according to the u.s. department of agriculture, the mean yield of corn per acre in the u.s. is 107 bushels with a standard deviation of 15 bushels. suppose 80 one-acre plots of corn are selected at random. (a) what is the probability that the mean yield of the 80 one-acre plots chosen will be between 104 and 111 bushels? (b) what is the probability that the mean yield of the 80 one-acre plots chosen will be greater than 105 bushels? (c) what is the probability that a randomly selected one-acre plot has a yield less than 100 bushels? (2) the average age of millionaires in the us is to be determined. twenty us millionaires are randomly selected to find a mean age of 58.5 years. assume the standard deviation of all us millionaires is 12.8 years. assume the population of ages of us millionaires is normally distributed. find a 95% confidence interval for the mean age of all millionaires. round your answer to 2 decimal places if necessary. interpret your answer in a sentence.

His results are in the table  are-

Southeast = 36.4%

Southwest  = 9.1%

West = 18.2%

Midwest = 20%

What is arithmetic?

Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.

Here, we have

: Leon is interested in the relationships between geographic region and political affiliation. Leon collected data about the political parties of US senators at that time and the regions of the US that they represent.

Southeast = 20/55 = 36.4%

Southwest = 5/55 = 9.1%

West = 10/55 = 18.2%

Midwest = 11/55 = 20%

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The height of the rocket at a time of 7.8 seconds is approximately 2,254 feet.

To find a quadratic regression equation, we need to fit a quadratic function of the form [tex]y = ax^2 + bx + c[/tex]to the data.

Using a calculator or spreadsheet software, we can find the coefficients that minimize the sum of the squared errors between the predicted values of y and the actual values of y.

The resulting quadratic regression equation is:

[tex]y = 88.6x^2 + 2.9x + 196.5[/tex]

To find the height of the rocket at a time of 7.8 seconds, we simply substitute x = 7.8 into the equation and evaluate:

[tex]y = 88.6(7.8)^2 + 2.9(7.8) + 196.5[/tex]

y ≈ 2,253.6

Rounding to the nearest foot, we get:

y ≈ 2,254 feet

Therefore, the height of the rocket at a time of 7.8 seconds is approximately 2,254 feet.

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Answer:

The volume of a cylinder is given by the formula: V = πr^2h where r is the radius and h is the height. Substituting the given values, we get: V = π(4 in)^2(3 in) V = π(16 in^2)(3 in) V = 48π in^3 Therefore, the volume of wax needed to make one candle is 48π cubic inches. To find the amount of wax needed for 120 candles, we can multiply the volume of one candle by the number of candles: Amount of wax = 48π in^3/candle × 120 candles Amount of wax = 5760π in^3 So the company will need 5760π cubic inches of wax to make 120 candles.

Answer: 14

Step-by-step explanation: you need to start by finding the factors of them and then what ever the biggest number is in both of them has to match but i got 14.

Answer:

(b/2)² + 1 = x² - 3x + (b/2)²

b²/4 + 1 = x² - 3x + b²/4

b²/4 - b²/4 = x² - 3x - 1

x² - 3x - 1 = 0

x² - 3x = 1

x² - 3x + 9/4 = 1 + 9/4

(x - 3/2)² = 13/4

x - 3/2 = ±(√13/4)

x = ±√13/2 + 3/2

so the number that must be added will be 3/2

The ball was thrown 11 m off the ground. The maximum height of the ball was 11 m. The height of the ball at 2.5s was 9.07 m. Yes, the football is in the air after 6s and the ball hits the ground at 8.67 s.

Height is the measure of vertical distance or the vertical extent of an object, person, or other thing from top to bottom. It is typically measured in units of meters or feet. Height is a measure of vertical distance or elevation above a given level, most commonly sea level. Height can also be determined by measuring the altitude of an object or location. Height is an important factor in many everyday contexts, such as architecture and sports.

To calculate the answers, we must first solve for the equation of the height of the ball h(t).

h(t) = -4.9(1-1.39)² + 11

1) The ball was thrown 11 m off the ground.

To calculate this, we set t = 0.

h(0) = -4.9(1-1.39)² + 11

h(0) = 11

Therefore, the ball was thrown 11 m off the ground.

2) The maximum height of the ball was 11 m.

To calculate this, we set the derivative of the equation, h'(t) = 0 and solve for t.

h'(t) = -9.8(1-1.39)

h'(t) = 0

1-1.39 = 0

1 = 1.39

Therefore, t = 1.

