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A rational function having the following vertical asymptotes x = -4 and x = 5 and x-intercept (-2, 0) is [tex]f(x) = \frac{x+2}{(x+4)(x-5)}[/tex].

A rational function is a function that can be expressed as the quotient of two polynomials. Vertical asymptotes occur when the denominator of the rational function is equal to zero. An x-intercept occurs when the rational function is equal to zero.

To find a rational function that satisfies the given conditions, we can use the information about the vertical asymptotes and x-intercept to create the function.

The vertical asymptotes occur at x = -4 and x = 5, so the denominator of the rational function must have factors of (x + 4) and (x - 5). The x-intercept occurs at (-2, 0), so the numerator of the rational function must have a factor of (x + 2).

Therefore, one possible rational function that satisfies the given conditions is:

[tex]f(x) = \frac{x+2}{(x+4)(x-5)}[/tex]

This function has vertical asymptotes at x = -4 and x = 5, and an x-intercept at (-2, 0), as required.

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After solving the problem we found that Wags is approximately 15 years old and Scratch is approximately 5 years old today.

The solution uses algebraic equations and substitution to solve for the ages of Wags and Scratch, which is based on the concept of solving systems of linear equations.

Let's assume that Wag's age today is represented by W, and Scratch's age today is represented by S.

From the first piece of information, we can set up the following equation:

W - 1 = 3 + 3(S - 1)

Simplifying:

W - 1 = 3S - 6

W = 3S - 5

From the second piece of information, we can set up the following equation:

S + 2 = (1/4)(W + 2) + 2

Simplifying:

S + 2 = (1/4)W + 2.5

Substituting W = 3S - 5:

S + 2 = (1/4)(3S - 5) + 2.5

Solving for S:

S + 2 = (3/4)S - (5/4) + 2.5

S + 2 = (3/4)S + 3/4

1/4 S = 1 1/4

S = 5

Substituting S = 5 into W = 3S - 5:

W = 10 + 5

W = 15

Therefore, Wags is approximately 15 years old and Scratch is approximately 5 years old today.

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To prove that the functions f(x)= (2)/(3)x-8 and g(x)=(3)/(2)x+12 are inverses, we can use composition of functions. We can do this by setting f(g(x)) = x and g(f(x)) = x and showing that they are equivalent.

First, let's look at f(g(x)). We can substitute g(x) into the equation for f(x), so we have: f(g(x)) = (2)/(3)((3)/(2)x+12)-8. Simplifying, we have: f(g(x)) = (4x+48)/6 - 8. Distributing the 4 and rearranging, we have: f(g(x)) = x.

Now let's look at g(f(x)). We can substitute f(x) into the equation for g(x), so we have: g(f(x)) = (3)/(2)((2)/(3)x-8)+12. Simplifying, we have: g(f(x)) = (2x-32)/3 + 12. Distributing the 2 and rearranging, we have: g(f(x)) = x.

Therefore, we have shown that f(g(x)) = x and g(f(x)) = x, which means that f(x) and g(x) are inverses of each other.

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The measures of those two angles are, 20° and 110°

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

Given that;

Two angles of a quadrilateral measure 180° and 50°.

And, The other two angles are in a ratio of 2:11.

Since, The other two angles are in a ratio of 2:11.

Hence, ,Measures of both angles are, 2x and 11x.

Now, We know that;

Sum of all interior angles of a quadrilateral are 360°.

Hence, We get;

⇒ 180 + 50 + 2x + 11x = 360

⇒ 230 + 13x = 360

⇒ 13x = 360 - 230

⇒ 13x = 130

⇒ x = 10

Thus, Measures of both angles are,

2x = 2 × 10 = 20°

and 11x = 11 × 10 = 110°

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Answer:

[tex]x=\frac{3}{2} - i\frac{\sqrt{7}}{2}[/tex]

[tex]x=\frac{3}{2} + i\frac{\sqrt{7}}{2}[/tex]

Step-by-step explanation:

We can use the quadratic formula to solve the equation:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = 1, b = -3, and c = 4. Substituting these values into the formula, we get:

x = (3 ± sqrt((-3)^2 - 4(1)(4))) / 2(1)

x = (3 ± sqrt(9 - 16)) / 2

x = (3 ± sqrt(-7)) / 2

Since the square root of a negative number is not a real number, the solutions to the quadratic equation are complex numbers. Therefore, the answer is:

[tex]x=\frac{3}{2}+i\frac{\sqrt{7} }{2}[/tex]

[tex]x=\frac{3}{2} - i\frac{\sqrt{7}}{2}[/tex]

The value of ST in the given triangle is is √73.

