hence the multiplication of the given expression gives 60x² - 28x - 8.

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2528.5 centimeters is approximately 0.0253 kilometers.

To convert centimeters to kilometers, we first need to convert centimeters to meters, then meters to kilometers.

1 kilometer = 1000 meters

1 meter = 100 centimeters

So, to convert 2528.5 centimeters to meters, we divide by 100:

2528.5 cm / 100 = 25.285 meters

And to convert meters to kilometers, we divide by 1000:

25.285 m / 1000 = 0.025285 km

Therefore, 2528.5 centimeters is approximately 0.0253 kilometers.

The answer can be rounded to 4 significant figures, which is the number of significant figures in the original measurement of 2528.5 centimeters:

0.0253 km (rounded to 4 significant figures)

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The complete factorization of the given expression is 2x³ - 36x² + 162x = 2x (x - 9)².

An algebraic expression is factorised when it is written as the product of its factors. These variables, factors, or algebraic expressions could be present. The breaking or decomposition of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorization or factoring in mathematics.

The steps to factorize the given expression is:

2x³ - 36x² + 162x

= 2x (x² - 18x + 81) Factor out 2x.

= 2x (x² - 2(9)(x) + (9)²) Rewrite the trinomial.

= 2x (x - 9) (x - 9) Use the whole square formula.

= 2x (x - 9)² simplify.

Hence, the complete factorization of 2x³ - 36x² + 162x = 2x (x - 9)².

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The number of longboards that will be defective based on the defective longboards in the month can be predicted to be 2, 500 longboards.

The number of longboards that would be defective for the other skateboarding company is 2, 446 longboards.

To find the defective longboards, first find the percentage of the longboards that were produced in the month. Then multiply this result with the number of longboards to be produced :

= 204 / 10, 200 x 125, 000

= 2 % x 125, 000

= 2, 500 longboards

The same procedure can be used for the other skateboarding company to find the defective products to be:

= 225 / 11, 500 x 125, 000

= 2, 446 longboards

The product for the second company therefore has fewer defects.

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Because in the first expression 4 was factored out to make it 4(2y-6). Before 4 was factored out the expressions were the same.

The largest force that can be supported before any member ruptures is 8, the maximum force that bar BC can support.

To solve this problem, we need to find the maximum force (stress) that each bar can withstand before rupturing.

This can be found using the stress-strain diagram.

For bar AB with a cross sectional area of 2.3 in^2, the maximum stress that it can withstand is given by the maximum slope of the stress-strain diagram, which is approximately 1.

For bar BC with a cross sectional area of 4 in^2, the maximum stress that it can withstand is given by the maximum slope of the stress-strain diagram, which is approximately 2.

Since the force (stress) is equal to the stress times the cross sectional area, the largest force that bar AB can support is given by 1 * 2.3 = 2.3 and the largest force that bar BC can support is given by 2 * 4 = 8.

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The dimensions of the rug are: width is 15 feet length is 21 feet.

Let's call the width of the strip of floor around the rug "x". Then the width of the rug would be 25 - 2x, and the length of the rug would be 34 - 2x.

The total area of the rug should not exceed 360 square feet, so we can set up an equation to represent this:

(25 - 2x)(34 - 2x) = 360

Expanding and simplifying the left-hand side:

25 × 34 - 2 × 25 × x - 2 × 34 × x + 4 × x² = 360

850 - 118x + 4x² = 360

4x² - 118x+ 490 =0

Now we have a quadratic equation that we can solve using the quadratic formula or by factoring. Either way, we find that x = 24.5 or x = 5. Since x represents the width of the strip of flooring, x must be positive. Therefore, x = 5 is the solution.

So the width of the strip of flooring is 5 feet, and the dimensions of the rug are:

width: 25 - 2 × 5 = 15 feet

length: 34 - 2 × 5 = 21 feet

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Answer:

Step-by-step explanation:

3 (w - 2) ≥ 10

3w - 6 ≥ 10

3w ≥ 16

w ≥ 16/3

The speed of Usain Bolt is given by the equation S = 23.25 miles per hour

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 equation be represented as S

Now , the value of S is

The speed at which Usain Bolt won the gold medal = 34.1 feet per second

So , the speed at which Usain Bolt won the gold medal in miles per hour is calculated by dividing the value by 1.467

Substituting the values in the equation , we get

The speed of Usain Bolt S = ( speed of Usain / 1.467 ) miles per hour

On simplifying the equation , we get

The speed of Usain Bolt S = 34.1 / 1.467

The speed of Usain Bolt S = 23.25 miles per hour

Therefore , the value of S is 23.25 mph

Hence , the equation is S = 23.25 miles per hour

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Answer:

last option

number of students in class plus new students

The correct options are a) ∠ R corresponds to ∠ P'QQ'  and d) side RQ corresponds to QQ'

Dilation means changing the size of an object without changing its shape. The size of the object may be increased or decreased based on the scale factor.

