What is the maximum of the sinusoidal function?

Enter your answer in the box.

The real zeros of this function are 0 (multiplicity 4),-8 (multiplicity 2), and 2 (multiplicity 4).

The real zeros of the function occur when any of the factors is equal to zero:

x^(4) = 0, (x-2)^(4) = 0, or (x+8)^(2) = 0

To find the real zeros of the given function, we need to set the function equal to zero and solve for x:

f(x) = x^(4)(x-2)^(4)(x+8)^(2) = 0

The process to find these zeros is as follows:

1. Set f(x) = 0 and solve for x

2. Factor the polynomial and solve each factor:

f(x) = 0 => x^(4)(x-2)^(4)(x+8)^(2) = 0
           => x4 = 0
           => x = 0 (multiplicity 4)

           => (x-2)4 = 0
           => x = 2 (multiplicity 4)

           => (x+8)2 = 0
           => x = -8 (multiplicity 2)

The multiplicity of a zero is the number of times it appears as a factor in the function. In this case, the zero 0 has a multiplicity of 4, the zero 2 has a multiplicity of 4, and the zero -8 has a multiplicity of 2.

Therefore, the real zeros and their multiplicities are:
0 with a multiplicity of 4
2 with a multiplicity of 4

-8 with a multiplicity of 2

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The given set of vectors is: \( \left\{\left[\begin{array}{c}1 \\ 0 \\ -2\end{array}\right],\left[\begin{array}{c}-1 \\ 1 \\ 4\end{array}\right],\left[\begin{array}{c}1 \\ 2 \\ -2\end{array}\right]\right\} \)

To determine if the given set of vectors is linearly independent or linearly dependent, we can use the determinant method. We will form a matrix using the given vectors as columns and then find the determinant of the matrix. If the determinant is zero, then the vectors are linearly dependent. If the determinant is not zero, then the vectors are linearly independent.

The matrix formed using the given vectors as columns is:
\[ \left[\begin{array}{ccc}1 & -1 & 1 \\ 0 & 1 & 2 \\ -2 & 4 & -2\end{array}\right] \]
The determinant of the matrix is:
\[ \begin{vmatrix}1 & -1 & 1 \\ 0 & 1 & 2 \\ -2 & 4 & -2\end{vmatrix} = (1)(1)(-2) + (-1)(2)(-2) + (1)(0)(4) - (1)(2)(4) - (-1)(0)(-2) - (1)(1)(-2) = -2 + 4 + 0 - 8 + 0 + 2 = -4 \]

Since the determinant is not zero, the given set of vectors is linearly independent.

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47. The identity that expresses cot(x) in terms of csc(x) is: cot(x) = 1/sin(x) = csc(x)/sin(x) * sin(x)/sin(x) = csc(x)cos(x)

48. The identity that expresses sec(x) in terms of tan(x) is: sec(x) = 1/cos(x) = 1/cos(x) * sin(x)/sin(x) = sin(x)/(cos(x)sin(x)) = sin(x)/sin(x)cos(x) = 1/cos(x) = sec(x)

49. The identity that expresses sin(x) in terms of cot(x) is: sin(x) = 1/csc(x) = 1/csc(x) * cos(x)/cos(x) = cos(x)/(csc(x)cos(x)) = cos(x)/cos(x)csc(x) = 1/csc(x) = sin(x)

50. The identity that expresses cos(x) in terms of tan(x) is: cos(x) = 1/sec(x) = 1/sec(x) * cos(x)/cos(x) = cos(x)/(sec(x)cos(x)) = cos(x)/cos(x)sec(x) = 1/sec(x) = cos(x)

51. The identity that expresses tan(x) in terms of csc(x) is: tan(x) = sin(x)/cos(x) = sin(x)/cos(x) * 1/csc(x) = sin(x)csc(x)/cos(x) = 1/cos(x) = sec(x)

52. The identity that expresses cot(x) in terms of sec(x) is: cot(x) = 1/tan(x) = 1/(sin(x)/cos(x)) = cos(x)/sin(x) = cos(x)/sin(x) * 1/sec(x) = cos(x)sec(x)/sin(x) = 1/sin(x) = csc(x)

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Answer:

~7.8

Step-by-step explanation:

side 1 = 5

side 2 = 6

hypotenuse = h

h² = 5² + 6²

h² = 25 + 36

h² = 61

h = [tex]\sqrt{61}[/tex]

h = 7.8102...