A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.

Here, we have

Given: In a triangle

TR = 8

SR = 3

ST =?

We apply here Pythagoras theorem and we get

SR² + TR² = ST²

3² + 8² = ST²

9 + 64 = ST²

73 = ST²

ST = √73

Hence, the value of ST is √73.

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Given vectors are, u = (6,1,6), v = (1,-1,4) and w = (1,4,7) and a vector (23, 10, 23).We need to express (23, 10, 23) as a linear combination of u, v and w.

To find the coefficients, we use matrix notation.
Let A be a matrix containing u, v and w as its column matrices, and X be a matrix containing the coefficients.
Then,AX = B Where A = [u | v | w]X = [a | b | c]B = (23, 10, 23)
Therefore, [u | v | w][a | b | c] = (23, 10, 23)⇒ a(u1) + b(v1) + c(w1) = 23a(6) + b(1) + c(1) = 23⇒ 6a + b + c = 23⇒ a(u2) + b(v2) + c(w2) = 10a(1) - b(1) + c(4) = 10⇒ a - b + 4c = 10⇒ a(u3) + b(v3) + c(w3) = 23a(6) - b(4) + c(7) = 23⇒ 6a - 4b + 7c = 23
The above system of linear equations can be solved using Gaussian elimination method.
The row echelon form of the augmented matrix [A | B] is, [6 1 1 | 23][1 -1 4 | 10][0 0 2 | 9]
The solution is, c = 9/2, b = -3, and a = 1/2.
Therefore, (23, 10, 23) = (1/2)u - 3v + (9/2)w can be written as a linear combination of u, v and w.

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To find the value of k for which the matrix A has one eigenvalue of multiplicity 2, we can use the characteristic equation of the matrix. The characteristic equation is det(A-λI)=0. For a given matrix, this equation will yield a polynomial equation of degree equal to the dimension of the matrix.

In this case, the equation is (λ+4)(λ-2) = 0. Solving this equation gives us two distinct eigenvalues of λ= -4 and λ=2. This means that for any value of k, the matrix A will have two distinct eigenvalues, and thus cannot have one eigenvalue of multiplicity 2. Therefore, the matrix A cannot have one eigenvalue of multiplicity 2 no matter what value of k is used.

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The probability of getting 4 heads is 0.205078125, the probability of getting 6 heads is 0.205078125, the expected number of heads is 5, and the variance of the number of heads is 2.5.

The probability of getting 4 heads or 6 heads in 10 coin tosses can be calculated using the binomial probability distribution formula:

P(X=x) = nCx * p^x * (1-p)^(n-x)

Where:
n = number of trials (10)
x = number of successes (4 or 6)
p = probability of success (0.5)
1-p = probability of failure (0.5)
nCx = combination of n things taken x at a time

For 4 heads:
P(X=4) = 10C4 * 0.5^4 * 0.5^(10-4)
P(X=4) = 210 * 0.0625 * 0.015625
P(X=4) = 0.205078125

For 6 heads:
P(X=6) = 10C6 * 0.5^6 * 0.5^(10-6)
P(X=6) = 210 * 0.015625 * 0.0625
P(X=6) = 0.205078125

The expected number of heads can be calculated using the formula:
E(X) = n * p
E(X) = 10 * 0.5
E(X) = 5

The variance of the number of heads can be calculated using the formula:
Var(X) = n * p * (1-p)
Var(X) = 10 * 0.5 * 0.5
Var(X) = 2.5
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The other three angle measures in the quadrilateral, in addition to <1 = 45 degrees, are: <2 = <3 = 135 degrees, and <4 = 45 degrees.

What is the quadrilateral?

A quadrilateral is a polygon with four sides and four angles. In other words, it is a closed figure that has four straight sides and four vertices (corners).

