3.
The box-and-whisker plot shown below represents
600 scores on a district geometry te

By trigonometric formulas, there are two solution sets for the trigonometric equation 2 · cos β = - √(3 · cos β) + sin β + 2 · cos β: β₁ ≈ 0.402π + 2π · k, β₂ = - 0.402π - 2π · k, where k is an integer.

In this problem we find the case of a trigonometric equation, whose roots must be found by algebra properties and trigonometric formulas. First, write the complete expression:

2 · cos β = - √(3 · cos β) + sin β + 2 · cos β

Second, simplify the expression by algebra properties:

√(3 · cos β) = sin β

Third, square both sides and simplify the expression:

3 · cos β = sin² β

3 · cos β = 1 - cos² β

cos² β + 3 · cos β - 1

Fourth, factor the resulting expression:

(cos β - 0.303) · (cos β + 3.303)

Fifth, determine the set of solutions:

cos β = 0.303

β₁ ≈ 0.402π + 2π · k, β₂ = - 0.402π - 2π · k, where k is an integer.

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Answer:

Step-by-step explanation:

Area of rectangle with sides a and b is ab.

We have one side x and area 100 m².

Therefore the second side is:

The perimeter is:

Proved

Given:

Area of rectangular pen = 100 m²

Lenth of one side of pen (a) = x m

To prove:

Perimeter (P) = 2x + 200/x

Least perimeter = 40 m

Solution:

Area of rectangle = ab

100 = x. b

b = 100/x

Perimeter= 2(a +b)

P = 2a + 2b

P = 2x + 2× 100/x

P = 2x + 200/x

To prove the least perimeter differentiate the perimeter P w.r.t. x,

dp/dx = 2 - 200/x²

Now equate the above function with zero,

2-200/x² = 0

200/x² = 2

x² = 100

x = ± 10

x = -10 is not valid as length can not be negative.

substitute x = 10, in parent function

P = 2x + 200/x

P = 2×10 + 200/10 = 20 + 20 = 40

Hence proved

P (Least perimeter) = 40

The simplified expression is 14+a.

Using the commutative and associative properties, we can simplify the expression (7+a)+7.

First, let's use the commutative property to rearrange the terms. The commutative property states that the order of addition or multiplication does not matter. In other words, a+b = b+a and a*b = b*a.

So, we can rearrange the terms in the expression to get:

(7+7)+a

Next, let's use the associative property to simplify the expression. The associative property states that the way we group terms in an addition or multiplication problem does not matter. In other words, (a+b)+c = a+(b+c) and (a*b)*c = a*(b*c).

So, we can group the terms in the expression to get:

14+a

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∫[infinity]0e^(−x^2/a)dx is 1/2√π/a, rounded to five decimal places is 0.38539.

To evaluate the integral ∫[infinity]0e^(−x^2/a)dx, given that ∫[infinity]0e^(−x^2)dx=1/2√π and a = 2.1, we use the substitution u = x^2/a. We then have du = 2x/a dx and the limits of the integral change to 0 to ∞/a. So, the integral becomes:

∫[∞/a]0 e^(-u) du

By the given information, we know that ∫[∞/a]0 e^(-u) du = 1/2√π/a.

Therefore, the answer to the integral ∫[infinity]0e^(−x^2/a)dx is 1/2√π/a, rounded to five decimal places is 0.38539.

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The length of the arc intercepted by a central angle of 6 radians in a circle of radius 53 is:

Arc length = radius x central angle

Arc length = 53 x 6

Arc length = 318

Therefore, the length of the arc intercepted by a central angle of 6 radians in a circle of radius 53 is 318.

The length of the arc intercepted by a central angle of 97° in a circle of radius 8 is:

First, we need to convert 97° to radians:

97° = (97/180) x π radians

97° = 1.69 radians (rounded to two decimal places)

Arc length = radius x central angle

Arc length = 8 x 1.69

Arc length ≈ 13.52 (rounded to two decimal places)

Therefore, the length of the arc intercepted by a central angle of 97° in a circle of radius 8 is approximately 13.52.

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The line of reflection that produces Z′(10, 7) is y - axis.

A triangle is a two - dimensional figure with three sides and three angles.

The sum of the angles of the triangle is equal to 180 degrees.