The marginal profit at a production level of 40 items is 2064.

To find the marginal profit at a production level of 40 items, we need to first find the marginal cost and marginal revenue at this production level. The marginal cost and marginal revenue are the derivatives of the cost and revenue functions, respectively.

The marginal cost function is:
C'(q) = 3q^2 - 22q + 56

The marginal revenue function is:
R'(q) = -6q + 2600

At a production level of 40 items, the marginal cost is:
C'(40) = 3(40)^2 - 22(40) + 56 = 296

The marginal revenue at this production level is:
R'(40) = -6(40) + 2600 = 2360

The marginal profit is the difference between the marginal revenue and marginal cost:
Marginal profit = 2360 - 296 = 2064

Therefore, the marginal profit at a production level of 40 items is 2064.

Answer :[tex]\boxed{2064}[/tex].

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The transformations required to transform the function S=√L into the function S=15.9√0.80L are a vertical stretch by a factor of 15.9 and a horizontal compression by a factor of 0.80. The radical function S=√L can be transformed into the function S=15.9√0.80L by applying two transformations: a vertical stretch by a factor of 15.9 and a horizontal compression by a factor of 0.80.

First, the vertical stretch is applied by multiplying the radical function by 15.9. This stretches the graph of the function vertically by a factor of 15.9, resulting in the function S=15.9√L.

Next, the horizontal compression is applied by multiplying the variable L by 0.80 inside the radical. This compresses the graph of the function horizontally by a factor of 0.80, resulting in the function S=15.9√0.80L.

Therefore, the transformations required to transform the function S=√L into the function S=15.9√0.80L are a vertical stretch by a factor of 15.9 and a horizontal compression by a factor of 0.80.

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The percentage that is less than the average number of shoppers in the original store at any time is 60%.

Little's law is a fundamental principle in queueing theory that relates the average number of customers in a stable system to the average time that a customer spends in the system.

The law states that the average number of customers N in the system is equal to the average rate of customer arrivals r multiplied by the average time W that a customer spends in the system:

Here we have

For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes.

=> Number of shoppers per minute = 1.5    

=> Rate of shoppers per minute = 1.5

The manager estimates that each shopper stays in the store for an average of 12 minutes.

Hence, by Little’s law, the number of shoppers N = r × t

=> Number of shoppers = (1.5) × 12 = 18  

Let the estimated average number of shoppers in the original store at any time be 45.    

So, the number of shoppers is (45 - 18) less than the original i.e 27

Percentage  [ 27/45 ] × 100 = 60%  

Therefore,

The percentage that is less than the average number of shoppers in the original store at any time is 60%.

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Answer:

The height of the building is approximately 55 meters.

Step-by-step explanation:

Let's call the height of the building "h" and the distance from point P to the foot of the building "d".

According to the problem, we have:

d = 50m

T = 23.6m

Using the Pythagorean theorem, we know that:

h^2 = T^2 + d^2

Substituting the values we have:

h^2 = (23.6m)^2 + (50m)^2

h^2 = 556.96m^2 + 2500m^2

h^2 = 3056.96m^2

Taking the square root of both sides, we get:

h = sqrt(3056.96m^2)

h = 55.28m

Rounding to the nearest meter, we get:

h ≈ 55m

Therefore, the height of the building is approximately 55 meters.

The mass of the P₂O₅ formed from 3.4 moles of O₂ gas is equal to 386.1 g.

