59. Standard Normal areas Find the proportion of observations in a standard Normal distribution that satisfies each of the following statements.
(a) z > -1.66
(b) -1.66​

(a) The exact distance between the points is 2√10.

(b) The midpoint of the line segment is (-7, 2).

(a) To find the distance between the two given points, use the distance formula, which is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²].

In this case, x₁ = -8, y₁ = 5, x₂ = -6, and y₂ = -1.

Plugging these values into the formula gives us:

d = √[(-6 - (-8))² + (-1 - 5)²]

d = √[(2)² + (-6)²]

d = √[4 + 36]

d = √40

d = 2√10

(b) The midpoint of a line segment can be found using the midpoint formula, which is:

(x₁ + x₂)/2, (y₁ + y₂)/2.

In this case, x₁ = -8, y₁ = 5, x₂ = -6, and y₂ = -1.

Plugging these values into the formula gives us:

midpoint = [(-8 + -6)/2, (5 + -1)/2]

midpoint = [(-14)/2, (4)/2]

midpoint = (-7, 2)

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Therefore , the solution of the given problem of triangle comes out to be  the shape is 55 square miles.

Because a triangular has two or more extra sections, it is a polygon. It's shape is a simple rectangle. A triangular shape can only be distinguished from a parallelogram by its sides A, B, and C. A singular plane rather than a cube is produced by Euclidean geometry when the edges are not perfectly collinear. A shape is referred to as triangular if it has three sides and three angles. An angle is the point where a quadrilateral's three sides meet. A triangle's sides add up to 180 degrees.

Here.

Two triangles with a base of 10 miles and heights of 7 miles and 4 miles can be formed from the composite design.

The following method can be used to determine a triangle's area:

A = (1/2)bh

where A represents area, B represents foundation, and H represents height.

We can calculate each triangle's area using the following formula:

=> Area of 1st triangle

=>  (1/2)(10)(7) = 35 square miles

=> Area of 2nd triangle

=>  (1/2)(10)(4) = 20 square miles

The two triangles' combined regions make up the composite shape's surface area.

55 square miles is the combined shape's area

=>  (35 + 20).

Consequently, the combined size of the two triangles in the shape is 55 square miles.

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Answer:

free

Step-by-step explanation:

28-28=0

The length of the sides x and y of the right-angled triangles using the trigonometric ratio of sine and cosine are:

11). x = 11√3 and y = 33

12). x = 4√7 and y = 8√7

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

recall that sin60° = √3/2 and cos60° = 1/2

For question 11:

sin 60° = y/22√3 {opposite/hypotenuse}

y = 22√3 × sin 60° {cross multiplication}

y = 22√3 × √3/2

y = 11 × 3

y = 33

cos 60° = x/22√3 {adjacent/hypotenuse}

x = 22√3 × cos 60° {cross multiplication}

x = 22√3 × 1/2

x = 11√3.

For question 12:

sin 60° = 4√21/y {opposite/hypotenuse}

y = 4√21/sin 60° {cross multiplication}

y = 4√21 ÷ √3/2

y = 4√21 × 2/√3

y = 4√7 × 2

y = 8√7

cos 60° = x/8√7 {adjacent/hypotenuse}

x = 8√7 × cos 60° {cross multiplication}

x = 8√7 × 1/2

x = 4√7

Therefore, the length of the sides x and y of the right-angled triangles using the trigonometric ratio of sine and cosine are:

11). x = 11√3 and y = 33

12). x = 4√7 and y = 8√7

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FR" - Rand GR" - Rare linear functions. That is they each preserve vector additions in their own dimension and ca multiplications.  G ∘ F is a linear function.

The question is asking you to show that the composition of two linear functions, G and F, is a linear function itself.
To show that G ◦ F is a linear function, we need to show that it satisfies the two properties of linearity: preservation of vector addition and preservation of scalar multiplication.

Let's first consider preservation of vector addition:

(G ◦ F)(u + v) = G(F(u + v)) [by definition of composition]

= G(F(u) + F(v)) [since F preserves vector addition]

= G(F(u)) + G(F(v)) [since G preserves vector addition]

= (G ◦ F)(u) + (G ◦ F)(v) [by definition of composition]

Therefore, G ◦ F preserves vector addition.