If <1 has a measure of 45 degrees, then the sum of the angle measures in the quadrilateral must be 360 degrees. Let's call the other three angles in the quadrilateral <2, <3, and <4.

Then we can use the fact that the opposite angles in a quadrilateral add up to 180 degrees to write two equations:

<1 + <3 = 180 (opposite angles)

<2 + <4 = 180 (opposite angles)

We also know that <1 = 45, so we can substitute that in the first equation:

45 + <3 = 180

Solving for <3, we get:

<3 = 135

Now we can substitute the values of <1 and <3 into the sum of angle measures equation:

45 + <2 + 135 + <4 = 360

Simplifying, we get:

<2 + <4 = 180

This is the same equation as the opposite angles equation, so we know that <2 and <4 are also opposite angles. Therefore, we can conclude that:

<2 = <3 = 135 degrees

<4 = 45 degrees

Hence, the other three angle measures in the quadrilateral, in addition to <1 = 45 degrees, are: <2 = <3 = 135 degrees, and <4 = 45 degrees.

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There are 720 different possibilities for selecting the executive board from a Student Council with 10 members. An answer is option (B) 720.

A permutation is to select an object then arrange it and it cares about the orders while a Combination is about only selecting an object without caring about the orders.

The number of ways to select the executive board is the number of permutations of 10 items taken 3 at a time.

That is:

P(10, 3) = 10!/(10-3)! = 10x9x8 = 720

Therefore, there are 720 different possibilities for selecting the executive board from a Student Council with 10 members. An answer is option (B) 720.

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The new ordered pair of A’ are (-1, -2.25). Hence the correct option is b.

A cοοrdinate system in geοmetry is a technique that uses οne οr mοre integers οr cοοrdinates tο establish the accurate pοsitiοning οf geοmetrical οbjects οn a manifοld, such as Euclidean space. When lοcating a pοint οr οbject οn a twο-dimensiοnal plane, cοοrdinates—pairs οf numbers—are utilized.

The x and y cοοrdinates serve as a representatiοn οf a pοint's lοcatiοn οn a 2D plane. a grοup οf numbers used tο denοte certain lοcatiοns. The figure usually cοnsists οf twο integers. The frοnt-tο-back and tοp-tο-bοttοm distances are represented by the first and secοnd numbers, respectively. When there are 12 units belοw and 5 abοve, as in (12.5), the ratiο is 1.

Tο cοnduct a dilatiοn with a scale factοr οf 1/2 arοund the οrigin, multiply each pοint's cοοrdinates by 1/2. If the initial cοοrdinates οf pοint A are (x, y), then the dilated pοint A' cοοrdinates are:

A' = (1/2 * x, 1/2 * y)

We negate the x-cοοrdinate while keeping the y-cοοrdinate cοnstant tο represent the dilated pοint A' οver the y-axis. As a result, the final cοοrdinates οf A" are:

A'' = (-1/2 * x, 1/2 * y)

As a result, Anew "'s οrdered pair is (1/2 * x, 1/2 * y)

= ((1/2 × -2), (1/2 × -5))

= (-1, -2.25)

Thus, the new ordered pair of A’ are (-1, -2.25). Hence the correct option is b.

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Correct statement: Two vectors are perpendicular if and only if the dot product of the two vectors is zero.

Explanation:

Two vectors are perpendicular if and only if the dot product of the two vectors is zero. This is equivalent to saying that the angle between the two vectors is 90 degrees. Therefore, the statement "Two vectors are perpendicular if and only if the angle between them is 90 degrees" is correct.

Every orthonormal set is linearly independent. This statement is also correct. An orthonormal set of vectors is a set of vectors that are pairwise perpendicular and have length 1.

Since the dot product of any two distinct vectors in an orthonormal set is 0, it follows that any linear combination of these vectors will also have a dot product of 0 with any other vector in the set.

Therefore, the only solution to a linear combination of these vectors being equal to the zero vector is for all the coefficients to be 0, which implies linear independence.

Every orthogonal set is linearly independent. This statement is also correct. An orthogonal set of vectors is a set of vectors that are pairwise perpendicular, but not necessarily of length 1.