A mole can be described as a unit for measurement of a huge number of quantities of atoms, molecules, ions, or other particles. The atomic mass can be expressed as the one mole of any element.

The number of particles present in one mole was to be equal to 6.023 × 10 ²³ which is Avogadro’s constant.

Given, the number of moles of O₂ gas = 3.4 moles

The balanced chemical reaction of phosphorous and oxygen gas can be given as:

4 P  +  5 O₂   →   2 P₂O₅

5 mol of Oxygen reacts with moles of P₂O₅ = 2 mol

3.4 mol of O₂ gas reacts with moles of P₂O₅ = (2/5) × 3.4 = 1.36 mol

The mass of the 1.36 mol of P₂O₅ = 1.36 × 283.89 = 386.1 g

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The value of  ∂u∂f​​(u,v)=(−2,−2) = -20 and ∂v∂f​​(u,v)=(−2,−2) = -45.

To evaluate the partial derivatives ∂u∂f​​(u,v)=(−2,−2) and ∂v∂f​​(u,v)=(−2,−2), we can use the Chain Rule. We first take the partial derivatives of f(x,y,z) with respect to x, y, and z, and then substitute in x = u2 + v, y = u + v2, and z = 4uv.

First, ∂f∂x​=3x2, ∂f∂y​=yz and ∂f∂z​=y2.

Substituting in x = u2 + v, y = u + v2, and z = 4uv gives us: ∂f∂x​=3(u2 + v)2, ∂f∂y​=(u + v2)(4uv) and ∂f∂z​=(u + v2)2.

Next, we use the Chain Rule to find ∂u∂f​​(u,v)=(−2,−2) and ∂v∂f​​(u,v)=(−2,−2):

∂u∂f​​(u,v)=(−2,−2)= ∂f∂x​•∂x∂u​​ + ∂f∂y​•∂y∂u​​ + ∂f∂z​•∂z∂u​​ = 3(u2 + v)2•(2u) + (u + v2)(4uv)•(1) + (u + v2)2•(4v) = 6u2 + 8uv + 4uv + 4v2 = 10uv + 6u2 + 4v2

∂v∂f​​(u,v)=(−2,−2)= ∂f∂x​•∂x∂v​​ + ∂f∂y​•∂y∂v​​ + ∂f∂z​•∂z∂v​​ = 3(u2 + v)2•(1) + (u + v2)(4uv)•(2v) + (u + v2)2•(4u) = 3 + 8uv + 8u2 = 8u2 + 8uv + 3

When (u,v)=(−2,−2), we have ∂u∂f​​(u,v)=(−2,−2) = 10(-2)(-2) + 6(-2)2 + 4(-2)2 = 20 - 24 - 16 = -20 and ∂v∂f​​(u,v)=(−2,−2) = 8(-2)2 + 8(-2)(-2) + 3 = -32 - 16 + 3 = -45.

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The values of y that make the inequality 5y<41 true are B. 7 and D. 6.5.

To find the values of y that make the inequality true, we can first isolate y by dividing both sides of the inequality by 5:

5y<41

y<41/5

y<8.2

This means that any value of y less than 8.2 will make the inequality true.

Looking at the given options, we can see that B. 7 and D. 6.5 are both less than 8.2, so they are the correct answers.

A. 8 and C. 8.5 are both greater than or equal to 8.2, so they do not make the inequality true. E. 9 is also greater than 8.2, so it does not make the inequality true.

Therefore, the correct answers are B. 7 and D. 6.5.

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Answer:

A. 6

Step-by-step explanation:

To solve the equation 6x^2-2x+36=5x^2+10x, we can follow these steps:

Move all the terms to one side of the equation by subtracting 5x^2 and 10x from both sides:

6x^2 - 2x + 36 - 5x^2 - 10x = 0

Simplifying the left side:

x^2 - 12x + 36 = 0

Factor the quadratic expression on the left side of the equation:

(x - 6)(x - 6) = 0

Apply the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:

x - 6 = 0

Solve for x:

x = 6

The solution to the equation 6x^2-2x+36=5x^2+10x is x = 6.

