The radius of a spherical balloon can be expressed as r= ³√3V / 4π inches, where r is the radius and V is the volume of the balloon in cubic inches. If air is pumped to inflate the balloon from 500 cubic inches to 800 cubic inches, by how many inches has the radius of the balloon increased?

a. What was the radius of the balloon originally?

b. What was the radius after inflating the balloon to 800 cubic inches?

c. How can you use the two radii to find the amount of increase?

To describe the family of quadratic functions that have zeros at r and s, we can start with the general form of a quadratic function:

f(x) = a(x - r)(x - s)
In this form, r and s are the zeros of the function. To sketch several members of the family in the coordinate plane, we can choose different values for a and plot the corresponding graphs.Let's start by considering a = 1. For example, if r = 2 and s = -3, the quadratic function becomes:

We can plot the graph of this function by choosing some values for x, calculating the corresponding y-values, and connecting the points. Repeat this process for different values of a to sketch several members of the family.

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The family of quadratic functions with zeros at $r$ and $s$ can be described by the equation $f(x) = a(x - r)(x - s)$. By varying the values of $a$, $r$, and $s$, we can sketch different members of this family in the coordinate plane.

The family of quadratic functions with zeros at $r$ and $s$ can be described by the equation:

\[f(x) = a(x - r)(x - s)\]

In this equation, $a$ is a constant that determines the shape and orientation of the quadratic function. The zeros of the function occur at the values of $x$ where $f(x) = 0$, which in this case are $r$ and $s$.

To sketch different members of this family in the coordinate plane, we can choose different values of $a$, $r$, and $s$. Let's illustrate this with an example:

Suppose we choose $a = 1$, $r = 2$, and $s = -3$. The equation becomes:

\[f(x) = (x - 2)(x - (-3))\]

Expanding this equation, we get:

\[f(x) = (x - 2)(x + 3) = x^2 + x - 6\]

To sketch the graph of this quadratic function, we can plot points by selecting different values of $x$ and calculating the corresponding $y$-values. For example, when $x = -2$, we have:

\[f(-2) = (-2)^2 + (-2) - 6 = 0\]

So the point $(-2, 0)$ lies on the graph. Similarly, we can choose other values of $x$ and find the corresponding $y$-values to plot more points. By connecting these points, we obtain the graph of the quadratic function.

To explore different members of the family, we can vary the values of $a$, $r$, and $s$. For instance, if we change $a$ to 2, the graph will become narrower. If we change $r$ and $s$ to -1 and 4, respectively, the graph will shift to the right.

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It is not reasonable to use the line of best fit to make predictions from the data.

A correlation coefficient of r = 0.15 simply suggests to us that the goodness of fit for the given data from the  linear regression analysis is

r² = 0.15² = 0.0225.

Now, when r² = 1, it tells us that the correlation between the dependent and independent variable is perfect, and that the dependent variable can be predicted reliably.

When r² = 0, it means that there is absolutely no correlation between the dependent and independent variable.

When r² > 0.9, it means that reasonable correlation exists between the dependent and independent variable.

A value of r²=0.0225 is extremely low, therefore it is not reasonable to use the line of best fit to make predictions.

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Complete question is:

A set of data has a correlation coefficient r = 0.15. Is it reasonable to use the line of best fit to make predictions from the data?

To find the constant rate of change of y with respect to x, we can use the formula for slope in a linear relationship. Therefore, when x = 2, the value of y is 9.

The formula for slope (m) is given by:

m = (y2 - y1) / (x2 - x1)

Given the points (-3, -1) and (3, 11), we can substitute the coordinates into the formula to find the slope:

m = (11 - (-1)) / (3 - (-3))

 = (11 + 1) / (3 + 3)

 = 12 / 6

 = 2

Therefore, the constant rate of change of y with respect to x is 2.

To express y in terms of x, we can use the point-slope form of a linear equation. Using the point (-3, -1), we have:

y - y1 = m(x - x1)

y - (-1) = 2(x - (-3))

y + 1 = 2(x + 3)

y + 1 = 2x + 6

y = 2x + 5

The formula expressing y in terms of x is y = 2x + 5.