Given that, a triangle PQR is being dilated to form a new triangle P'Q'Q following a rule: (x, y) → (2x, 2y)

This show that, the scale factor is 2 and the dilation is an enlargement,

We know that, dilation transformation makes similar figures, and in similar figures, the corresponding angles and ratio of corresponding sides are equal,

Here, the corresponding angles are:

∠ P = ∠ P'

∠ PQR = ∠ P'Q'Q

∠ R = ∠ P'QQ'

And, the corresponding side:

P'Q'/PQ = P'Q/PR = Q'Q/QR

Therefore, true statements are;

a) ∠ R corresponds to ∠ P'QQ'  

d) side RQ corresponds to QQ'

Hence, the correct options are a) ∠ R corresponds to ∠ P'QQ'  and d) side RQ corresponds to QQ'

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When 2, 325 is rounded to the nearest thousand, the result would be 2, 000.

To round a number to the nearest thousand, identify the place value of the rounding digit, which is the thousandth place (the third digit from the right).

Then, look at the digit to the right of the rounding digit. If this digit is 5 or greater, then round up the rounding digit by adding one to it. If the digit is less than 5, keep the rounding digit as it is.

The number 2, 325 would therefore be rounded to 2, 000 because the digit " 3 " is less than 5.

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The average gas mileage for new vehicles sold in Switzerland was the greatest in the year 2013.

The equation models the average gas mileage of new vehicles sold in Switzerland as a parabolic function, y=-0.2(x-5)²+35.

To find the year in which the average gas mileage was the greatest, we need to find the vertex of this parabolic function.

The x-coordinate of the vertex is given by x = 5 + (-b/2a) = 5 - (-0.2/2(-0.2)) = 5 - (0.2/0.4) = 5 - 0.5 = 4.5 years since 2010, or the year 2013. The y-coordinate of the vertex can be found by substituting x = 4.5 years into the equation, y = -0.2(4.5-5)² + 35 = -0.2(0.5)² + 35 = 0.2 * 0.25 + 35 = 0.05 + 35 = 35.05 mpg.

So the average gas mileage for new vehicles sold in Switzerland was greatest in the year 2013, with a value of 35.05 mpg.

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Answer:

Option A) The volume is multiplied by 1/8.

Step-by-step explanation:

For the first volume problem involving the rectangular pyramid, the normal volume formula is V = 1/3 (l)(w)(h).  Since all 3 dimensions are being multiplied by 1/2, this involves multiplying the original formula by (1/2)(1/2)(1/2) which, put together, is multiplying by 1/8, so the volume will be 1/8 of what it was with the original dimensions.

The volume of the rectangular pyramid is multiplied by ( 1/8 )

The volume of the rectangle is given by the product of the length of the rectangle and the width of the rectangle and the height of the rectangle

Volume of Rectangle = Length x Width x Height

Volume of Rectangle = Area of Rectangle x Height

Given data ,

Let the length of the rectangle be L

Let the width of the rectangle be W

Let the height of the rectangle be H

Now , the volume of rectangle be V = L x W x H

And , when the length , width and height are multiplied by ( 1/2 ) , we get

L' = L/2

W' = W/2

H' = H/2

So , the volume V' = ( L' x W' x H' )

The volume of rectangular pyramid V' = ( 1/8 ) V

Hence , the volume is multiplied by ( 1/8 )

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The Taylor expansion of cos(x) around x=0 is [tex]cos(x) = 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) +[/tex]......

We would use this expansion to calculate the cosine of a given angle with an error tolerance of 0.1%.

To do this, we must first write a function taylorcosine() which takes two arguments: the angle (in radians) and the relative percentage error. The function should then return two values: the Taylor approximation for the cosine of the angle, and the number of terms required in the Taylor series to reach this level of error.

For example, if we want to find the approximate value of cos(3π) with an error tolerance of 0.1%, we would call the function as follows:

[tex]val, nsteps = taylorcosine(3π, 1e-3)[/tex]

To find the percentage relative error, we must multiply the relative error by 100.