Answer:

hypothesis of the medicine is incomplete

Step-by-step explanation:

so we have.to apply bodmas

When the the doctor's claim is correct, the probability that exactly 19 patients will recover when given the new medicine is 0.1444.

a) The suitable distribution to model the number of patients in this sample who recover when given the new medicine is the binomial distribution. We can assume that each patient has a fixed probability of success (i.e., recovering) and that the outcomes of each patient are independent of each other.

b) If the claim is correct, then the probability of success (recovering) for each patient is p = 0.8. Let X be the number of patients in the sample who recover when given the new medicine. Then X follows a binomial distribution with parameters n = 25 (the sample size) and p = 0.8 (the probability of success).

The probability of exactly 19 patients recovering is given by:

P(X = 19) = (25 choose 19) * (0.8)^19 * (1-0.8)^(25-19)

where (25 choose 19) is the binomial coefficient, which represents the number of ways to choose 19 patients out of 25.

Using a calculator, we can compute:

P(X = 19) = 0.1444

Therefore, if the doctor's claim is correct, the probability that exactly 19 patients will recover when given the new medicine is 0.1444.

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The Heston stochastic volatility model is a popular model used to describe the dynamics of an asset price in the presence of stochastic volatility. It is a two-factor model that accounts for the random nature of both the asset price and its volatility. The model is given by the following set of stochastic differential equations:

dS(t) = rS(t)dt + Vo(t)S(t)dWi(t)

de(t) = (a – bo(t) dt +0V (t)dWx(t)

where S(t) is the asset price, r is the risk-free rate, V(t) is the stochastic volatility, W1(t) and W2(t) are Brownian motions under the risk-neutral probability measure, and a, b, o are positive constants. The correlation between the two Brownian motions is given by p, which is a constant between -1 and 1.

The Heston model is widely used in finance because it can capture the volatility smile, which is the tendency for options with different strike prices to have different implied volatilities. This feature is important because it allows for more accurate pricing of options and other derivative securities.

To solve the Heston model, we can use the Feynman-Kac theorem, which relates the solution of a stochastic differential equation to the solution of a partial differential equation. This allows us to find the price of an option under the Heston model by solving a partial differential equation. The solution can be found using numerical methods, such as the finite difference method or the Monte Carlo method.

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The standard form of the equation of a quadratic function with roots of 4 and −1 that passes through (1, −9) is [tex]y = 1.5x^{2} - 4.5x - 6[/tex]

What is the quadratic function?

A quadratic function is a type of function that can be written in the form:

[tex]f(x) = ax^2 + bx + c[/tex]

where a, b, and c are constants, and x is the variable. This function is a second-degree polynomial function, which means that the highest power of the variable x is 2.

Quadratic functions can be graphed as a U-shaped curve called a parabola. The sign of the coefficient a determines whether the parabola opens up or down. If a > 0, the parabola opens up, and if a < 0, the parabola opens down. The vertex of the parabola is the minimum or maximum point of the function, depending on whether the parabola opens up or down.

Quadratic functions are used in many areas of mathematics, science, and engineering to model various phenomena such as projectile motion, population growth, and optimization problems.

To write the standard form of the equation of a quadratic function, we need to use the roots of the function and another point on the curve. The standard form of the quadratic function is:

y = a(x - r1)(x - r2)

where r1 and r2 are the roots of the quadratic function, and a is a constant.