To find the value of y when x = 2, we substitute x = 2 into the equation:

y = 2(2) + 5

y = 4 + 5

y = 9

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The point on the graph of the inverse function g corresponding to the point (a, b) on the graph of the one-to-one function f is (b, a).

Let g denote the inverse function of f. Then the point (g(y), y) lies on the graph of f if and only if the point (y, g(y)) lies on the graph of g. If (a, b) is a point on the graph of f, then g(a) = b, so (a, b) and (b, a) lie on the graphs of f and g, respectively. Therefore, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of g.

The inverse of a one-to-one function is also a function and can be obtained by interchanging the variables and solving for y. Given a point (a, b) on the graph of the one-to-one function, it implies that f(a) = b. The inverse function g is obtained by interchanging x and y, so g(b) = a. Hence, the point on the graph of g is (b, a) if (a, b) is a point on the graph of f.

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No, the F-test statistic cannot be less than 1 when it represents the larger of two sample variances.

The F-distribution is always a positive distribution with a range of values greater than zero. The F-test value is always greater than or equal to 0, but it is not always less than or equal to 1.

The numerator, representing the bigger sample variance, is always greater than or equal to the denominator, representing the smaller sample variance. Consequently, the F-ratio, which is the ratio of the larger to the smaller variance, is always greater than or equal to 1. Thus, an F-test statistic that is less than 1 is mathematically incorrect.

Therefore, an F-test statistic is always a positive number that is greater than or equal to zero and greater than or equal to 1. This is because the numerator of an F-distribution is always greater than or equal to the denominator.

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The rationalized form of (3 + √8) / (2 - 2√8) is - (11 + 4√8)/14.

Given the number with irrational denominator is,

(3 + √8) / (2 - 2√8)

So the numerator is = 3 + √8

and the denominator is = 2 - 2√8

Now calculating the product when multiplying denominator and numerator with 2 + 2√8 are,

(2 - 2√8) (2 + 2√8)

= (2)² - (2√8)² [Since, a² - b² = (a + b) (a - b)]

= 4 - 4*8

= 4 - 32

= -28

and,

(3 + √8) (2 + 2√8)

= 6 + 2√8 + 6√8 + 2√8*√8

= 6 + 8√8 + 16

= 22 + 8√8

Rationalizing the given number we get,

(3 + √8)(2 + 2√8)  / (2 - 2√8)(2 + 2√8)

= (22 + 8√8)/(-28)

= - (11 + 4√8)/14

Hence the rationalized form is = - (11 + 4√8)/14.

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Conjecture: The pattern in the sequence of follow-up visits is that they occur in a cyclical pattern, alternating between months that are five months apart and months that are seven months apart. Next item in the sequence: Aug.

Conjecture: The pattern in the sequence of follow-up visits is that they occur in a cyclical pattern, alternating between months that are five months apart and months that are seven months apart.

Based on this conjecture, the next item in the sequence would be August.

Explanation: Let's analyze the given sequence of follow-up visits: Dec., May, Oct., Mar.,...

The first observation is that the months in the sequence are not consecutive. Instead, they follow a cyclical pattern. The pattern alternates between months that are five months apart and months that are seven months apart.

Starting with December (12th month), the next follow-up visit occurs in May (5 months apart). The subsequent follow-up visit happens in October (7 months apart from May). Following this, the pattern continues with the next visit in March (5 months apart from October).

Based on this pattern, the next follow-up visit should occur in a month that is seven months apart from March. Counting seven months forward from March, we arrive at August. Therefore, the next item in the sequence would be August.

It is important to note that this conjecture assumes the pattern will continue in the same cyclical manner. However, without further information or confirmation, it is always advisable to gather additional data or consult the source that provided the sequence to ensure the accuracy of the conjecture and predict future items in the sequence.

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Therefore, the probability that the team is comprised of 2 men and 2 women is approximately 0.4545, which is true.

To calculate the probability that the team is comprised of 2 men and 2 women, we need to determine the number of favorable outcomes (teams with 2 men and 2 women) and the total number of possible outcomes (all possible teams).

The number of ways to choose 2 men from 6 men is given by the combination formula: C(6, 2) = 6! / (2! * (6-2)!) = 15.

Similarly, the number of ways to choose 2 women from 6 women is also C(6, 2) = 15.