∠A + ∠B + ∠C = 180°

Given is a Triangle XYZ is drawn with vertices X(−2, 4), Y(−9, 3), Z(−10, 7). Determine the line of reflection that produces Z′(10, 7).

The rule for a reflection over the y -axis is (x, y) → (−x, y). We can write the reflection as -

Z(- 10, 7) → Z'(10, 7)

Therefore, the line of reflection that produces Z′(10, 7) is y - axis.

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The line of reflection that produces Z′(10, 7) is y - axis.

What is a triangle?

A triangle is a two - dimensional figure with three sides and three angles.

The sum of the angles of the triangle is equal to 180 degrees.

∠A + ∠B + ∠C = 180°

Given is a Triangle XYZ is drawn with vertices X(−2, 4), Y(−9, 3), Z(−10, 7). Determine the line of reflection that produces Z′(10, 7).

The rule for a reflection over the y -axis is (x, y) → (−x, y). We can write the reflection as -

Z(- 10, 7) → Z'(10, 7)

Therefore, the line of reflection that produces Z′(10, 7) is y - axis.

After the multiplication of the given decimal numbers and their power, we got the value as 105,000.

One of the number types in algebra that has a whole integer and a fractional portion separated by a decimal point is a decimal. The decimal point is the dot that appears between the parts of a whole number and a fraction. Between integers, decimal numbers are used to express the numerical value of complete and partially whole quantities.

Simply change the decimal place to the right by the number of 0s in the power of 10 when multiplying a decimal by a power of 10. Simply change the decimal place to the left by the number of 0s in the power of 10 when dividing a decimal by a power of 10.

We are asked to solve the question:

      (2.5 X 10⁻²)(4.2 X 10⁶)

Now we have to solve this following the PEDMAS rule.

So we have to solve the parentheses first.

2.5 * 10⁻²

Here the exponent has negative power.

10⁻²  = 1/10² = 1/100

Then,

2.5 * 10⁻²  = 2.5 * 1/100 = 0.0025

This is because when the number is divided by the power of 10, the decimal point moves to the left by the value of the power of 10.

Here the power is 2. So it moved two decimal places to the left.

4.2 * 10⁶

Here the decimal number is multiplied by the power of 10. So the decimal point moves  6 places to the right.

4.2 * 10⁶  = 4200000

Now after solving parentheses, we get

0.0025 * 4200000 = 25 * 10⁻⁴ * 4200000 = 105,000,000 * 10⁻⁴

                                                              = 105,000

Therefore after the multiplication of the given decimal numbers and their power, we got the value as 105,000.

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The linear dependence between the vectors is -3[10 02] + 1[21 3−1] + 2[11 3−3] = 0.

The linear dependence between the following vectors in Mat 2×2 (R) [10 02],[21 3−1],[11 3−3] can be found by solving the equation a[10 02] + b[21 3−1] + c[11 3−3] = 0 for a, b, and c. This gives us the following system of equations:10a + 21b + 11c = 02a + 3b + 3c = 02b - c = 0Solving this system of equations, we get a = -3, b = 1, and c = 2. Therefore, the linear dependence between the vectors is -3[10 02] + 1[21 3−1] + 2[11 3−3] = 0. This means that the vectors are linearly dependent and can be written as a linear combination of each other.

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Diameter of flu virus particle is 8.1 × 10⁻⁸ meter

Scientific notation is a way of representing numbers that are too large or too small to be easily written in decimal form because they require very long strings of digits to  be written. This is sometimes called scientific or standard exponential format, or British standard format.

Given,

Diameter of water molecule = 2.7 × 10⁻¹⁰ meters

Flu virus particle is 300 times bigger than water molecule

Diameter of flu virus

= 300 × 2.7 × 10⁻¹⁰

= 810 × 10⁻¹⁰

= 8.1 × 10⁻⁸ meter

Hence, 8.1 × 10⁻⁸ meter is the diameter of the Flue virus particle.

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An example of a compound interest problem and the solution problem is:

"Sarah deposits $10,000 into a savings account with a 5% annual interest rate compounded annually. How much will she have in the account after 3 years?"

Solution: Sarah will have $11,576.25 in the account after 3 years.

The compound interest on the same sum at the same rate for the same period, compounded annually, is:

$6,646.34

Sarah deposits $10,000 into a savings account with a 5% annual interest rate compounded annually. How much will she have in the account after 3 years?