Now let's consider preservation of scalar multiplication:

(G ◦ F)(ku) = G(F(ku)) [by definition of composition]

= G(kF(u)) [since F preserves scalar multiplication]

= kG(F(u)) [since G preserves scalar multiplication]

= k(G ◦ F)(u) [by definition of composition]

Therefore, G ◦ F preserves scalar multiplication.

Since G ◦ F satisfies both properties of linearity, it is a linear function that preserves vector addition and scalar multiplication.

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The restrictions on the variable in the rational equation (1)/(x-4)-(2)/(x+5)=(2x-1)/(x^(2)+x-20) are x != 4 and x != -5.

The restrictions on the variable in the rational equation are x != 4 and x != -5. This is because the denominators of the rational expressions (x-4) and (x+5) cannot equal zero, as division by zero is undefined.

To find the restrictions, we can set each denominator equal to zero and solve for x:

x - 4 = 0 => x = 4

x + 5 = 0 => x = -5

So, the restrictions are x != 4 and x != -5.

In summary, the restrictions on the variable in the rational equation (1)/(x-4)-(2)/(x+5)=(2x-1)/(x^(2)+x-20) are x != 4 and x != -5.

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Khan and Jorman's scores will be the same in 4.545 weeks, and their score at that time will be 786.36 using the equal values.

With the equal values approach, two expressions are made equal, and the variable that makes them equal is then solved for. When solving systems of equations with two variables, this is often done in order to determine the values of the two variables that will simultaneously make the equations true.

Let us suppose the number of weeks = y.

Let us suppose the score = x.

Given that, Khan's current score is 750 and is rising 8 points per week.

Khan's score: y = 8x + 750

Jormans current score is 650 but is growing by 30 points each week:

Jorman's score: y = 30x + 650

Their scores will be equal at:

8x + 750 = 30x + 650

22x = 100

x = 4.545

Substitute the value of x in the equation to obtain y:

y = 8(4.545) + 750

y = 786.36

Hence, Khan and Jorman's scores will be the same in 4.545 weeks, and their score at that time will be 786.36.

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Evaluating the limit given by [tex]{ lim_{(x- > 9)} } (7x-63)/(x^2-2x-63)[/tex], we will obtain as a result [tex]7/16[/tex]

To evaluate the limit, we need to simplify the expression first. We can do this by factoring the numerator and denominator of the fraction:

[tex]lim_{(x->9)} (7x-63)/(x^2-2x-63) = lim_{(x->9)} (7(x-9))/((x-9)(x+7))[/tex]

Next, we can cancel out the [tex](x-9)[/tex] terms in the numerator and denominator:

[tex]lim_{(x->9)} (7)/(x+7)[/tex]

Now, we can plug in the value of x that the limit is approaching (9) into the expression:

[tex]lim_{(x->9)} (7)/(x+7) = (7)/(9+7) = (7)/(16)[/tex]

Therefore, the limit of the expression as x approaches 9 is 7/16.

Answer: [tex]{ lim_{(x->9)} } (7x-63)/(x^2-2x-63) = 7/16[/tex]

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If the length of the flag is 12 ft more than its width, then the width of the flag is 30ft and length of the flag is 42ft.

Let "w" represent the width of the flag.

We know that the length of the flag is 12 feet more than its width, which means:

⇒ Length = w + 12

We also know that the perimeter of the flag is 144 feet,

The Perimeter of flag can be expressed as:

⇒ Perimeter = 2(Length + Width),

Substituting the expression for Length from above,

We get,

⇒ 144 = 2(w+12 + w)

Simplifying the equation:

We get,

⇒ 144 = 2(2w + 12)

⇒ 72 = 2w + 12

⇒ 60 = 2w

⇒ w = 30

So, width of flag is 30 feet. and Length = w + 12,

⇒ Length = 30 + 12

⇒ Length = 42

Therefore, the dimensions of the flag are 42feet by 30feet.

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The value of secant theta is √2. The solution has been obtained by using trigonometry.

Trigonometry, a subfield of mathematics, is the study of the sides, angles, and connections of the right-angle triangle.

We are given that tangent theta = -1

We know that tan θ = sin θ / cos θ

So,

⇒ -1 = sin θ / cos θ

⇒ -cos θ  = sin θ

We know that angle θ lies in the 4th quadrant i.e. between 3π/2 and 2π.

In the 4th quadrant, at 7π/4, the above is true.