Therefore, the number of favorable outcomes is the product of the number of ways to choose 2 men and 2 women: 15 * 15 = 225.

Now, let's calculate the total number of possible outcomes. We need to choose 4 members from a total of 12 members, which is given by C(12, 4) = 12! / (4! * (12-4)!) = 495.

The probability that the team is comprised of 2 men and 2 women is the ratio of favorable outcomes to the total number of outcomes: 225/495 = 45/99 ≈ 0.4545 (rounded to four decimal places).

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The 90% confidence interval for the mean GPA of all accounting students at the university is (2.765, 2.835). We are given a random sample of 36 accounting students with a mean of 2.80 and a standard deviation of 0.12.

To construct the confidence interval, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √Sample Size)

Since the population is normally distributed, we can use the z-distribution. For a 90% confidence level, the critical value is approximately 1.645.

Plugging in the values, we have:

Confidence Interval = 2.80 ± (1.645) * (0.12 / √36)

Calculating the values, we get:

Confidence Interval = 2.80 ± 0.035

Therefore, the 90% confidence interval for the mean GPA of all accounting students at the university is (2.765, 2.835). This means that we are 90% confident that the true mean GPA of all accounting students falls within this interval.

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The equation Paloma solved is 6x + 5 = 3(2x + 1). After solving the equation, she obtained the result 2 = 0.

The meaning of her result, 2 = 0, is that there is no solution to the equation. In other words, the equation is inconsistent or contradictory. When the left side of the equation does not equal the right side for any value of x, the result is that the equation has no solution.

In this case, Paloma likely made an error in her calculations or made an incorrect assumption along the way, leading to an inconsistent result. The equation 6x + 5 = 3(2x + 1) should have a valid solution, but her result of 2 = 0 indicates that there is no solution that satisfies the equation.

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The expression (4+√-25)√(-100) can be simplified using the imaginary unit i to -4i.

Let's break down the expression step by step. First, we simplify the square root of -25, which can be written as √(-1 * 25). Applying the property of the square root, we get √(-1) * √(25) = i * 5 = 5i.

Next, we have (4+5i)√(-100). Similar to the previous step, we simplify the square root of -100 as √(-1 * 100). Again applying the square root property, we have √(-1) * √(100) = i * 10 = 10i.

Now, we combine the two expression  : (4+5i) * 10i. Using the distributive property, we multiply each term inside the parentheses by 10i. This gives us [tex]40i + 50i^2.[/tex]

Since[tex]i^2[/tex] is equal to -1, we can substitute it into the expression: 40i + 50(-1) = 40i - 50 = -50 + 40i.

Finally, we rearrange the terms to obtain the simplified form: -50 + 40i = -4i. Therefore, the simplified expression is -4i.

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The surface area of the triangular prism is given by the expression b × h + 3bh + 2lw, where b, h, l, and w are the dimensions of the prism in centimeters.

To calculate the surface area of a triangular prism, we need to determine the areas of each of its faces and sum them up.

A triangular prism has two triangular bases and three rectangular faces.

Let's denote the dimensions of the triangular prism as follows:

- The base of the triangular face is denoted as b (in centimeters).

- The height of the triangular face is denoted as h (in centimeters).

- The length of one of the rectangular faces is denoted as l (in centimeters).

- The width of one of the rectangular faces is denoted as w (in centimeters).

The surface area of a triangular prism can be calculated using the formula:

Surface Area = 2 × Area of Triangular Base + Perimeter of Base × Height + 2 × Length × Width

The area of a triangle can be calculated using the formula:

Area of Triangle = (base × height) / 2

So, the surface area of the triangular prism is:

Surface Area = 2 × (0.5 × b × h) + (b + b + b) × h + 2 × l × w

Surface Area = b × h + 3bh + 2lw

Therefore, the surface area of the triangular prism is given by the expression b × h + 3bh + 2lw, where b, h, l, and w are the dimensions of the prism in centimeters.

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If "a" is an m × n matrix whose columns do not span Rm, then the equation ax = b is inconsistent for some b in Rm.

In linear algebra, the concept of spanning refers to the ability of a set of vectors to reach or cover an entire space. In this case, the columns of matrix "a" do not span Rm, which means they do not cover the entire m-dimensional space.