To solve this problem, we can use the formula for compound interest: A = P(1 + r)^n, where A is the final amount, P is the principal amount, r is the annual interest rate, and n is the number of years.

To find the compound interest on the same sum at the same rate for the same period, we can use the same formula for compound interest: A = P(1 + r)^n.

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Answer:

V=103.

Step-by-step explanation:

V=πr2h

=π·2.82

·4.2≈103

.44636

Answer:

Step-by-step explanation:

the correct answer is 10.39 cubic feet

Bob was on the top rung of his ladder (rung 30) and Roy was on the 15th rung of his ladder when they were at the same height.

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

Let's assume that Bob started at the top of his ladder and Roy started at the bottom of his ladder.

If both ladders have 30 rungs, then they have a total height of 30 - 1 = 29 rungs.

If Bob went down two rungs every second and Roy went up one rung every second, then they were changing their height at a rate of -2

Let's use "t" to represent the time they had been climbing. Then, we can write an equation to represent the heights of Bob and Roy at this moment:

Bob's height = 30 - 2t

Roy's height = t

30 - 2t = t

Solving for "t", we get:

t = 15

This means that Bob and Roy were at the same height after 15 seconds of climbing.

Bob's height = 30 - 2t = 30 - 2(15) = 0

Roy's height = t = 15

Therefore, Bob was on the top rung of his ladder (rung 30) and Roy was on the 15th rung of his ladder when they were at the same height.

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Answer:

Step-by-step explanation:

Diameter is the width of the cup and 7 is the height.

Process of elimination rules out A as it is too big.

The base of the cup is a circle.

  Formula for area of a circle is Pi (3.14) times the radius squared.

1. Find radius

Diameter is 2x the radius, so if we have a 4.4 diameter, the radius will be 2.2.

2. Square radius

2.2x2.2=

4.84

3. Times Pi

Multiply 4.84 by 3.14 (Pi) to get

15.1976

15.1976 is the area of the base. Now we multiply that by 7!

15.1976 x 7=

This is our answer, but we need to round it to the nearest hundredth, the question tells us.

106.38 is the final answer.

Answer:

x=18

Step-by-step explanation:

Being that the angles are corresponding they are going to equal the same thing.

(7x+7)=133

Now just follow the laws of PEMDAS

7x+7=133

-7 -7

7x=126

÷7 ÷7

x=18

The distance between Dallas, TX and Austin, TX is 312 kilometers.

To convert miles to kilometers, we need to multiply the distance in miles by the conversion factor, which is 1.6.

So, to find the distance between Dallas, TX and Austin, TX in kilometers, we need to multiply 195 miles by 1.6:

Distance in kilometers = 195 miles * 1.6 = 312 kilometers

Therefore, the distance between Dallas, TX and Austin, TX is 312 kilometers.

To understand how this conversion works, we need to understand what a mile and a kilometer represent. A mile is a unit of length that is used primarily in the United States and is defined as 5,280 feet. A kilometer, on the other hand, is a unit of length that is used in most other countries and is defined as 1,000 meters.

Since the conversion factor between miles and kilometers is 1.6, it means that for every 1 mile, there are 1.6 kilometers. This conversion factor is derived from the fact that 1 mile is approximately equal to 1.60934 kilometers.

Therefore, to convert any distance in miles to kilometers, we need to multiply the distance in miles by 1.6. Conversely, to convert a distance in kilometers to miles, we need to divide the distance in kilometers by 1.6.

Therefore, the distance between Dallas, TX and Austin, TX is 312 kilometers.

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Yes, the two points (-2,-3) and (2,-1) belong to the same linear function.

To determine the function, we can use the slope-intercept form of an linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.

First, we need to find the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Plugging in the values of the given points, we get:

m = (-1 - (-3)) / (2 - (-2)) = 2 / 4 = 1/2

Now, we can use one of the given points and the slope to find the y-intercept. Let's use the point (2,-1) and plug in the values into the equation y = mx + b:

-1 = (1/2)(2) + b

Solving for b, we get:

b = -1 - 1 = -2

Therefore, the equation of the linear function is y = (1/2)x - 2.

So, the two points (-2,-3) and (2,-1) belong to the same function, which is y = (1/2)x - 2.

