Write the polynomial as the product of linear factors. g(x) = x3 – 3x2 + x + 5 g(x) =________ List all the zeros of the function. (Enter your answers as a comma-separated list.) X =__________

The coordinates of the centroid (,)(△) of triangle ABC are (-1, 8/3).

To find the coordinates of the centroid, we can use the formula:

(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices A, B, and C, respectively.

Plugging in the coordinates of A (0, 4), B (0, 2), and C (-3, 2), we get:

(x, y) = ((0 + 0 - 3)/3, (4 + 2 + 2)/3) = (-1, 8/3)

In a two-dimensional plane, the centroid of a triangle is the point where the three medians of the triangle intersect. A median is a line segment drawn from a vertex of the triangle to the midpoint of the opposite side. The centroid divides each median into two segments, with the ratio of the length of the segment closer to the vertex to the length of the segment closer to the opposite side being 2:1.

In a three-dimensional space, the centroid of a solid can be found by dividing the solid into smaller parts, finding the centroid of each part, and then averaging them. The centroid of a solid is the point where the lines connecting the centroids of each part intersect.

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Complete Question: -

Give the coordinates of the centroid of triangle ABC, which is denoted by (,)(△) and is the point where the medians of the triangle intersect for a (0 , 4) , b (0 ,2) , and C  (-3 ,2 ).

In other words, the quotient is 2x+4 and the remainder is 10x+7.

To solve this problem, we will use polynomial long division. The steps are as follows:

1. Divide the first term of the dividend (4x^(3)) by the first term of the divisor (2x^(2)) to get the first term of the quotient (2x).
2. Multiply the first term of the quotient (2x) by the divisor (2x^(2)+4x) to get (4x^(3)+8x^(2)).
3. Subtract the result from step 2 (4x^(3)+8x^(2)) from the dividend (4x^(3)+16x^(2)+18x+7) to get the remainder (8x^(2)+18x+7).
4. Repeat steps 1-3 with the new dividend (8x^(2)+18x+7) and the same divisor (2x^(2)+4x) until the degree of the remainder is less than the degree of the divisor.

The final quotient and remainder are as follows:

Quotient: 2x+4
Remainder: 10x+7

So, the final answer is:

(4x^(3)+16x^(2)+18x+7)-:(2x^(2)+4x) = 2x+4+(10x+7)-:(2x^(2)+4x)

In other words, the quotient is 2x+4 and the remainder is 10x+7.

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a)CD and AA

b)CD + AA = 190,000

c)0.06CD + 0.11AA = 18,900

d)AA = 187,333.17.

(a) Let CD be the amount invested in Certificates of Deposit and AA be the amount invested in AA bonds.

(b) The total amount invested is given by the equation CD + AA = 190,000.

(c) The total amount earned from the investment is given by the equation 0.06CD + 0.11AA = 18,900.

(d) To solve this system of equations using elimination, we multiply the first equation by 0.11 and the second equation by 0.06 and add the resulting equations together, yielding the equation 12.06CD + 10.11AA = 214,084. We then subtract the original equation from this new equation to get 9.06CD = 24,084. Solving for CD, we get CD = 2,666.83. We can then substitute this value into either of the original equations to get AA = 187,333.17.

(e) Sally should invest $2,666.83 in Certificates of Deposit and $187,333.17 in AA bonds to earn exactly $18,900 per year.

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The variable costs to produce one widget is 6.00

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

E = 6.00 q + 11,000

The variable cost is the slope of the relation

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

Fixed cost = 11000

Variable cost = 6.00

The above parameters mean that

The variable cost is 6.00

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1) a) ω = 2πf = 2π(50) = 100π radians per second

b) 96.78m/s.

2) (a) T = 1/f = 1/12 seconds,

(b) ω = 2πf = 2π(12) = 24π radians per second.

3) original length of the pendulum is 9.0m.

4)  original length of the spiral spring is 20cm.

5)  the initial velocity of the football is 17.32m/s At the highest

Angular speed is the speed of the object in rotational motion. Distance traveled is represented as θ and is measured in radians. The time taken is measured in terms of seconds. Therefore, the angular speed is articulated in radians per second or rad/s.

1. (a) The angular speed of a body vibrating at 50 cycles per second is:

ω = 2πf = 2π(50) = 100π radians per second

(b) The stone is projected horizontally, so its initial vertical velocity is zero. The time taken for the stone to fall from the top of the tower to the ground is given by:

t = √(2h/g) = √(2×75/10) = √15

The horizontal distance traveled by the stone is:

d = vxt = (25m/s)×(√15) ≈ 96.78m

Therefore, the speed with which the stone strikes the ground is approximately 96.78m/s.