Similar to the previous argument, the dot product of any two distinct vectors in an orthogonal set is 0, which implies linear independence.

R^n has an orthonormal basis. This statement is also correct. An orthonormal basis for a vector space is a basis consisting of orthonormal vectors. It can be shown that any finite-dimensional inner product space has an orthonormal basis, and R^n is a finite-dimensional inner product space with the standard dot product.

The zero vector is perpendicular to all other vectors. This statement is false. The zero vector is orthogonal to all other vectors, but not necessarily perpendicular, as the concept of perpendicularity requires the vectors to have a nonzero length.

Any orthonormal set in R^n is a basis. This statement is also correct. An orthonormal set of n vectors in R^n must span the entire space, as any vector in the space can be expressed as a linear combination of these vectors.

Furthermore, any set of n linearly independent vectors in R^n is a basis, so the fact that an orthonormal set is linearly independent (as shown in statement 2) implies that it is a basis.

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The composite functions and their domains are:
(a) (fog)(x) = 30x + 8, domain: all real numbers
(b) (gof)(x) = 30x + 40, domain: all real numbers
(c) (fof)(x) = 36x + 56, domain: all real numbers
(d) (gog)(x) = 25x, domain: all real numbers

The composite functions are formed by substituting one function into another. We can find the composite functions for f(x)=6x+8 and g(x)=5x by following these steps:

(a) (fog)(x) = f(g(x)) = f(5x) = 6(5x) + 8 = 30x + 8

The domain of (fog)(x) is the set of all real numbers, since there are no restrictions on the values of x.

(b) (gof)(x) = g(f(x)) = g(6x+8) = 5(6x+8) = 30x + 40

The domain of (gof)(x) is also the set of all real numbers, since there are no restrictions on the values of x.

(c) (fof)(x) = f(f(x)) = f(6x+8) = 6(6x+8) + 8 = 36x + 56

The domain of (fof)(x) is also the set of all real numbers, since there are no restrictions on the values of x.

(d) (gog)(x) = g(g(x)) = g(5x) = 5(5x) = 25x

The domain of (gog)(x) is also the set of all real numbers, since there are no restrictions on the values of x.

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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 baker will use 2.5 cups of flour.

Multiplication is a mathematical operation that involves combining two or more numbers to find their product or the total number of objects in equal groups. It is represented using the multiplication symbol (*) or a dot (·). For example, 2 * 3 = 6, which means that two groups of three objects will give you a total of six objects.

To solve this problem, we need to find the amount of flour required for each cupcake and cookie, and then multiply by the total number of cupcakes and cookies.

For the cupcakes:

1.5 cups of flour make 18 cupcakes

1 cup of flour will make 12 cupcakes (divide both sides by 1.5)

To make 30 cupcakes, we need 2.5 cups of flour (multiply both sides by 2.5)

For the cookies:

3 cups of flour make 36 cookies

1 cup of flour will make 12 cookies (divide both sides by 3)

To make 60 cookies, we need 5 cups of flour (multiply both sides by 5)

Therefore, to make 30 cupcakes and 60 cookies, the baker will need:

2.5 cups of flour for the cupcakes

5 cups of flour for the cookies

Total: 2.5 + 5 = 7 cups of flour

Therefore,  the baker will use 2.5 cups of flour.

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Answer:

Step-by-step explanation: the answer is 20, 48, 52

From the equation of the least squares regression line, we can say that the height of the flower compared to the soil line, in the beginning, is 3.16 units and it grows by 0.56 units each day.

A mathematical equation is a formula that uses the equals symbol (=) to connect two expressions and express their equality. Two expressions joined by an equal sign form a mathematical statement known as an equation. The expression on the left and the expression on the right is shown to be equal in relation to one another.

LHS = RHS (left-hand side = right-hand side) appears in all mathematical equations. You can solve equations to determine an unknown variable's value, which corresponds to an unknown quantity. It is not an equation if there is no "equal to" symbol in the statement. It will be taken into account as an expression.

Given,

The predicted height = 0.56

Days in soil = 3.16

The y-axis of the plot is the height of the line compared to the soil line.