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The value of 8#(8#8) is (A) 1/2

An operation \# for positive real numbers is defined by the equation a#b=ab/a+b. The value of 8#(8#8) is (A) 1/2. To solve, use the equation given:

8#(8#8) = (8*8)/(8+8)

= 64/16

= 1/2

Hope this helps!

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The number of laps increases by 50 percent.

The number of laps Jane can run has increased by 50%. To find this, we can use the formula:

Percentage Increase = (New Value - Old Value) / (Old Value) × 100

In this case, the new value is 6 (the number of laps Jane can now run) and the old value is 4 (the number of laps Jane used to be able to run). Plugging these values into the formula gives us:

Percentage Increase = (6 - 4) / (4) × 100
Percentage Increase = 2 / 4 × 100
Percentage Increase = 0.5 × 100
Percentage Increase = 50%

Therefore, the number of laps Jane can run has increased by 50%.

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Answer:

a) x = 132

b) x = 125

Step-by-step explanation:

Angles in a quadrilateral sum to 360 degrees. Given this, we can determine the missing angles by subtracting the known angles from 360:

a) 360 - 48 - 48 - 132 = 132

b) 360 - 108 - 68 - 59 = 125

Answer:

slope is undefined

Step-by-step explanation:

We have 2 coordinates (-4,0) and (-4,1)

Slope m = (y2 - y1)/(x2 - x1) = (1 - 0)/(-4 - -4) = 1/0 or undefined

The undefined slope is the slope of a vertical line

-6x+25y+66z=232 is the equation of the plane and distance between the point D and the plane is  157/70.

lets find the equation of the line AB and BC and equations of the lines will be of the form

[tex]  \text{ $\frac{x-a}{l}$ = $\frac{y-b}{m}$ = $\frac{z-c}{n}$ } [/tex]

So, the equation of AB is

[tex]  \text{ $\frac{x}{2}$ = $\frac{y-4}{18}$ = $\frac{z-2}{7}$ } [/tex]

So, the equation of AB is

[tex]  \text{ $\frac{x-3}{-1}$ = $\frac{y+2}{24}$ = $\frac{z}{9}$ } [/tex]

let b1 andb2 are the direction vectors of the two above lines.

b1= 2i + 18j + 7k

b2= -1i + 24j + 9k

now n= Det( i       j       k)

                   2      18     7

                  -1       24    9

or, n= -6i + 25j + 66k

the point (0,4,2) lies on the plane. the equation of the plane passing through (0,4,2) and perpendicular to a line with a direction ratio (-6, 25, 66) is

-6x+25(y-4)+66(z-2)=0

or, -6x+25y+66z=232.

now formula of distance between the point (x0,y0,z0) and the plane is

[tex] \frac{Ax0+By0+Cz0+D}{√A^2+B^2+C^2} [/tex]

so for the point D and the plane is

( -6*0 + 25*1 + 66*2 - 132)/√5017 = 157/70.

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(a) Stratified random sampling ensures a representative sample, and Ys is an unbiased estimate of the population mean y.

(b) Proportional allocation allocates sample size in each stratum proportionally to its size, while optimum allocation determines sample size based on variance and cost in stratified random sampling.

(a) Stratified random sampling is a method of sampling that involves dividing the population into smaller groups or strata, and then randomly selecting a sample from each stratum. This is done to ensure that the sample is representative of the population as a whole. The formula for the expected value of the sample mean, E(Ys), in stratified random sampling is:
E(Ys) = Σ wi * E(Yi)
where wi is the weight of the ith stratum, and E(Yi) is the expected value of the sample mean in the ith stratum.
Since Ys is an unbiased estimate of the population mean y, we can prove that E(Ys) = y by substituting y for E(Yi) in the formula:
E(Ys) = Σ wi * y
= y * Σ wi
= y * 1
= y
Therefore, E(Ys) = y, proving that Ys is an unbiased estimate of the population mean y.

(b) Proportional allocation is a method of allocating the sample size in stratified random sampling, where the sample size in each stratum is proportional to the size of the stratum in the population.

This ensures that each stratum is represented in the sample in proportion to its size in the population.

Optimum allocation is another method of allocating the sample size in stratified random sampling, where the sample size in each stratum is determined based on the variance of the stratum and the cost of sampling from the stratum.

This ensures that the sample size in each stratum is optimized to minimize the variance of the sample mean and the cost of sampling.