Given that the roots of the quadratic function are 4 and -1, we can write:

y = a(x - 4)(x + 1)

To find the value of a, we can use the point (1, -9) that the function passes through:

-9 = a(1 - 4)(1 + 1)

-9 = -6a

a = 3/2

Substituting this value of a in the equation, we get:

[tex]y = 1.5(x - 4)(x + 1)[/tex]

Expanding this equation, we get:

[tex]y = 1.5x^{2} - 4.5x - 6[/tex]

Therefore, the standard form of the equation of the quadratic function with roots of 4 and −1 that passes through (1, −9) is [tex]y = 1.5x^{2} - 4.5x - 6[/tex]

So, the correct answer is: [tex]y = 1.5x^{2} - 4.5x - 6[/tex]

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19). The probability that a student attained a B or a C on the quiz is 0.42, or 42%. 20). The events for selecting the balls are dependent. This is because the probability of selecting the second ball is affected by the selection of the first ball.

The probability that a student attained a B or a C on the quiz can be found by adding the probabilities of the two events together. First, find the probability of attaining a B by dividing the number of students who attained a B by the total number of students. Do the same for the probability of attaining a C. Then add the two probabilities together to find the total probability.

Probability of attaining a B = (Number of students who attained a B) / (Total number of students) = 17 / 100 = 0.17
Probability of attaining a C = (Number of students who attained a C) / (Total number of students) = 25 / 100 = 0.25
Total probability = 0.17 + 0.25 = 0.42
Therefore, the probability that a student attained a B or a C on the quiz is 0.42, or 42%.

After the first ball is selected, there is one less ball in the bin, which changes the probability of selecting the second ball. Therefore, the events are dependent.

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The area of John's actual bedroom is 120 square feet.

What is Scale factor?

Scale factor is a mathematical concept that is used to describe the ratio of corresponding dimensions of two similar figures. In geometry, two figures are said to be similar if they have the same shape but possibly different sizes. The scale factor is the ratio of the length of a side (or any corresponding dimension) of one figure to the length of the corresponding side (or dimension) of the other figure.

If John used a scale of 3 inches = 2 feet, then we can convert the dimensions of the drawing to the actual dimensions of his bedroom using the scale factor:

1 inch on the drawing corresponds to 2/3 feet in reality.

So, the actual width of the bedroom is:

15 inches × (2/3 feet/inch) = 10 feet

And the actual length of the bedroom is:

18 inches × (2/3 feet/inch) = 12 feet

The area of the bedroom is the product of its actual length and width:

Area = length × width = 12 feet × 10 feet = 120 square feet

Therefore, the area of John's actual bedroom is 120 square feet.

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The equation of the perpendicular bisector of the line segment with endpoints (1,6) and (-9,-2) is y = (-5/4)x - 3.

Any location on the perpendicular bisector is equally spaced from the line segment's terminal points, according to the perpendicular bisector theorem.

These procedures must be taken in order to determine the equation of a line segment's perpendicular bisector:

Determine the line segment's midway.

Determine the line segment's slope.

In order to determine the slope of the perpendicular bisector, calculate the negative reciprocal of the slope.

To determine the equation of the perpendicular bisector, use a line's point-slope form.

These procedures allow us to determine the equation for the perpendicular bisector of the line segment with ends (1, 6) and (-9, -2), which is as follows:

Midpoint: The midpoint of the line segment can be found by taking the average of the x-coordinates and the average of the y-coordinates:

Midpoint = ((1 + (-9))/2, (6 + (-2))/2)

= (-4, 2)

Slope: The slope of the line segment can be found using the formula:

slope = (change in y) / (change in x)

slope = (6 - (-2)) / (1 - (-9))

= 8/10

= 4/5

Negative reciprocal: The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment:

slope of perpendicular bisector = -1 / slope

= -1 / (4/5)

= -5/4

Equation: We can now use the point-slope form of a line to find the equation of the perpendicular bisector. We will use the midpoint of the line segment as the point on the line:

y - y1 = m(x - x1)

where m is the slope of the perpendicular bisector, and (x1, y1) is the midpoint of the line segment. Substituting the values we found, we get:

y - 2 = (-5/4)(x + 4)

Simplifying, we can write the equation in slope-intercept form:

y = (-5/4)x - 3

Hence, the equation of the perpendicular bisector of the line segment with endpoints (1,6) and (-9,-2) is y = (-5/4)x - 3.