When we have an equation of the form ax = b, where "a" is an m × n matrix, x is an n-dimensional vector, and b is an m-dimensional vector, the equation represents a system of linear equations. The matrix "a" defines the coefficients of the variables, and b represents the constant terms.

If the columns of matrix "a" do not span Rm, it implies that the system of linear equations represented by ax = b does not have a solution for some values of b in Rm. This means that there exist m-dimensional vectors "b" for which the equation is inconsistent, meaning there is no solution that satisfies the equation.

Geometrically, this can be visualized as the columns of matrix "a" not covering the entire space Rm, resulting in a gap or missing region. Consequently, there are certain vectors "b" that lie outside the range or subspace spanned by the columns of "a," leading to an inconsistent system.

In summary, if the columns of matrix "a" do not span Rm, then the equation ax = b is inconsistent for some vectors "b" in Rm, meaning there is no solution that satisfies the equation for those particular values of "b."

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Answer:-5

Step-by-step explanation:

Answer: 30/(-6) = -5

The correct option is (B).

The given line is 3x - y = 6. We can transform this equation to the slope-intercept form y = mx + b, where m is the slope of the line and b is the y-intercept.

So, we have:3x - y = 6-y = -3x + 6y = 3x - 6/3 = 3x - 2

Thus, the slope of the given line is 3.

Since the line we're interested in is perpendicular to the given line, the slope of the new line will be the negative reciprocal of the slope of the given line.

So, the slope of the new line is:-1/3

Now, we have the slope of the new line and one point it passes through: (-3, 4).

We can use the point-slope form of the equation of a line to write the equation of the new line.

y - y1 = m(x - x1),where m is the slope of the line, and (x1, y1) is a point on the line.

Plugging in the values, we get: y - 4 = -1/3(x - (-3))y - 4 = -1/3(x + 3)y - 4 = -1/3x - 1y = -1/3x + 3

Thus, the equation of the line that passes through (-3, 4) and is perpendicular to 3x - y = 6 is : y = -1/3x + 3.

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In step 2, she didn’t use an x and y from the same coordinate pair

We can easily understand by the attachment of the question:

From the figure,

We have to find the slope

We know that, The formula of slope is:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

The formula to calculate the slope between two points and substitute the values:

[tex]m=\frac{5-1}{3-5}[/tex]

m = 4/-2

m = -2

The equation of the line in point slope form is:

[tex]y_2-y_1=m(x_2-x_1)[/tex]

Take the point (3,5)

m = -2

y - 5 = -2(x - 3)

y - 5 = -2x + 6

y = -2x + 6 + 5

y = -2x + 11

Therefore,  In step 2, she didn’t use an x and y from the same coordinate pair.

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The complete question is:

Helena wrote the equation using point-slope form for the line that passes through the points (5, 1) and (3, 5). Analyze the steps of her work using the point-slope form to write the equation. equation In which step did Helena make an error? In step 1, she substituted the x values for y and the y values for x. In step 2, she didn’t use an x and y from the same coordinate pair. In step 3, she solved for the wrong variable. Helena did not make an error.

The solution to the system of equations ΔXYZ is approximately X = 8.80, Y = 21.59, and Z = 8.53.

To solve the system of equations, we can use the method of substitution. Let's start by solving for one variable in terms of the others.

From the equation XY = 190, we can express X in terms of Y by dividing both sides by Y:

X = 190 / Y

Similarly, from the equation YZ = 184, we can express Z in terms of Y:

Z = 184 / Y

Substituting these expressions into the third equation, ZX = 75, we have:

(184 / Y) * (190 / Y) = 75

To simplify this equation, we can cross-multiply:

184 * 190 = 75 * Y^2

Simplifying further:

34960 = 75 * Y^2

Dividing both sides by 75:

Y^2 = 466.1333

Taking the square root of both sides:

Y ≈ ±21.59

Since we're looking for a positive value, we take Y ≈ 21.59.

Substituting this value back into the expressions for X and Z, we find:

X = 190 / 21.59 ≈ 8.80

Z = 184 / 21.59 ≈ 8.53

Therefore, ΔXYZ ≈ (-15, 10, 5).

The values are found by solving the three given equations using the method of substitution.