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The zero vector [0, 0, 0], the length would be √(0^2 + 0^2 + 0^2) = 0.

Examples of 3 element vectors include:
- [1, 2, 3]
- [-1, 0, 5]
- [3, -2, 1]
- [0, 0, 0]
- [4, 4, 4]
- [-2, -2, -2]

There are infinitely many 3 element vectors that have a length of 0. A vector with a length of 0 is called the zero vector, and it has all of its components equal to 0. In the case of a 3 element vector, the zero vector would be [0, 0, 0].

Examples of 3 element vectors include:
- [1, 2, 3]
- [-1, 0, 5]
- [3, -2, 1]
- [0, 0, 0]
- [4, 4, 4]
- [-2, -2, -2]

Remember that the length of a vector is calculated using the formula √(x1^2 + x2^2 + x3^2), where x1, x2, and x3 are the components of the vector. So, for the zero vector [0, 0, 0], the length would be √(0^2 + 0^2 + 0^2) = 0.

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Answer:

Yes

What is horizontal distance ?

The distance between two points is understood to mean the horizontal distance, regardless of the relative elevation of the two points.

How to calculate horizontal distance?

Horizontal distance can be expressed as x = Vtx = Vtx=Vt. Vertical distance from the ground is described by the formula y = – 1 2 g t 2 y = – \frac{1}{2}g t^2 y=–21gt2, where g is the gravity acceleration, and h is an elevation.

Step by step explanation:

As long as the ramp is no more than .9965 feet high, then yes

If the ramp is .9965 feet high then its horizontal distance is 12 X .9965 feet or 11.958 feet

Using Pythagoras’ Theorem, the actual length of the ramp would be the square root of (11.958 X 11.958 + .9965 X .9965)

Or the square root of (142.9934 + .9930)

Or SQRT (143.987)

= 11.999 feet

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The leading coefficient of f(x) is positive, which means that the graph will be pointing upwards on both ends. The graph is attached below.

Graphs are a not unusual place technique to visually illustrate relationships within side the statistics. The reason for a graph is to provide statistics that are too severe or complex to be defined appropriately within side the textual content and in much less space.

To sketch the graph of f(x) = x^4 - 3x^3 + x - 3, we can use the information gathered from analyzing the function:

Using this information, we can sketch the graph of f(x) as follows:

Putting all of this information together, we can sketch the graph of f(x) as shown below:

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The solution to the exponential equation [tex]4^-3v-2=4^-v[/tex]is v = -1.

To solve the exponential equation 4^(-3v-2)=4^(-v) using like bases and the one-to-one property,

Identify the like bases on both sides of the equation. In this case, the like base is 4.

Use the one-to-one property of exponential functions, which states that if two exponential functions with the same base are equal, then their exponents must also be equal. So, we can set the exponents equal to each other:

-3v - 2 = -v

Solve for the variable, v. We can do this by isolating the variable on one side of the equation:

-3v + v = 2

-2v = 2

v = -1

Check our answer by plugging it back into the original equation:

4^(-3(-1)-2) = 4^(-(-1))

4^(3-2) = 4^(1)

4^(1) = 4^(1)

4 = 4

So the solution to the exponential equation [tex]4^-3v-2=4^-v[/tex]is v = -1.

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Answer:

The decimal part of 40 is 0.40. 102 is not a decimal part of 40, so the answer is 0.

To explain further, a decimal part of a number is a fractional part of the number expressed as a decimal. For example, the decimal part of 40 is 0.40, which is the same as 40/100 or 4/10. 102 is not a fractional part of 40, so it cannot be expressed as a decimal part of 40.

The fraction of the euro is 8/100

one Euro will cost in 8 rands

When we talk about exchange rates, we're essentially talking about the value of one currency compared to another. In this case, we're comparing the South African rand to the euro.

We know that one South African rand is valued at 0.125 of one euro. To figure out what fraction of the euro one South African rand is worth, we can simply divide the value of one rand by the value of one euro:

1 rand ÷ 0.125 euro = 8/100 euro

So, one South African rand is worth 8/100 or 0.08 (which is equivalent to 8%) of one euro.

To calculate what one euro would cost in rands, we can use the inverse of the exchange rate we were given:

1 euro ÷ 0.125 rand/euro = 8 rand

So, one euro would cost 8 South African rand.

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