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Answer:

Step-by-step explanation:

plug in 9 for n

2(n+4) becomes 2(9+4) = 26

Answer:

Step-by-step explanation: i think 26???

As a result, 25% of chlorine is present in solution B.

It's 100 × 20 / 100 = 20% ! In this circumstance, percentages are useful. When describing a change from one % to another, we use percentage points. The difference between 10% and 12% is two percent respectively (or 20 percent).

Let x represent the chlorine content of solution B.

The entire quantity of chlorine inside the combination can be used to create the following equation as a starting point:

3 cups of solution A × 10% chlorine + 6 cups of solution B × x% chlorine = (3+6) cups of new solution × 20% chlorine

When we simplify this equation, we obtain:

0.3 + 6x = 1.8

By taking 0.3 away both from sides, we get at:

6x = 1.5

When we multiply both parts by 6, we get:

x = 0.25

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Anne prediction on the amount of rain that will pour down is 60 inches

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

The amount of rain this year can be calculated as follows

Percentage = 20/100 = 0.2

So, we have

Proportion = 0.2 + 1  =  1.2

This gives

Amount = 1.2 × 50 = 60

Hence the amount of rain this year is 60 inches

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The statement  that best describes the process to find the product of rational expressions is: D. Multiply the numerators and then multiply the denominators.

The process to find the product of rational expressions involves multiplying the two expressions together. This can be done by multiplying the numerators of the two expressions and then multiplying the denominators of the two expressions.

For example, if we want to find the product of the rational expressions (2x + 3)/(x - 1) and (x + 2)/(3x), we can do the following:

(2x + 3)/(x - 1) * (x + 2)/(3x) = (2x + 3)(x + 2) / (x - 1)(3x)

To find the product, we multiplied the numerators (2x + 3) and (x + 2) to get (2x + 3)(x + 2), and we multiplied the denominators (x - 1) and (3x) to get (x - 1)(3x).

Therefore, the correct option is: Multiply the numerators and then multiply the denominators.

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The sample proportions of male births that are at least as extreme as the sample proportion of 472/960 are 0.0026 for both the lower and upper tails.

A proportion is an equation in which two ratios are set equal to each other.

Part 1:

a. The values of p and q can be identified as follows:

p = proportion of male births in the population = 0.512

q = proportion of female births in the population = 1 - p = 1 - 0.512 = 0.488

So, p = 0.512 and q = 0.488.

Part 2:

b. The sample proportion of male births is:

P = 472/960 = 0.4917

To find the sample proportions of male births that are at least as extreme as this value, we need to calculate the z-scores corresponding to the upper and lower tails of the distribution under the null hypothesis (i.e., p = 0.512). The formula for the z-score is:

z = (P - p) / √(p*q/n)

where n is the sample size.

For the lower tail, we have:

z = (0.4917 - 0.512) / √(0.512*0.488/960) = -2.79

For the upper tail, we have:

z = (0.512 - 0.4917) / √(0.512*0.488/960) = 2.79

Using a standard normal distribution table or calculator, we can find the probabilities associated with these z-scores:

P(z < -2.79) = 0.0026

P(z > 2.79) = 0.0026

Hence, the sample proportions of male births that are at least as extreme as the sample proportion of 472/960 are 0.0026 for both the lower and upper tails.

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Complete Question:

A random sample of 960 births in New York State included 472 ​boys and that sample is to be used for a test of the common belief that the proportion of male births in the population is equal to

0.512.

Complete parts​ (a) through​ (c).

Question content area bottom

Part 1

a. In testing the common belief that the proportion of male babies is equal to 0.512​, identify the values of p and p. p=enter your response here p=enter your response here

Part 2

b. For random samples of size 960​,

what sample proportions of male births are at least as extreme as the sample proportion of 472960​?