So, at θ = 7π/4, we get

cos (7π/4) = √2/2

We know that

sec θ = 1 / cos θ

So,

sec θ = √2

Hence, the value of secant theta is √2.

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The equation of the line shown is y = 2x.

In Mathematics, a proportional relationship can be defined as a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:

y = kx

Where:

Next, we would determine the constant of proportionality (k) as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 2/1 = 4/2

Constant of proportionality, k = 2.

Therefore, the required equation is given by:

y = kx

y = 2x

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Answer:

Should be the second answer.

Step-by-step explanation:

Answer: 0.66 cents per pen

Step-by-step explanation:

$4.65 for a pack of 7 pens. You want to divide $4.65 by 7 and you get 0.66

The solution is, the value of x is, x=10.

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.

here, we have,

from the given figure, we get,

the two triangles are similar

and, DE is the bisector of other  two sides,

so, we know from the property of  triangles, that,

DE = 1/2 * AB

we have,

DE = 5

and. AB =x

so, we get,

5 = x/2

or, x = 10

Hence, The solution is, the value of x is, x=10.

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The probability that none of the 3 detectors would work if there was smoke in the factory is 0.25³, or 0.015625 (1.56%).

Probability is a branch of mathematics that deals with the likelihood of certain outcomes or events occurring. Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1. An event that is certain to occur has a probability of 1, while an event that is impossible to occur has a probability of 0. The probability of an event occurring can be calculated by dividing the number of successful outcomes by the total number of possible outcomes.

This probability indicates that the factory does have a safety problem, as there is a 1.56% chance that none of the smoke detectors would sound an alarm if there was smoke present. This presents a significant risk to the factory, as it could lead to an undetected fire that may cause damage to the facility or put employees in danger.

The company should look into ways to reduce the risk of the smoke detectors not working. One way to do this would be to increase the number of smoke detectors in the factory room, as the more detectors there are, the lower the probability that none of them will work. The company could also consider using more reliable smoke detectors that have a higher probability of working correctly when needed. Ultimately, the company should seek to reduce the risk of none of the detectors working by taking steps to increase the probability that at least one of them will sound an alarm in the event of smoke or a fire.

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The area of rectangular field is   [tex]\frac{9}{20} \ \text{miles}^2[/tex].

The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area. In essence, the area of a rectangle is equal to the sum of its length and breadth.

Here the given rectangle , length= [tex]\frac{9}{10}[/tex]miles and Breadth = [tex]\frac{1}{2}[/tex]miles

Now using area of rectangle formula,

=> A = length× breadth square unit.

=> A = [tex]\frac{9}{10}\times\frac{1}{2}[/tex]

=> A =   [tex]\frac{9}{20} \ \text{miles}^2[/tex]

Hence the area of rectangular field is  [tex]\frac{9}{20} \ \text{miles}^2[/tex].

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Both f(g(x)) and g(f(x)) equal x, we can determine that the functions f(x) and g(x) are inverses.

To determine if the functions f(x) and g(x) are inverses using composition, we need to find f(g(x)) and g(f(x)) and see if they both equal x.

First, let's find f(g(x)):
f(g(x)) = f((3)/(2)x+12)
= (2)/(3)((3)/(2)x+12)-8
= (2)/(3)(3)/(2)x + (2)/(3)(12) - 8
= x + 8 - 8
= x

Now, let's find g(f(x)):
g(f(x)) = g((2)/(3)x-8)
= (3)/(2)((2)/(3)x-8) + 12
= (3)/(2)(2)/(3)x - (3)/(2)(8) + 12
= x - 12 + 12
= x

Since both f(g(x)) and g(f(x)) equal x, we can determine that the functions f(x) and g(x) are inverses.

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Answer: it can change the amount of force that you can apply on an object.

Step-by-step explanation:

Answer:

Simple machine makes work go faster

Step-by-step explanation:

If you replace human Labour with simple machine, work is completed faster. It increases output

Answer: This is a classic problem in mathematics known as the arithmetic series or the Gauss sum.