The x-axis gives the number of days.

from the equation of the least squares regression line, we get the above predicted height and the days in the soil.

y  = mx+b is the equation of a line.

Then, we can say that,

m = slope = 0.56

b = y-intercept = 3.16

Therefore from the equation of the least squares regression line, we can say that the height of the flower compared to the soil line, in the beginning, is 3.16 units and it grows by 0.56 units each day.

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(1, 7) and (10, 1) would be the two points that a line of fit would go through to best fit the data.

What is Scatter Plot ?

A scatter plot is a graph that shows the relationship between two sets of data. Each dot on the plot represents a single data point, and the position of the dot corresponds to the values of the two variables being plotted.

In this specific scatter plot, we have 10 data points represented by dots. To find the two points that a line of best fit would go through, we want to look for a pattern or trend in the data. Ideally, the line of best fit should pass as close as possible to all of the data points, but this is not always possible.

One common method for finding the line of best fit is to choose two points that seem to be close to the middle of the data and that the line passes through. This is because we want the line to be a good representation of the overall trend in the data.

Looking at the scatter plot provided, we can see that there is a general trend of the data points sloping downward from left to right. If we draw a line that passes through the points (3, 7) and (9, 1), we can see that it closely follows the trend of the data points. Therefore, these are the two points that a line of best fit would go through to best fit the data.

Therefore, (1, 7) and (10, 1) would be the two points that a line of fit would go through to best fit the data.

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The equation are: x = -1 radians (-57.3°) and x = 11 radians (626.9°).

To solve this equation, use the identity sin2x + cos2x = 1 and apply it to the left side of the equation.

2 sin2x + 20 cos x = 6

2 (1 - cos2x) + 20 cos x = 6

2 - 2 cos2x + 20 cos x = 6

2 cos2x - 20 cos x + 2 = 6

cos2x - 10 cos x + 1 = 0

Next, solve the resulting quadratic equation using the quadratic formula: x = [-b ± √(b2 - 4ac)]/2a. In this case:

x = [-(-10) ± √((-10)2 - 4(1)(1))]/2(1)

x = [10 ± √(100 - 4)]/2

x = [10 ± √(96)]/2

x = (10 ± 4√6)/2

x = (10 ± 12)/2

x = 5 ± 6

We then use the interval [0,2π) to calculate the exact radian values for x. The two solutions in this interval are:

x = 5 - 6 = -1

x = 5 + 6 = 11

For reference, the angle corresponding to -1 radians is -57.3° and the angle corresponding to 11 radians is 626.9°.
To check the solution, graph the two sides of the equation on Desmos, with the interval [0,2π). The graph will show the two points of intersection (marked with circles) which correspond to the two solutions.

In conclusion, the exact values of x which satisfy the equation are: x = -1 radians (-57.3°) and x = 11 radians (626.9°).

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Yes, if u1, u2, and u3 are linearly independent, then v1, v2, and v3 will also be linearly independent.  


To show this, assume that v1, v2, and v3 are linearly dependent. This means that there are scalars a,b, and c, such that:

a*v1 + b*v2 + c*v3 = 0

Since v1 = u1 + u2, v2 = u1 + u3, and v3 = u2 + u3, the equation above can be rewritten as:

a*(u1 + u2) + b*(u1 + u3) + c*(u2 + u3) = 0


Simplifying, this gives us:

(a + b + c)*u1 + (a + c)*u2 + (b + c)*u3 = 0


But since u1, u2, and u3 are linearly independent, the coefficients (a + b + c), (a + c), and (b + c) must all be equal to 0. This implies that a = b = c = 0, meaning that the original equation must be equal to 0. This means that v1, v2, and v3 are linearly independent.

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I'm not 100%

sure

:))))))))))))))))))))))))

If the balloon can hold 14 passengers at a time, then number of rounds to empty the que is 9.

In order to find out number of rounds it will take to empty the queue of 124 people, we divide the total number of people by the number of people that can be carried in each round,

⇒ Number of rounds = (Total number of people)/(Number of people per round);

⇒ Number of rounds = 124/14,

⇒ Number of rounds = 8.86 (rounded to two decimal places)

Since we can't have a fraction of a round, we round up to the nearest whole number.