Substituting t = 1 into the equation for h(t), we get:

h(1) = -4.9(1-1.39)² + 11

h(1) = 11

Therefore, the maximum height of the ball was 11 m.

3) The height of the ball at 2.5s was 9.07 m.

To calculate this, we substitute t = 2.5 into the equation for h(t).

h(2.5) = -4.9(1-1.39)² + 11

h(2.5) = 9.07

Therefore, the height of the ball at 2.5s was 9.07 m.

4) Yes, the football is in the air after 6s.

To calculate this, we substitute t = 6 into the equation for h(t).

h(6) = -4.9(1-1.39)² + 11
+ 11

h(8.67) = 0

Therefore, the ball hits the ground at 8.67 s.

In conclusion, the ball was thrown 11 m off the ground. The maximum height of the ball was 11 m. The height of the ball at 2.5s was 9.07 m. Yes, the football is in the air after 6s and the ball hits the ground at 8.67 s.

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We can find y by solving for the length of CN: CN = Sy - BC = 600 - 0 = 600 So the values of x and y are:
x = 0 , y = 600

First, we can look at the corresponding sides of triangle ABC and triangle MNC. Since we know that these triangles are

similar (as indicated by the ~ symbol), we can set up the following proportion:

AB/AM = BC/CN

We can substitute in the given values for these lengths:

4x/2x = (Lv + Sy)/Sy

Simplifying this equation, we can cancel out the common factor of 2x on the left side:

2 = (Lv + Sy)/Sy

Multiplying both sides by Sy, we can isolate the sum of Lv and Sy:

2Sy = Lv + Sy

Combining like terms, we get:

Sy = Lv

So we now know that Sy is equal to Lv, which is 600.

To solve for x and y, we can use the fact that corresponding sides are in proportion. Specifically, we can look at the ratio of the length of AB to the length of AM:

AB/AM = 4x/Lv

Substituting in the values we know, we get:

AB/AM = 4x/600

We can also look at the ratio of the length of BC to the length of CN:

BC/CN = 2x/Sy

Substituting in the values we know, we get:

BC/CN = 2x/600

Since AB and BC are corresponding sides in triangle ABC and AM and CN are corresponding sides in triangle MNC, we

know that these ratios must be equal:

4x/600 = 2x/600

Simplifying this equation, we get:

4x = 2x

Subtracting 2x from both sides, we get:

2x = 0

Dividing both sides by 2, we get:

x = 0

Since x is 0, we can't solve for it any further. However, we can solve for y by using the ratio we found earlier:

AB/AM = 4x/600 = 0/600 = 0

Since AB is 0, we know that BC is also 0 (since the sum of the lengths of the sides of a triangle must be greater than 0).

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Answer:

12.7

Step-by-step explanation:

3x+10x-15=180

13x=165

x=12.69.....

x=12.7

The difference in the account balance between a simple interest and a compound interest over an 8-year time period, rounded to the nearest cent, is $12,189.02.

To calculate the difference in the account balance between simple interest and compound interest over 8 years, we first need to determine the final balance for each case.

Simple Interest:

Simple interest is calculated as a percentage of the initial principal (or the original amount deposited) and is paid only on the principal amount. The formula for simple interest is:

I = P * r * t

where I is the interest earned, P is the principal amount (in this case, $5,000), r is the annual interest rate (as a decimal), and t is the time period in years.

For this problem, the interest rate is given as 31%, which is equivalent to 0.31 as a decimal. Therefore, we have:

I = $5,000 * 0.31 * 8 = $12,400

The final balance for simple interest is the sum of the principal and the interest earned:

Final balance (simple interest) = $5,000 + $12,400 = $17,400

Compound Interest:

Compound interest is interest that is calculated on both the principal amount and any accumulated interest. The formula for compound interest is:

A = P * (1 + r/n)^(n*t)

where A is the final amount, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times per year that interest is compounded, and t is the time period in years.

For this problem, the interest rate is 31% and the account is compounded annually, so n = 1. Therefore, we have:

A = $5,000 * (1 + 0.31/1)^(1*8) = $29,589.02

The final balance for compound interest is the amount we just calculated:

Final balance (compound interest) = $29,589.02

The difference in the two account balances is:

$29,589.02 - $17,400 = $12,189.02

Therefore, the difference in the account balance between a simple interest and a compound interest over an 8-year time period, rounded to the nearest cent, is $12,189.02.