2. (a) The period of the motion is:

T = 1/f = 1/12 seconds

(b) The angular speed of the motion is:

ω = 2πf = 2π(12) = 24π radians per second

(c) The velocity at the middle of oscillation is zero, since this is the point of maximum displacement and therefore maximum potential energy.

(d) The acceleration at the end of oscillation is given by:

a = -ω²x = -(24π)²(0.02) = -11.52 m/s²

where x is the amplitude of the motion.

3. The period of a simple pendulum is given by:

T = 2π√(L/g)

where L is the length of the pendulum and g is the acceleration due to gravity. Therefore, we can write:

17 = 2π√(L/g) (1)

8.5 = 2π√[(L-1.5)/g] (2)

Dividing (2) by (1), we get:

1/2 = √[(L-1.5)/L]

Squaring both sides and simplifying, we get:

L = 9.0m

Therefore, the original length of the pendulum is 9.0m.

4. (a) Let the original length of the spiral spring be x. Then, using Hooke's law, we have:

5N = k(25cm - x) (1)

10N = k(30cm - x) (2)

where k is the spring constant. Solving for k in equation (1), we get:

k = 5N/(25cm - x)

Substituting into equation (2), we get:

10N = [5N/(25cm - x)](30cm - x)

Simplifying and solving for x, we get:

x = 20cm

Therefore, the original length of the spiral spring is 20cm.

(b) The pressure of the trapped air column in the capillary tube is balanced by the pressure of the atmosphere. When the tube is held horizontally, the length of the air column is the same as its length in the vertical position. Therefore, the length of the air column is 15cm.

When the tube is held vertically with the open end underneath, the length of the air column is given by:

L = 76cm - 20cm = 56cm

Therefore, the length of the air column is 56cm.

5. The horizontal and vertical components of the initial velocity of the football are:

v₀x = v₀cos(60°) = (20m/s)cos(60°) = 10m/s

v₀y = v₀sin(60°) = (20m/s)sin(60°) = 17.32m/s

At the highest

Hence, the answer to each question:

1) a) ω = 2πf = 2π(50) = 100π radians per second

b) 96.78m/s.

2) (a) T = 1/f = 1/12 seconds,

(b) ω = 2πf = 2π(12) = 24π radians per second.

3) original length of the pendulum is 9.0m.

4)  original length of the spiral spring is 20cm.

5)  the initial velocity of the football is 17.32m/s At the highest

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No, the function g will be the same as the graph of f, translated to the right by 2.

What is transformation?

A point, line, or geometric figure can be changed in four different ways that are all collectively referred to as transformations. The original shape of the object is called the Pre-Image and the final shape and position of the object is the Image under the transformation.

f(x + a)horizontally shift the graph of f(x) left by a units.

The graph of f(x) is horizontally shifted by a units right side by f(x - a).

The graph of f(x) is vertically shifted upward by a unit by f(x)+a.

f(x)- a one-unit vertical shift downward of the f(x) graph.

The given function is

g(x) = f(x-2)

According to rule,

The function g will be the same as the graph of f, translated to the right by 2.

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The distance is 35 units and the average speed is 5 units per min.

To find the distance that the bug ran, we need to take the difference between the final position and the initial position, it gives:

distance = -12 - (-47) = 12 + 47 = 35 units

And the average speed is the quotient between the distance and the time, so:

Average speed = distance/time.

And here we know that.

distance = 35 units.

time=  7 minutes.

Then we can replace these in the formula above to get:

s = (35 units)/7min = 5 units per min.

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The equation of line in the point-slope form is y - 2 = -1(x + 2)

The equation of a straight line is a relationship between x and y coordinates, The point-slope form of the equation of a straight line is,

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

Given that,

A graph having straight line,

and it can be seen in the graph it is passing through (-2, 2) & (-1, 1)

Slope m = (y₂-y₁)/(x₂-x₁)

              = (2 - 1)/ (-2 -(-1))

              =  1/-1

              =   -1

So now taking point (-2, 2) and slope is -1

The point slope form of the equation is:

y - 2 =  -1 (x - (-2))

y - 2 = -1(x + 2)

Hence, the equation is y - 2 = -1(x + 2)

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The required measure of q in the given triangle is 3.