We can also use the bisection method to determine the roots of a function. To do this, we must write a function myBisection() which takes

three arguments: the function handle, an array containing the lower and upper limits of the starting interval, and the maximum allowable relative error. The return value of the function is the estimate of the root.

For example, if we want to find the root of f(x) = x^2 - 3x + 2, with an initial bracket of [1.3, 0] and an error tolerance of 1%, we would call the function as follows:

root = [tex]myBisection(f, [1.3, 0], 1%)[/tex]

The error is defined as the absolute value of the ratio of the difference between the old and the new estimates of the root and the new estimate of the root.

Error = [tex]|x_new/x_new - x_old|/x_new[/tex]

By writing the taylorcosine() and myBisection() functions, we can calculate the cosine of an angle with a given error tolerance, and determine the roots of a function with a given error tolerance, respectively.

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Yes, that is correct. A system of linear equations with fewer equations than unknowns is often referred to as an underdetermined system.

In this type of system, there are infinitely many solutions or no solutions, and it is not possible to determine a unique solution for the variables.

When a system of equations has more variables than the equation, therefore, it has fewer rules to curb the variables, giving the variables more freedom to take up values from their respective domain(Usually it is a set of real numbers), in such a condition the system of equations has infinite solutions. This kind of system is known as an underdetermined system.

Hence, the system of equations doesn't have a unique solution.

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The circumference of the circle is 2 * pi * 9 = 56.5 feet. So, the degree measure is (6/56.5) x 360 = 51.1 degrees. Rounded to two decimal places, the degree measure is 51.1 degrees.

To find the degree measure of an angle that subtends an arc of length 6 feet on a circle of radius 9 feet, use the formula: degree measure = (arc length / circumference) x 360 degrees.

The degree measure of the angle that would subtend an arc of length 6 feet on a circle of radius 9 feet can be found by using the formula:

degree measure = (arc length / circumference) * 360

The circumference of the circle is 2 * pi * 9 = 18 * pi

So, degree measure = (6 / (18 * pi)) * 360 = (6 / 18) * (360 / pi) = 60 / pi

To convert this to degrees, we can use an approximation of pi as 3.14, so

degree measure = 60 / 3.14 = 19.11

Rounding to two decimal places, the degree measure is 19.11, which can be rounded to 19.11 degrees.

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(i) the coordinates of the vertex S = Q + QS = (-1, -1) + (2, 2) = (1, 1).

(ii) the coordinates of the point of intersection of the diagonals PR and QQS is point (-1, 0).

(i) The coordinates of the vertex S can be found by shifting the vector PQ to QS. The vector PQ is given by Q - P = (-1 - (-3), -1 - (-1)) = (2, 0). The vector QS can be found by shifting PQ by 2 units vertically, so QS = (2, 2). To find the coordinates of S, we add the vector QS to the point Q: S = Q + QS = (-1, -1) + (2, 2) = (1, 1).

(ii) The point of intersection of the diagonals PR and QS can be found by finding the midpoint of each diagonal and checking if they are equal. The midpoint of PR is given by ( (1 + -3)/2 , (1 + -1)/2 ) = (-1, 0). The midpoint of QS is given by ( (1 + -1)/2 , (1 + 1)/2 ) = (0, 1). Since the midpoints are equal, the diagonals PR and QS intersect at the point (-1, 0).

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Answer: 1:2.5

Step-by-step explanation:

Step by step explanation:

12:30

÷ by 12

= 1:2.5

The correct option is : The price of Juan’s bond will decrease. Because similar bonds are offered a year later with a 3% coupon rate.

For the stated question:

Then,  the price of Juan’s bond will decrease.

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The solution is (FoF) (0) = -8, when f(x) = 3x-2.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

f(x) = 3x-2

i.e. fof(x)

 =3(3x-2)-2

=9x-8

so. fof(0)

   =9*0-8

   = -8

Hence, The solution is (FoF) (0) = -8, when f(x) = 3x-2.

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The probability of finding a particle in a one-dimensional box of length a between 0 and a/2 is[tex](2/a)^3/3[/tex].

The probability of finding a particle in a one-dimensional box of length a between 0 and a/2 can be calculated using the equation [tex]P = (1/h)^3 * (∫0^a/2Ψ^2dx)[/tex]. The equation is derived from the equation of probability density of finding a particle in a given state which is given as [tex]P = (1/h)^3 * ∫Ψ^2dx[/tex]. In this equation, h is the Planck's constant, and Ψ is the wave function of the particle.