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Answer:

y = [tex]\frac{4}{3}[/tex]x - 3

Step-by-step explanation:

Oof... I haven't done this in a while, but here you go! Slope-intercept form is y=mx+b, and in this one, the y-intercept is -3, just to get that out of the way. Now, the slope is the rise over the run, and I'm going to use the points (3, 1) and (6, 5) to measure the rise and run. The change in y, or the rise, is 4, while the change in x, or the run, is 3, so the slope is 4/3. Hope this was helpful!

Answer:

[tex]y = \dfrac{4}{3}x - 3[/tex]

Step-by-step explanation:

The slope-intercept form of a line is defined as:

[tex]y=mx+b[/tex],

where [tex]m[/tex] is the line's slope, and [tex]b[/tex] is the y-coordinate of its y-intercept.

We can see that the slope of the line is:

slope = rise / run = 4/3

(See the attached image for a model of rise and run.)

We can also see that the line touches the y-axis when y = -3, so this is the y-coordinate of the line's y-intercept.

With these pieces of information, we can craft the line's equation in slope-intercept form:

[tex]y = \dfrac{4}{3}x - 3[/tex]

One-third of the worker bees forage for pollen and nectar.

What is percentage?

Percentage is a way of expressing a number as a fraction of 100. It is often denoted by the symbol "%". For example, 50% means "50 out of 100" or "50/100" or "0.5 as a decimal". Percentages are commonly used to express the proportion or relative size of a quantity with respect to another quantity. They can be used in a wide range of contexts, including finance, statistics, science, and everyday life.

Here out of the 18,000 worker bees, 6,000 forage for pollen and nectar.

To calculate the part of worker bees that forage for pollen and nectar, we can use the following formula:

Part = (Number of bees foraging / Total number of worker bees) x 100

Plugging in the numbers, we get:

Part = (6,000 / 18,000) x 100

Part = 0.333 x 100

Part = 33.3%

Therefore, 33.3% or approximately one-third of the worker bees forage for pollen and nectar.

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The provided arithmetic expression's explicit and recursive formulas are as follows:

Recursive formula = [tex]a_{n+1}=a_{n-7}[/tex]

Explicit formular is Tn = 22 - 7n

An explicit formula can be used to determine the value of any phrase in a sequence. The natural numbers 1, 2, 3, 4, and so on make up the integers. If you have a recursive formula for the sequence and know the value of the (n-1)th term in the series, you can use it to calculate the value of the nth term. While explicit formulae provide the value of a specific term depending on the location, recursive formulas supply the value of a specific phrase based on the preceding term. This is the primary difference among recursive and explicit formulations.

Given:

[tex]a_1[/tex] = 15

[tex]a_2[/tex] = 22 = 15 - 7 = [tex]a_1[/tex] - 7

Recursive formular = [tex]a_{n+1}=a_{n-7}[/tex]

a = 15, d = -7

Fn = a + (n - 1)d = 15 + (n - 1)-7 = 15 - 7n + 7 = 22 - 7n

Explicit formular is Tn = 22 - 7n

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Graph attached below,

(a) The probability that both John and Jane watch the show is 0.15 or 15% (rounded to 2 decimal places).
(b)The probability that Jane watches the show, given that John does, is 0.3 or 3.

(c) Since these probabilities are equal, John and Jane watch the show independently of each other.

The answer is Yes.

John and Jane watch the show independently of each other if

(a) To find the probability that both John and Jane watch the show, we can use the conditional probability formula:

P(A and B) = P(A|B) * P(B),

where A represents John watching the show and B represents Jane watching the show.
Given, P(A|B) = 0.5 and P(B) = 0.3.
P(A and B) = 0.5 * 0.3 = 0.15.
Therefore, the probability that both John and Jane watch the show is 0.15 or 15% (rounded to 2 decimal places).
(b) To find the probability that Jane watches the show, given that John does, we can use the conditional probability formula: P(B|A) = P(A and B) / P(A).
Given, P(A and B) = 0.15 and P(A) = 0.5.
P(B|A) = 0.15 / 0.5 = 0.3.
Therefore, the probability that Jane watches the show, given that John does, is 0.3 or 30% (rounded to 3 decimal places).
(c) John and Jane watch the show independently of each other if P(A and B) = P(A) * P(B).

In this case, P(A and B) = 0.15 and P(A) * P(B) = 0.5 * 0.3 = 0.15.

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On the number line, the empty set, denoted by an empty interval or a pair of crossed-out points, would be the solution set.