Trigonometric ratios are mathematical functions used to relate the angles of a right-angled triangle to the lengths of its sides. There are three basic trigonometric ratios: sine (sin), cosine (cos), and tangent (tan), which can be defined as follows:

Sine (sin) = opposite/hypotenuse

Cosine (cos) = adjacent/hypotenuse

Tangent (tan) = opposite/adjacent

Here,
Applying the Sine rule,
sin45 = q/3√2
q = 3

Thus, the required measure of q in the given triangle.

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The figure for the given net diagram is square pyramid.

Net diagram is a 2-dimensional plane figure which can be folded to form a 3-dimensional figure. Or we can say net diagrams are the figures which obtained by unfolding some 3D figures.

The given net diagram has a square and 4 triangles.

If we fold all the faces and make a solid figure, we get square pyramid.

Therefore, the figure is square pyramid.

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The absolute maximum value of the function on the interval is approximately 2.28, and the absolute minimum value is approximately 0.37. Both of these values are expressed in terms of the notiral number, e.

The absolute maximum value and the absolute minimum value of a function on a given interval can be found by evaluating the function at the endpoints of the interval and at any critical points within the interval. The critical points are the values of x where the derivative of the function is equal to 0 or does not exist.

First, let's find the derivative of the function:
\( f'(x)=e^{\operatorname{xin}(x)} \cdot \operatorname{xin}'(x) \)

Using the chain rule, the derivative of \( \operatorname{xin}(x) \) is:
\( \operatorname{xin}'(x)=\operatorname{cos}(x)-\operatorname{xin}(x) \cdot \operatorname{sin}(x) \)

So, the derivative of the function is:
\( f'(x)=e^{\operatorname{xin}(x)} \cdot (\operatorname{cos}(x)-\operatorname{xin}(x) \cdot \operatorname{sin}(x)) \)

To find the critical points, we need to set the derivative equal to 0:
\( e^{\operatorname{xin}(x)} \cdot (\operatorname{cos}(x)-\operatorname{xin}(x) \cdot \operatorname{sin}(x))=0 \)

Since \( e^{\operatorname{xin}(x)} \) is never equal to 0, we can focus on the second factor:
\( \operatorname{cos}(x)-\operatorname{xin}(x) \cdot \operatorname{sin}(x)=0 \)

This equation cannot be solved algebraically, so we will use a graphing calculator to find the approximate values of x that make the equation true. The values are approximately 0.86 and 4.71.

Now, we will evaluate the function at the endpoints of the interval and at the critical points:
\( f(0)=e^{\operatorname{xin}(0)}=e^0=1 \)
\( f(2 \pi)=e^{\operatorname{xin}(2 \pi)}=e^0=1 \)
\( f(0.86)=e^{\operatorname{xin}(0.86)} \approx 2.28 \)
\( f(4.71)=e^{\operatorname{xin}(4.71)} \approx 0.37 \)

The absolute maximum value of the function on the interval is approximately 2.28, and the absolute minimum value is approximately 0.37. Both of these values are expressed in terms of the notiral number, e.

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The height of the rectangle is

⇒ Height = 1/2 ft

A rectangle is a 2 - dimension figure with 4 sides, 4 corners and 4 right angles. And, Opposite sides of the rectangle are equal and parallel to each other.

We have to given that;

Length of the rectangle = 5/3 ft

Area of the rectangle = 5/6 ft

We know that;

Area of rectangle = Length x Height

Hence, We get;

5/6 ft = 5/3 ft x Height

5/6 x 3/5 = Height

Height = 1/2 ft

Thus, The height of the rectangle is;

⇒ Height = 1/2 ft

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Answer:

We can use the formula for the volume of a cylinder:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

Plugging in the given values, we get:

6,430.72 = π(16)^2h

Simplifying:

6,430.72 = 256πh

Dividing both sides by 256π:

h = 6,430.72 / (256π)

h ≈ 8.05

Therefore, the height of the cylinder is approximately 8.05 yards.

The formulas for the exponential functions that pass through the two points given are:
f(x) = 6 * 7^x
g(x) = 1280 * 0.125^x

To find the formula for an exponential function that passes through two points, we can use the general form of an exponential function, which is f(x) = a * b^x, where a and b are constants. We can plug in the x and y values from the two points and solve for a and b.

For part a:
f(x) = a * b^x
6 = a * b^0 (from point (0,6))
294 = a * b^2 (from point (2,294))

From the first equation, we can see that a = 6. We can substitute this value into the second equation:
294 = 6 * b^2
49 = b^2
b = 7

So the formula for the exponential function that passes through the two points is f(x) = 6 * 7^x.