To find out how many days the person can donate with a total of Rs. 210, we need to sum the sequence of donations until we reach the total amount of Rs. 210. The sequence of donations is:

1 + 2 + 3 + 4 + 5 + ... + n

The sum of the sequence can be expressed as:

n(n+1)/2

So we need to solve the equation:

n(n+1)/2 = 210

n(n+1) = 420

n^2 + n - 420 = 0

We can solve this quadratic equation using the quadratic formula:

n = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = 1, and c = -420.

n = (-1 ± sqrt(1^2 - 4(1)(-420))) / 2(1)

n = (-1 ± sqrt(1 + 1680)) / 2

n = (-1 ± sqrt(1681)) / 2

n = (-1 ± 41) / 2

Since we are looking for a positive integer value of n, we can discard the negative solution:

n = (41 - 1) / 2

n = 40 / 2

n = 20

Therefore, the person can donate for a maximum of 20 days with a total donation of Rs. 210, starting with Rs. 1 on the first day and increasing by Rs. 1 each day.

Step-by-step explanation:

If the dividend is 872 and the quotient is 62 and remainder is 4, the divisor is equal to 14.

In Mathematics, a quotient can be defined as a mathematical expression that is typically used for the representation of the division of a number by another number.

In Mathematics, dividend can be calculated by using this mathematical expression:

Dividend = divisor × quotient + residual

Substituting the given data points into the dividend formula, we have the following;

Dividend = divisor × quotient + residual

872 = divisor × 62 + 4

Divisor = (872 - 4)/62

Divisor = 868/62

Divisor = 14

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Complete Question;

If the dividend is 872 and the quotient is 62 and remainder is 4. What is the divisor?

when dividing x^(4)-3x^(3)-7x^(2)-80 by x-5, the remainder will be -5.

The given polynomial equation is x^(4)-3x^(3)-7x^(2)-80 and we have to divide it by x-5 to find the remainder.

Set up the long division as follows:
```
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
```

Divide the first term of the dividend (x^(4)) by the first term of the divisor (x) to get the first term of the quotient (x^(3)):
```
x^(3)
_______
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
```

Multiply the first term of the quotient (x^(3)) by the divisor (x-5) and write the result below the dividend:
```
x^(3)
_______
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
    x^(4) - 5x^(3)
```

Subtract the result from the dividend to get the new dividend:
```
x^(3)
_______
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
    x^(4) - 5x^(3)
    --------------
         2x^(3) - 7x^(2)
```

Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor:
```
x^(3) + 2x^(2) + 3x + 15
______________________
x-5 | x^(4) - 3x^(3) - 7x^(2) + 0x - 80
    x^(4) - 5x^(3)
    --------------
         2x^(3) - 7x^(2)
         2x^(3) - 10x^(2)
         ---------------
               3x^(2) + 0x
               3x^(2) - 15x
               ------------
                    15x - 80
                    15x - 75
                    --------
                         -5
```

The remainder is -5. Therefore, when dividing x^(4)-3x^(3)-7x^(2)-80 by x-5, the remainder will be -5.

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The simplified expression is (x+1)^2/(2x-1).

To simplify the expression ((x)/(x+1)+1)*(1+x)/(2x-1), we can begin by multiplying the numerator and denominator of the first fraction by x+1 to eliminate the fraction in the numerator:

((x)/(x+1)+1)*(1+x)/(2x-1)

= ((x)/(x+1)*(x+1)/(x+1) + (x+1)/(x+1)) * (1+x)/(2x-1)

= (x(x+1)/(x+1) + (x+1)/(x+1)) * (1+x)/(2x-1)

= (x + 1) * (1+x)/(2x-1)

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

Now, the expression is simplified to (x^2 + 2x + 1)/(2x-1).

We can further simplify this expression by factoring the numerator as (x+1)^2:

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

Note that I did not solve for any variables or values, as the original expression did not provide an equation or specific values to solve for. Instead, I simplified the expression to its most simplified form.

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Answer:29 1/4

Step-by-step explanation:

Because the formula for the volume of a rectangular prism is (w)(h)(l).

Answer:

Let's call the weight of the book that was removed "x".

The total weight of the twelve books can be represented as 12 times the average weight of 2.75 pounds:

12(2.75) = 33

After the book is removed, there are only 11 books left, and their total weight can be represented as 11 times the new average weight of 2.70 pounds:

11(2.70) = 29.7

We can set up an equation using these two expressions and the weight of the book that was removed:

33 - x = 29.7

Solving for x, we get:

x = 33 - 29.7

x = 3.3

Therefore, the weight of the book that was removed was 3.3 pounds.

The solution is, that central angle Ф is, 3.49 radians.

Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length.

here, we have,

The arc length formula is s = rФ,

where r is the radius and Ф is the central angle.  