A mathematical assertion that establishes the equal of two expressions is called an equation. Variables, can serve as placeholders with unknowable values, and multipliers, and that are constants can multiply the variables, are frequently included. Determine you significance of the variables necessary make the answer true by solving the equation. The significant early (=), plus sign (+), and debit sign (-), among several other mathematical symbols, can be used to indicate equations. They are employed in many disciplines, such as mathematics, mathematics, chemistry, construction, and economics, to model the interactions between variables and to address issues.

given

Due to the fact that a real number's absolute value is never negative, the equation |x+8|=-2 has no solutions.

As a result, it is impossible for x+8 to have an absolute value of -2, which is a negative number.

On the number line, the empty set, denoted by an empty interval or a pair of crossed-out points, would be the solution set.

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Therefore, the sphere described here cannot hold all of the water from the cylinder.

Diameter is a straight line passing through the center of a circle or a sphere, and connecting two points on its circumference. It is the longest distance between any two points on a circle or a sphere, and it is equal to twice the radius. In other words, if you know the diameter of a circle or a sphere, you can find its radius by dividing the diameter by 2, and vice versa, if you know the radius, you can find the diameter by multiplying the radius by 2.

The volume of the cylinder can be calculated as follows:

Radius of cylinder = Diameter[tex]/2 = 3/2 = 1.5 cm[/tex]

Volume of cylinder[tex]= \pi r^2h = \pi (1.5)^2(8) = 56.55 cm^3[/tex]

To determine if a sphere with a radius of 2 cm can hold all of the water from the cylinder, we can calculate its volume as follows:

Volume of sphere [tex]= 4/3\pi r^3 = 4/3\pi (2)^3 = 33.51 cm^3[/tex]

Since the volume of the cylinder is greater than the volume of the sphere, the sphere cannot hold all of the water from the cylinder. Therefore, the sphere described here cannot hold all of the water from the cylinder.

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Answer:

If three dogs can eat three bones in three days, we can assume that each dog eats one bone in three days. Therefore, in one day, one dog can eat one-third of a bone.

If nine dogs were to eat in the same efficiency, we can multiply the number of dogs by the fraction of bones each can eat per day. So, nine dogs can eat 3 times more than 3 dogs in a day, which is 3 x 3 = 9 bones. In one day, each dog can eat one-third of the bone, and therefore nine dogs will eat 9 x 1/3 = 3 bones.

This means that in nine days, nine dogs can eat 9 x 3 = 27 bones.

Answer:

14

Step-by-step explanation:

A = length x width

l = length

w = 3l + 2

A = l(3l + 2) = 56

3l² + 2l - 56 = 0

Now you have a polynomial you can solve by factoring or using the quadratic equation to find the 2 roots of l:

using the quadratic equation with a = 3, b = 2, c = -56

l = 4, -14/3

Disregarding the negative root,

l = 4

Therefore, w = 3(4) + 2 = 14

Answer:

a 2.009, 2.15, 2.7

b 3.2, 3.342, 3.45

c 17.05, 17.1, 17.125, 17.42

Answer:

1)55 and for 2) 35

Step-by-step explanation:

All you have to do is and for perimeter.

Answer:

Step-by-step explanation:

5)

Draw a right triangle with a hypotenuse of 60 km, a base angle of 60°, and the base leg of x.

Find x, the distance it traveled to the east:

cos60° = x/60

x = 60(cos60) = 30 km

The first quartile of the data set is 25.

Box plot is a simple way of representing statistical data on a plot in which a rectangle is drawn to represent the second and third quartiles, usually with a vertical line inside to indicate the median value.

The first quartile and third quartile are the lower and upper side of the rectangle respectively.

In this case, the first quartile of the data set is 25.

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The value of variable x=4 units in the given figure.

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The side opposite to the right angle is called the hypotenuse, while the other two sides are called the legs or catheti. The lengths of the legs and the hypotenuse of a right triangle are related by the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

From the given figure;

Tan60°=p/1

p=1.732 units

The smaller triangle is right angle

h²=p²+b²

h=√1²+1.732²

h=2units

In the bigger triangle, angle subtended by base is 60 (vertically opposite angle)

Using the trigonometric function

Cos60°=2/x

1/2=2/x

x=4units

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We can start by finding the length of the arc \overset{\LARGE\frown}{TS} using the fact that the length of the arc \overset{\LARGE\frown}{RS} is \frac{4}{3}\pi times the radius of the circle. Since angle RQS is 120 degrees, arc \overset{\LARGE\frown}{TS} is \frac{1}{3} of the circumference of the circle:

arc \overset{\LARGE\frown}{TS} = \frac{1}{3} (2\pi R) = \frac{2}{3}\pi R

Next, we can find the length of the chord TS using the Law of Cosines:

TS^2 = TR^2 + RS^2 - 2(TR)(RS)\cos(\angle TRS)

Since \angle TRS is 120 degrees, we have:

TS^2 = R^2 + (4/3)^2R^2 - 2(R)(4/3)R(-1/2)

Simplifying this expression, we get:

TS^2 = \frac{25}{9}R^2

Taking the square root of both sides, we get:

TS = \frac{5}{3}R

Now we can find the height of the shaded region by drawing the altitude from the center of the circle to chord TS. This altitude bisects chord TS and is also perpendicular to it, so it divides TS into two segments of equal length:

Height = \frac{1}{2}(TS) = \frac{5}{6}R

Finally, we can find the area of the shaded region by subtracting the area of triangle RST from the area of sector RQS:

Area of sector RQS = (120/360)\pi R^2 = \frac{1}{3}\pi R^2

Area of triangle RST = (1/2)(RS)(height) = (1/2)(4/3)R(\frac{5}{6}R) = \frac{5}{9}R^2

Area of shaded region = Area of sector RQS - Area of triangle RST = \frac{1}{3}\pi R^2 - \frac{5}{9}R^2 = \frac{2}{9}\pi R^2

Therefore, the area shaded below is \frac{2}{9}\pi times the square of the radius R of the circle.

As per the given data, in circle QQ, the area shaded below is (1/27)π.

The circumference is the perimeter of a circle or ellipse in geometry. That is, the circumference would be the circle's arc length if it were opened up and straightened out to a line segment.

To find the area shaded below, we need to first find the area of sector RQS and then subtract the area of triangle RQS to get the shaded area.

The length of arc RS is given as 4/3π. Since the circumference of the circle is 2πr, where r is the radius, we have:

2πr = 4/3π

r = 2/3

So, the radius of the circle is 2/3.

Next, we can use the formula for the area of a sector, which is given as:

A = [tex](1/2)r^2\theta[/tex]

where r is the radius of the sector and θ is the central angle in radians.

In this case, θ is 120 degrees or 2/3π radians, so we have:

A(sector RQS) = [tex](1/2)(2/3)^2(2/3\pi)[/tex] = 4/27π

Now, we need to find the area of triangle RQS. To do this, we can use the formula for the area of a triangle, which is given as:

A = (1/2)bh

Where b is the base of the triangle and h is its height.

Since RQ is the base of triangle RQS and is also a radius of the circle, its length is 2/3. To find the height of the triangle, we can draw a perpendicular from point S to line QR and call the point of intersection T.

Since angle RQS is 120 degrees, angle RQT is 30 degrees (since the sum of the angles in a triangle is 180 degrees), and we have:

sin 30 = h/2/3

h = 1/3

So, the area of triangle RQS is:

A(triangle RQS) = (1/2)(2/3)(1/3) = 1/9

Finally, we can find the shaded area by subtracting the area of triangle RQS from the area of sector RQS:

A(shaded) = A(sector RQS) - A(triangle RQS) = (4/27π) - (1/9) = (1/27)π

Therefore, the area shaded below is (1/27)π.

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The calculated value of the distance between points A and B is 2√29 units.

We can use the distance formula to find the distance between points A and B:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁, y₁) = A = (-2, -10) and (x₂, y₂) = B = (-6, 0).

Substituting these values into the formula, we get:

d = √[(-6 - (-2))² + (0 - (-10))²]

d = √[(-4)² + 10²]

d = √(16 + 100)

d = √116

d = 2√29

Therefore, the distance between points A and B is 2√29 units.

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Answer:

a=671000

Step-by-step explanation:

woah

The formula for the number of internet host computers as an exponential function of t is: n(t) = 35.3 × e^((1/7) × ln(n(7) / 35.3) × t) and the values of a and k in the model are k = (1/7) × ln(n(7) / n(0)) and a = n(0) = 35.3 million

a. To find the values of a and k, we can use the information given in the question. Let n(0) be the number of host computers in July 1998, which is 35.3 million. Let n(7) be the number of host computers in July 2005, which is 352.1 million. We can use these values to solve for a and k.

n(0) = a × e^(k0) = a

n(7) = a × e^(k7)

Dividing the second equation by the first equation, we get:

n(7) / n(0) = e^(k×7)

Taking the natural logarithm of both sides, we get:

ln(n(7) / n(0)) = k×7

Therefore, k = (1/7) × ln(n(7) / n(0)) and a = n(0) = 35.3 million.