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The measure of m∠5 = m∠6, and this equality can be justified by the Vertical Angles Theorem.

The Vertical Angles Theorem states that when two lines intersect, the pairs of vertical angles formed are congruent. In other words, if two lines intersect at a point, then the angles opposite each other (across the intersection) are equal in measure. This theorem is based on the geometric property that vertical angles are formed by a pair of opposite rays.

Given, that m∠5 = m∠6.

The measure of the numbered angle is not provided in the question, so it cannot be determined without additional information. However, if we assume that m∠5=m∠6, then the theorem that justifies this work is the Vertical Angles Theorem. According to the Vertical Angles Theorem, when two lines intersect, the vertical angles formed are congruent.

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The value of x if equation is 17 e⁴ˣ = 85 is given by (ln 5)/4.

Hence the correct option is given by  (h).

Given the equation is,

17 e⁴ˣ = 85

On dividing both sides by 17 we get,

e⁴ˣ = 85 / 17

e⁴ˣ = (17 * 5)/17

e⁴ˣ = 5 [Eliminating the likewise terms]

To remove e we have to take ' ln ' function to the both sides then we get,

ln (e⁴ˣ) = ln 5

4 x = ln 5

x = (ln 5)/4, dividing both sides by 4

Hence the correct option is given by (h).

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The solution to the given system of equations is x = 1 and y = -3.

To solve the system, we can use the method of elimination or substitution. Let's use the elimination method here.

We have the following system of equations:

-2x + y = -5 (Equation 1)

4x + y = -2 (Equation 2)

To eliminate the y variable, we can add Equation 1 and Equation 2 together:

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

2x = -7

Dividing both sides of the equation by 2, we get:

x = -7/2 = 1

Substituting the value of x into Equation 1, we can solve for y:

-2(1) + y = -5

-2 + y = -5

y = -5 + 2

y = -3

Therefore, the solution to the system of equations is x = 1 and y = -3.

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To make your data more conclusive by conceding some sources of experimental error, you can acknowledging and accounting for potential sources of experimental error, such as measurement uncertainties, instrument limitations, and environmental factors.

In scientific experiments, acknowledging and conceding potential sources of experimental error means recognizing and accepting factors that could introduce uncertainty or variation in the data. This includes factors like imprecise measurements, limitations of the instruments or equipment used, and environmental conditions that may affect the results.

By openly acknowledging these potential errors, researchers can demonstrate transparency and a thorough understanding of the limitations of their study, leading to more robust and conclusive data analysis.

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The correct answer would be c. 1.5, since it is the only value between 2 and 0.12. To find the value of x that would make the inequality 2 > x > 12% true, we need to convert 12% to a decimal.

12% is equal to 0.12 in decimal form.

So, the correct answer would be c. 1.5,

since it is the only value between 2 and 0.12.

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The values of x that would make the inequality true are 0.3 and 1.5 (option b and c)

The given inequality is 2 > x > 12%. Let's break it down to understand it better.

First, we have 2 > x. This means that x is less than 2. In other words, any value of x that is less than 2 would make this part of the inequality true.

Next, we have x > 12%. This means that x is greater than 12%. To compare x with a percentage, we need to convert it to a decimal. 12% is equal to 0.12 as a decimal. So, any value of x that is greater than 0.12 would satisfy this part of the inequality.

Now, let's combine the two parts of the inequality. We have 2 > x > 0.12.

To find the values of x that make the entire inequality true, we need to find the overlapping range between the two parts. In this case, any value of x that is greater than 0.12 but less than 2 would satisfy the inequality.

Looking at the answer choices given:

a. lo (This appears to be a typo, assuming it's supposed to be "10".) 10 is not within the range of 0.12 to 2, so it would not make the inequality true.

b. 0.3: 0.3 is within the range of 0.12 to 2, so it would make the inequality true.

c. 1.5: 1.5 is also within the range of 0.12 to 2, so it would make the inequality true.

d. 5: 5 is greater than 2, so it would not make the inequality true.

Therefore, the values of x that would make the inequality true are 0.3 and 1.5 (option b and c).

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The volume of the sphere with an area of the great circle equal to 98.5m² is approximately 244.7 cubic meters.

To find the volume of a sphere, we need to know the radius or the area of the great circle. In this case, we are given the area of the great circle, which is 98.5m².