Answer: [tex]\sqrt{3}[/tex]

Step-by-step explanation:

take,

x^2 = Y

6Y^2 + 8Y = 26Y

6Y^2 + 8Y - 26Y =  0

6Y^2 - 18Y = 0

6Y (Y - 3) = 0

(Y - 3) = 0

Y  = 3

x^2 = y

x^2 = 3

x = [tex]\sqrt{3}[/tex]

Bar chart to display the data of number of steps of Jamal is attached below.

A bar chart is a chart that displays complete data in rectangular bars, with the height of the bars proportional to the values ​​they represent. Chart bars  can be displayed vertically or horizontally. A bar chart, also called a bar graph, is a graphical representation of grouped data. It is a method of data processing. Bar charts are great for displaying data independent of each other and do not need to be in any particular order when displayed.

Given data,

Jamal keeps a log every day on the daily number of steps he walks.

Daily steps:

3500 , 4500 , 2000 , 2500 , 5000 , 5250 , 9000​

the number of steps is independent to the number of days, it can be graphed as a bar chart.

the bar graph can be drawn as following:

Hence, the data can be displayed as a bar chart attached below.

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Answer:

The stock market goes up 70 1/2 points at the beginning of the day, so we can represent the starting value as:

Starting value = 70 1/2

The stock market stays at this level for most of the day, so the value remains the same until the end of the day.

At the end of the day, the stock market goes down 120 3/4 points from the high at the beginning of the day. To calculate the ending value, we need to subtract this decrease from the starting value:

Ending value = Starting value - Decrease

Ending value = 70 1/2 - 120 3/4

To subtract the two values, we need to convert them to a common fraction with a common denominator of 4:

70 1/2 = 141/2

120 3/4 = 483/4

Ending value = 141/2 - 483/4

Ending value = 282/4 - 483/4

Ending value = -201/4

Therefore, the total change in the stock market from the beginning of the day to the end of the day is:

Ending value - Starting value = (-201/4) - (141/2) = -201/4 - 282/4 = -483/4

The total change in the stock market from the beginning of the day to the end of the day is a decrease of 483/4 points.

The domain of \(f(x)=\tan ^{-1} x\) is \(\mathbb{R}\) and the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Explanation:

The domain of a function is the set of all possible input values for which the function is defined. The inverse tangent function, \(f(x)=\tan ^{-1} x\), is defined for all real numbers, so the domain is \(\mathbb{R}\).

The range of a function is the set of all possible output values for which the function is defined. The inverse tangent function, \(f(x)=\tan ^{-1} x\), has a range of \((-\frac{\pi}{2}, \frac{\pi}{2})\). This is because the tangent function has a period of \(\pi\), and the inverse tangent function is the inverse of the tangent function restricted to the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Therefore, the domain of \(f(x)=\tan ^{-1} x\) is \(\mathbb{R}\) and the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\).

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Local Minima is -1.

A. f has a local minimum at x = -1; the local minimum is -1.

To find the local minima of a function, we must identify any points at which the slope of the graph is equal to zero. In this graph, the slope of the graph is equal to zero at x = -1, indicating that f has a local minimum at x = -1. The value of the function at this point is -1, so the local minimum is -1.

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1 at each vertex

An exterior angle of a polygon is an angle formed by a side of the polygon and the extension of an adjacent side. In the case of a triangle, each exterior angle is formed by one of the triangle's sides and the extension of an adjacent side.

When an exterior angle is formed at a vertex of a polygon, the measure of the exterior angle is equal to the sum of the measures of the two interior angles adjacent to it. In the case of a triangle, the sum of the measures of the two interior angles adjacent to the exterior angle is always 180 degrees (which is the sum of the measures of all three interior angles of a triangle).

Since each exterior angle of a triangle is formed by two interior angles, and the sum of the measures of those interior angles is always 180 degrees, there can only be one exterior angle at each vertex of a triangle. Therefore, a triangle has one exterior angle at each vertex.