Therefore, the formula for the number of internet host computers as an exponential function of t is:

n(t) = 35.3 × e^((1/7) × ln(n(7) / 35.3) × t)

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the equation y=1 does not represent a direct variation, whereas the other three equations given do represent direct variation.

What is Direct variation?

Direct variation is a relationship between two variables in which one is a constant multiple of the other. This means that if one variable increases, the other variable will also increase by a proportional amount. In other words, direct variation can be represented by the equation y=kx, where k is the constant of variation.

Out of the four equations given, the equation y=1 does not represent a direct variation. This is because the equation is not a function of x. In other words, no matter what value of x we choose, y will always be equal to 1. This means that the two variables, x and y, are not related to each other in any way. Therefore, there is no constant k that can be multiplied by x to get y.

On the other hand, the other three equations do represent direct variation. For example, in the equation y=x, y is equal to x multiplied by the constant k=1. Similarly, in the equation y=-5x, y is equal to x multiplied by the constant k=-5. Finally, in the equation y=0.9x, y is equal to x multiplied by the constant k=0.9.

In summary, the equation y=1 does not represent a direct variation, whereas the other three equations given do represent direct variation.

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To graph the solution set of the system of inequalities y>x²-2 and y ≥ -x²+5, which is the area between the two curves, above the curve of f(x) = [tex]x^2 - 2[/tex]and above or on the curve of g(x) = [tex]-x^2 + 5[/tex].

To graph the solution set to the system of inequalities y>x²-2 and y ≥ -x²+5, you can follow these steps:

Graph the functions f(x) = x² - 2 and g(x) = -x² + 5 on the same coordinate plane.

Shade the region above the curve of f(x) = x² - 2 since the inequality y > x² - 2 indicates that the values of y are greater than the corresponding values of x² - 2.

Shade the region above or on the curve of g(x) = -x² + 5 since the inequality y ≥ -x²+5 indicates that the values of y are greater than or equal to the corresponding values of -x² + 5. This region can be represented as the area bounded by the curve of g(x) and the x-axis.

This shaded region is the area between the two curves, above the curve of f(x) = x² - 2, and above or on the curve of g(x) = -x² + 5. This region can be described as the set of all points (x, y) in the coordinate plane that satisfy the two inequalities simultaneously.

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The functions f(x) = x2 – 2 and g(x) = –x2 + 5 are shown on the graph.

Explain how to modify the graphs of f(x) and g(x) to graph the solution set to the following system of inequalities How can the solution set be identified?

y>x²-2

y ≥ -x²+5

The quadratic function f has a Minimum value of -3 at x = 2, taken from the given graph.

A parabola, a U-shaped curve, is the shape of a quadratic function's graph. The graph's vertex, which is an extreme point, is one of its key characteristics.

Given data:

As, the U shaped graph opens upwards, it has the minimum value at the turning point.

Thus, the quadratic function f has a Minimum value of -3 at x = 2, taken from the given graph.

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We have verified the identity of the expression 1 + cos(x) / cot(x) + cos(x) = sec(x) using the Pythagorean Identity.

The mathematical subject of trigonometry is the study of the connections between the angles and sides of triangles.

let's rewrite the expression using sine and cosine:

1 + cos(x) / cot(x) + cos(x) = sec(x)

cot(x) is the same as cos(x)/sin(x), so we can substitute:

1 + cos(x) / (cos(x)/sin(x)) + cos(x) = sec(x)

Simplifying the expression in the denominator:

1 + cos(x) * (sin(x)/cos(x)) + cos(x) = sec(x)

Canceling out the cos(x) terms in the numerator:

1 + sin(x) + cos(x) = sec(x)

Now, let's use the Pythagorean Identity:

sin²(x) + cos²(x) = 1

Divide both sides by cos²(x):

tan²(x) + 1 = sec²(x)

Substitute tan(x) for sin(x)/cos(x):

(sin²(x)/cos²(x)) + 1 = sec²(x)

Multiply both sides by cos²(x):

sin²(x) + cos²(x) = cos²(x)sec²(x)

Substitute cos²(x) with 1 - sin²(x):

sin²(x) + (1 - sin²(x)) = (1 - sin²(x))sec²(x)

Simplify:

1 = sec²(x)

Taking the square root of both sides:

1 = sec(x)

Therefore, the identity is verified.

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