The formula to find the volume of a sphere is V = (4/3)πr^3, where V represents the volume and r is the radius of the sphere.

To find the radius, we need to determine the radius of the great circle. The radius of the great circle is equal to the radius of the sphere.

The formula for the area of a circle is A = πr^2, where A represents the area and r is the radius.

We are given the area of the great circle as 98.5m². Using the formula, we can solve for the radius of the great circle:

98.5 = πr^2

Solving for r, we get r ≈ √(98.5/π) ≈ 5.6 meters.

Substituting the radius value into the volume formula, we have:

V = (4/3)π(5.6)^3 ≈ 244.7 cubic meters.

Therefore, the volume of the sphere with an area of the great circle equal to 98.5m² is approximately 244.7 cubic meters when rounded to the nearest tenth.

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The angle bisector from a vertex to the opposite side of a triangle is unique. We get AD = BE by subtracting BD from both sides of AB + BD = BE.

Proposition 4.4 states that in every triangle, there exists an angle bisector from a vertex to the opposite side and the angle bisector is unique.

Proof:

Let ΔABC be the given triangle and suppose AD is the angle bisector from A to BC. To prove that AD is unique, suppose BE is another angle bisector from B to AC.

We need to show that AD = BE.

Construct the incenter I of ΔABC. Draw lines AI and BI. AI bisects ∠A and BI bisects ∠B.

So, ∠AIB = 90° + ½ ∠C (proved in proposition 4.3).

Again, ∠AIB = ∠AID + ∠BIE [angle addition property of equality]

Therefore, ∠AID + ∠BIE = 90° + ½ ∠C...(1)

Since AD is the angle bisector from A to BC, we have

∠BAD = ∠CAD

Also, BE is the angle bisector from B to AC, so

∠EB = ∠EC

Adding these two equations, we get

∠BAD + ∠EB = ∠CAD + ∠EC...(2)

Since ∠A + ∠B + ∠C = 180° (angle sum property of a triangle),

∠C/2 + ∠B/2 + ∠A/2 = 180° (angle bisector property)

Therefore, ∠A/2 + ∠B/2 = 90° - ∠C/2

So, 2∠A/2 + 2∠B/2 = 180° - ∠C...(3)

Adding equations (1), (2), and (3), we get

∠AID + ∠BIE + ∠BAD + ∠EB + 2∠A/2 + 2∠B/2 = 360° - ∠C/2 + 180° + ∠C/2 + 180° - ∠C

∠AID + ∠BIE + ∠BAD + ∠EB + ∠A + ∠B = 360°(simplifying)

Therefore,

∠AID + ∠BIE + ∠BAD + ∠EB = 180°...(4)

Now, consider ΔABD and ΔEBD.

∠ABD = ∠EBD (∵ AD and BE are angle bisectors)

∠BAD = ∠BED [from equation (2)]

Therefore, ΔABD ≅ ΔEBD (∵ SAS congruence criterion)

So, BD = BD [by CPCTC]

Thus, we get AD = BE [by subtracting BD from both sides of AB + BD = BE]

Hence, the angle bisector from a vertex to the opposite side of a triangle is unique.

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The following are the percentage increase or decrease of each employee to the nearest whole number;

Ruiz = 13%

Aria = 36%

Jack = 6%

The employee who had an increase of at least 10% and sold $425 or more in product is Ruiz.

Ruiz:

Percent increase = change in sales / march sales × 100

= ($450 - $400) / $400 × 100

= $50/$400 × 100

= 13%

Aria:

Percent increase = change in sales / march sales × 100

= ($420 - $310) / $310 × 100

= $110/$310 × 100

= 36%

Jack:

Percent decrease = change in sales / march sales × 100

= ($460 - $432) / $460 × 100

= $28/$460 × 100

= 6%

Therefore, Ruiz is the only employee who receives a bonus.

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A n × n matrix is called a square matrix.

A square matrix is a matrix that has an equal number of rows and columns.

In mathematics, m × m matrix is called the square matrix of order m. If we multiply or add any two square matrices, the order of the resulting matrix remains the same.

A n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations,

An inverse matrix should be a square matrix.