We know s and r and need to calculate Ф.

now, we get,

From s = rФ

we get Ф = central angle = s/r

Here, that central angle Ф is,

(31.4 cm) / 9 cm

= 3.49 radians

Hence, The solution is, that central angle Ф is, 3.49 radians.

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Answer: If you want to live in a city with greater variability of temperature, you should look for cities with a continental climate. Continental climates are characterized by hot summers and cold winters, with large temperature fluctuations between day and night and between seasons.

Cities located in the interior of large landmasses, far from large bodies of water, are more likely to have a continental climate. Examples of cities with a continental climate include Moscow (Russia), Calgary (Canada), and Chicago (USA). These cities experience wide temperature swings throughout the year, with hot summers and cold winters.

On the other hand, cities located near large bodies of water tend to have a more moderate climate, with smaller temperature fluctuations. Examples of such cities include San Francisco (USA), Sydney (Australia), and London (UK). These cities experience mild summers and winters, with relatively constant temperatures throughout the year.

Step-by-step explanation:

The distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.

In mathematics, an expression is a combination of numbers, variables, and operators, which when evaluated, produces a value. An expression can contain constants, variables, functions, and mathematical operations such as addition, subtraction, multiplication, and division.

To find the distance from a point to a line, we need to find the length of the perpendicular segment from the point to the line.

The line [tex]$y=\frac{1}{2}x-3$[/tex]  can be rewritten in slope-intercept form as [tex]$y = \frac{1}{2}x - 3$[/tex], so its slope is [tex]\frac{1}{2}$.[/tex]

A line perpendicular to this line will have a slope that is the negative reciprocal of [tex]\frac{1}{2}$[/tex], which is [tex]-2$.[/tex]

We can then use the point-slope form of a line to find the equation of the perpendicular line that passes through the point [tex]$(1,2)$[/tex]:

[tex]$y - 2 = -2(x - 1)$[/tex]

Simplifying, we get:

[tex]$y = -2x + 4$[/tex]

Now we need to find the point where the two lines intersect, which will be the point on the line [tex]$y = \frac{1}{2}x-3$[/tex] that is closest to [tex]$(1,2)$[/tex]. We can do this by setting the equations of the two lines equal to each other and solving for [tex]$x$[/tex]:

[tex]$\frac{1}{2}x - 3 = -2x + 4$[/tex]

Solving for [tex]$x$[/tex], we get:

[tex]$x = \frac{14}{5}$[/tex]

To find the corresponding [tex]$y$[/tex] value, we can substitute this value of [tex]$x$[/tex] into either of the two line equations. Using [tex]$y = \frac{1}{2}x-3$[/tex], we get:

[tex]$y = \frac{1}{2} \cdot \frac{14}{5} - 3 = -\frac{7}{5}$[/tex]

Therefore, the point on the line [tex]$y = \frac{1}{2}x-3$[/tex] that is closest to [tex]$(1,2)$[/tex] is [tex]$\left(\frac{14}{5}, -\frac{7}{5}\right)$[/tex].

Finally, we can use the distance formula to find the distance between [tex]$(1,2)$[/tex] and [tex]$\left(\frac{14}{5}, -\frac{7}{5}\right)$[/tex]:

[tex]$\sqrt{\left(\frac{14}{5} - 1\right)^2 + \left(-\frac{7}{5} - 2\right)^2} \approx 3.7$[/tex]

Rounding to the nearest tenth, the distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.

Therefore, the distance from the point [tex]$(1,2)$[/tex] to the line [tex]$y=\frac{1}{2}x-3$[/tex] is approximately 3.7 units.

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The measurements are shown in the solution

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

Given are right triangles, we need to find the value of x in each,

We will use Trigonometric ratios to find the same,

The sine is the ratio between the length of the opposite leg and the length of the hypotenuse.

The cosine is the ratio between the length of the adjacent leg and the length of the hypotenuse.

The tangent is the ratio between the length of the opposite leg divided by the length of the adjacent leg.

1) sin 21° = 11/x

x = 30.7

2) cos 34° = x/28

x = 23.21

3) tan 49° = 33/x

x = 28.68

4) T =  cos⁻¹ (11/23)

=  61.42°

5) T = sin⁻¹(21/44)

= 28.50°

6) T = tan⁻¹(8/12)

= 33.7°

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