For a particle in a one-dimensional box of length a, the wavefunction can be written as [tex]Ψ(x) = (2/a)^1/2 * sin (nπx/a)[/tex]. Substituting this in the equation for probability density, the probability of finding a particle between 0 and a/2 is [tex]P = (1/h)^3 * (2/a)^3 * (1/3 - 0)[/tex]. Therefore, the probability of finding a particle in a one-dimensional box of length a between 0 and a/2 is [tex](2/a)^3/3[/tex].

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Note that the rates that are unit rates are given as follows:

Note that a unit rate is a rate in which the second term equals 1. It represents a comparison of a quantity to 1.

Since:

Therefore, it is right or correct to state that 2/4,  9/1, and 10:1 are unit rates.

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The statement is false, as it is possible for three vectors to not span Rn but still lie on the same plane, such as the three vectors (1,1,1), (2,2,2), and (3,3,3), which all lie on a plane that passes through the origin.

(a) In order for three vectors un, U2, U3 to span Rn, all possible values of n must be 3 or greater. This is because three vectors can only span a 3-dimensional space or higher. Therefore, any n greater than or equal to 3 will work.

(b) Since {u, uy, u3} is a spanning set for Rn, then {u, u, + u2, u, -u3} is also a spanning set for Rn. This is because adding or subtracting any of the vectors in the original set will still form a set of three vectors that span Rn.

(c) This statement is always true. If three vectors ui, uy, u3 are linearly independent, then any linear combination of those three vectors (such as u,, u, + u2, u, -u3) will also be linearly independent.

(d) This statement is also always true. If three vectors ui, uy, u3 are linearly independent, then any linear combination of those three vectors (such as u,, u, + u2, u, -u3) will also be linearly independent.

(e) This statement is false. It is possible for three vectors ui, uy, u3 to not span Rn but still lie on the same plane. For example, the three vectors (1,1,1), (2,2,2), and (3,3,3) do not span R3, but they all lie on the same plane that passes through the origin (x+y+z=0).

The statement is false, as it is possible for three vectors to not span Rn but still lie on the same plane, such as the three vectors (1,1,1), (2,2,2), and (3,3,3), which all lie on a plane that passes through the origin.

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The numerical value of x in the similar pair of triangle is 10.

A ratio is simply the relation between two amounts showing how many times a value is contained within another value.

From the diagram,

Side A of the small triangle = 10

Side B of the small triangle = x + 2

Side A of the Big triangle = 10 + x

Side B of the big triangle = x + 14

Hence;

Ratio of Side A to Side B of the small triangle = 10 / x + 2

Ratio of Side A to Side B of the Big triangle = 10 + x / x + 14

Since the triangle area similar, equate the two ratio and solve for x.

10 / ( x + 2 ) = ( 10 + x ) / ( x + 14 )

Cross multiply

10( x + 14 ) = ( 10 + x )( x + 2 )

Expand using FOIL method

10x + 140 = 10x + 20 + x² + 2x

10x + 140 = x² + 12x + 20

x² + 12x + 20 = 10x + 140

x² + 12x + 20 - 10x - 140 = 0

x² + 2x - 120 = 0

Factor the left side of the equation

( x - 10 )( x + 12 ) = 0

Hence;

x - 10 = 0 and x + 12 = 0

x = 10 and x = -12

Since the dimension of the triangle cannot be negative,

x = 10

Therefore, the value of x is 10.

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Answer:

a: 1/8 and 1/4. b: 1/2 1/4. c: 1/1 and 5/12. d: 1 1/2 and 3/4

Step-by-step explanation:

You had to find 2 fractions that add up to a number between the two numbers listed.

The statement 8x 8y 8z; (z < 0 ^ x < y) ) yz > xz is true, since the statement x < y and z < 0 both hold true, and thus yz > xz must also be true. The negation of this statement would be 8x 8y 8z; (z ≥ 0 ∨ x ≥ y) ) yz ≤ xz.

The first-order language of (directed) graphs is L = {r}, where r is a binary relation symbol. The only terms are the variables x , y, z. Atomic formulas have the form (x, y, z). These formulas represent relationships between the terms, such as x being adjacent to y and z.

These relationships can be used to model different types of graphs. For example, to model a directed graph, the binary relation x->y can be used to represent a directed edge from x to y. These formulas can be combined and manipulated to express more complex formulas, such as the perpendicularity relation between two lines or the perpendicular bisector theorem. These formulas allow for solving problems related to geometry and algebra in graphs.

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