Therefore, the balloon will need to make 9 rounds to empty the queue of 124 people.

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The given question is incomplete, the complete question is

There are 124 people waiting to take a ride in a hot air balloon. The balloon can hold 14 passengers at a time, how many rounds will it take to empty the que?

Answer:

1st one: (x-7)

2nd: (-x+7)

3rd: (5x+3)

Step-by-step explanation:

1: you subtract like terms so first you do 3x-2x which is x and do -2-5 and get -7 so its (x-7)

2: do 2x-3x and get (-x) and then do 5-(-2) and get 7 so the answer is (-x+7)

3: first do 2x+3x and get 5x and then do 5+(-2) and get 3 so the answer is (5x+3)

Answer:

Step-by-step explanation:

( 3x - 2 ) - ( 2x + 5 ) = 3x - 2 - 2x - 5  = x - 7

( 2x + 5 ) - ( 3x - 2 ) = 2x + 5 - 3x + 2 = - x + 7

( 2x + 5 ) + ( 3x - 2 ) = 2x + 5 + 3x - 2 = 5x + 3

Answer:

Let's call the number of packs of skimmed milk "S" and the number of packs of full cream milk "F".

We know that the total number of packs of milk is 1350:

F + L + S = 1350

We also know that there are 150 more packs of skimmed milk than low fat milk:

S = L + 150

And we know that there are 465 packs of low fat milk:

L = 465

We can substitute L=465 into the equation S=L+150 to get:

S = 465 + 150 = 615

Now we can use the first equation to solve for F:

F + L + S = 1350

F + 465 + 615 = 1350

F = 270

Therefore, there are 270 packs of full cream milk.

Answer:

Step-by-step explanation:

To graph the equation 3(x - 2) = 5(x + 1), we can first simplify it by expanding the brackets:

3x - 6 = 5x + 5

Next, we can isolate the variable on one side of the equation. We can do this by subtracting 3x from both sides and adding 6 to both sides:

-11 = 2x

x = -11/2

Now we have the x-coordinate of the point where the graph of the equation intersects the x-axis. To find the y-coordinate of this point, we can substitute x = -11/2 into one of the original equations and solve for y:

3(x - 2) = 5(x + 1)

3(-11/2 - 2) = 5(-11/2 + 1)

-33/2 - 6 = -55/2 + 5

-33/2 - 6 + 55/2 = 5

-33/2 + 44/2 = 5

11/2 = 5

The slope of the linear equation is; 0.5

The y-intercept of the Linear Equation is: 4.75

Yes the equation is proportional

The equation we are given is expressed as;

y = ¹/₂x + 4³/₄

When x = 0,

y = ¹/₂(0) + 4³/₄

y = 4.75

When x = 1,

y = ¹/₂(1) + 4³/₄

y = 5.25

When x = 2,

y = ¹/₂(2) + 4³/₄

y = 5.75

When x = 3,

y = ¹/₂(3) + 4³/₄

y = 6.25

If we go on and on till x = 10, we will see the constant difference in y for every change in x is 0.5

Thus, slope = 0.5

y-intercept = 4.75

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The first monthly charge has about $78.78 allotted in the direction of reducing the principal.

The total buy charge of the house is $96,400 and the Spinoses made a 25% down payment, because of this they paid $96,400 x 0.25 = $24,100 in advance.

Therefore, the final quantity that they financed is $96,400 - $24,100 = $72,300.

They financed this amount at 5.50% for 30 years, which gives us the following method for calculating the monthly payment (P):

P = (r * PV) / (1 - (1 + r)^(-n))

in which:

Substituting the values, we get:

P = (0.00458 * 72,300) / (1 - (1 + 0.00458)^(-360))

P ≈ $410.66

We recognise that the monthly payment is $410.66 and we will calculate the interest portion of the primary monthly price as follows:

interest = balance * monthly interest price

interest = $72,300 * (5.50% / 12) ≈ $331.88

To calculate the amount of the primary monthly charge this is used to lessen the fundamental, we subtract the interest component from the total monthly payment:

$410.66 - $331.88 ≈ $78.78

Thus, the first monthly charge has about $78.78 allotted in the direction of reducing the principal.

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