The inverse of the matrix can be calculated by two approaches:

1) Adjoint method

2) Row/Column elementary method

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The given inequality is[tex]\(\frac{\left(\sqrt{9-x}-\left|3x-4\right|\right)}{\sqrt{x+2}-\left|3x-4\right|}\leq1\)[/tex]. The solution to this inequality is[tex]\(x\leq\frac{4}{3}\)[/tex].

To solve the inequality, let's analyze the terms separately. The expression[tex]\(\sqrt{9-x}\)[/tex] will be real only if[tex]\(9-x\geq0\)[/tex], which means [tex]\(x\leq9\).[/tex]
Next, consider the term[tex]\(\left|3x-4\right|\)[/tex]. The absolute value[tex]\(\left|3x-4\right|\)[/tex] will be positive or zero .

So we can consider two cases[tex]: \(3x-4\geq0\) and \(3x-4 < 0\).[/tex]
Case 1[tex]: \(3x-4\geq0\)[/tex]
In this case, the absolute value term becomes[tex]\(3x-4\)[/tex], and the inequality becomes [tex](\frac{\left(\sqrt{9-x}-(3x-4)\right)}{\sqrt{x+2}-(3x-4)}\leq1\)[/tex].

Simplifying further, we get[tex]\(\sqrt{9-x}\leq\sqrt{x+2}\)[/tex].
Squaring both sides of the inequality, we have[tex]\(9-x\leq x+2\), which gives \(2x\geq7\) or \(x\geq\frac{7}{2}\).[/tex]
Case 2: [tex]\(3x-4 < 0\)[/tex]
In this case, the absolute value term becomes \(-(3x-4)\), and the inequality becomes[tex]\(\frac{\left(\sqrt{9-x}+(3x-4)\right)}{\sqrt{x+2}+(3x-4)}\leq1\)[/tex].

Simplifying further, we get[tex]\(\sqrt{9-x}\geq\sqrt{x+2}\)[/tex].
Squaring both sides of the inequality,

We  [tex]have \(9-x\geq x+2\),[/tex]which gives [tex]\(2x\leq7\) or \(x\leq\frac{7}{2}\).[/tex]
Combining both cases, we find that [tex]\(x\leq\frac{7}{2}\)[/tex] satisfies the inequality.
Therefore, the solution to the given inequality is[tex]\(x\leq\frac{4}{3}\)[/tex]

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The hands on a clock form congruent acute angles at 1:05, 2:10, and 3:15.

To determine the times when the hands on a clock form congruent acute angles, we need to consider the hour hand and the minute hand. The minute hand moves 12 times faster than the hour hand.

At 1:05, the minute hand is on the 1, while the hour hand is slightly past the 1. The angle between the hour and minute hands is acute and congruent.

At 2:10, the minute hand is on the 2, while the hour hand is slightly past the 2. Again, the angle between the hour and minute hands is acute and congruent.

At 3:15, the minute hand is on the 3, while the hour hand is slightly past the 3. The angle between the hour and minute hands is once again an acute and congruent angle.

In each of these cases, the hour hand is slightly ahead of the corresponding hour mark, and the minute hand points directly at the hour mark. This results in the formation of congruent acute angles between the hands on the clock.

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The equation 2m² + 5m + 3 = 0 can be solved using the quadratic formula. The solutions for the equation are m = -1 and m = -1.5.

To solve the quadratic equation 2m² + 5m + 3 = 0 using the quadratic formula, we need to identify the values of the coefficients a, b, and c. In this case, a = 2, b = 5, and c = 3. The quadratic formula is given by:

m = (-b ± √(b² - 4ac)) / (2a)

Substituting the values of a, b, and c into the formula, we get:

m = (-(5) ± √((5)² - 4(2)(3))) / (2(2))

 = (-5 ± √(25 - 24)) / 4

 = (-5 ± √1) / 4

Since the discriminant (b² - 4ac) is equal to 1, we have two solutions. Simplifying further:

m = (-5 + 1) / 4 = -4 / 4 = -1

m = (-5 - 1) / 4 = -6 / 4 = -1.5

Therefore, the solutions to the equation 2m² + 5m + 3 = 0 are m = -1 and m = -1.5. Both values satisfy the equation and make it equal to zero when substituted back into it.

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