For part b:
g(x) = a * b^x
1280 = a * b^0 (from point (0,1280))
20 = a * b^2 (from point (2,20))

From the first equation, we can see that a = 1280. We can substitute this value into the second equation:
20 = 1280 * b^2
0.015625 = b^2
b = 0.125

So the formula for the exponential function that passes through the two points is g(x) = 1280 * 0.125^x.

In conclusion, the formulas for the exponential functions that pass through the two points given are:
f(x) = 6 * 7^x
g(x) = 1280 * 0.125^x

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[tex]-\dfrac{3}{4}x-\dfrac{1}{3}+\dfrac{7}{8}x-\dfrac{1}{2} = -\dfrac{3}{4}x+\dfrac{7}{8}x -\dfrac{1}{3}-\dfrac{1}{2} = \\[/tex]

[tex]= -\dfrac{3(2)}{4(2)}x+\dfrac{7}{8}x -\dfrac{1(2)}{3(2)}-\dfrac{1(3)}{2(3)} = -\dfrac{6}{8}x+\dfrac{7}{8}x -\dfrac{2}{6}-\dfrac{3}{6} \\[/tex]

[tex]= \dfrac{-6+7}{8}x -\dfrac{2+3}{6} = \dfrac{1}{8}x -\dfrac{5}{6} \\[/tex]

The end behaviour of g(x) = log₁,₃(-x + 4) is as x approaches 4, g(x) approaches ∝ and as x approaches 4, g(-∝) approaches -∝

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

g(x) = log₁,₃(-x + 4)

The maximum input value of the above function is 4

This is so because

-x + 4 = 0

x = 4

Set the value of x in the function to 4

So, we have

g(4) = log₁,₃(4 + 4)

g(4) => ∝

Set the value of x in the function to -∝

So, we have

g(-∝) = log₁,₃(-∝ + 4)

g(-∝) => -∝

Hence, the end behaviour is as x approaches 4, g(x) approaches ∝

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Answer:

m<6 = 78°

Step-by-step explanation:

I hope your drawing of this problem looks somewhat like this:

                                                                ^  t

                                                              /

                                                            /

                                                   1     /    2

<----------------------------------------------------------------------------->   ℓ

                                               3     /    4

                                                    /

                                         5       /    6

<------------------------------------------------------------------------------>  m

                                      7       /   8

                                            /

                                          /

                                        V

Sorry about the terrible drawing, but I hope I got the angle numbers correctly written.

Angles 3 and 6 are are called alternate interior angles.

They are "interior angles" because they are on the inside of lines l and m. They are "alternate angles" because they are on different sides of the transversal, t.

There is a Geometry theorem about this situation.

Theorem:

If parallel lines are cut by a transversal, then alternate interior angles are congruent.

In this case, since the pair of angles 3 and 6 is a pair of alternate interior angles, then by the theorem above, they are congruent.

m<6 = m<3

Since m<3 = 78°, then m<6 = 78°.

The population mean is greater than $21.

The 95% confidence interval is $13.3896, $34.1304. As $21 is outside of the confidence interval, we can conclude that the population mean is greater than $21.

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The given system of equations can be written as: y=-3x^2+12x  3x+y=12

To solve this system, we can use the substitution method. We will solve the first equation for y and substitute this value in the second equation to solve for x. From the first equation, we have y=-3x^2+12x. Substituting this in the second equation, we get 3x+(-3x^2+12x)=12. Simplifying, we get -3x^2+15x=12.

Now, we can solve this equation for x. To do this, we can divide both sides of the equation by -3 to get x^2+5x/3=4/3. Factoring the left side, we get (x+5/3)(x-4/3)=0. From here, we can find the two solutions for x: x=-5/3 and x=4/3. Substituting these values in the first equation, we get the two solutions for y: y=-17/3 and y=20/3. Therefore, the two solutions for the system of equations are (x=-5/3, y=-17/3) and (x=4/3, y=20/3).

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The tabs can be completed in the following way:

1. Fraction for 25% = 25/100 = 5/20 = 1/4

2. 90 percent: Decimal 0.90: Fraction: 90/100 =9/10

3. 60 percent: Decimal 0.60: Fraction: 60/100 = 6/10 = 3/5

4. 35 percent: Decimal 0.35: Fraction: 35/100 = 7/20

5. 33 percent: Decimal 0.33: Fraction: 33/100

6. 65 percent: Decimal 0.65: Fraction: 65/100 = 13/20

A percentage is a value that is obtained as a fraction of the number 100. In the above question, we are given a list of values in percentages and told to convert them to decimal and fraction forms.

To do this, we need to divide the number 90 by 100 for the second expression and express this as a decimal. The resultant figure is 0.9. When converted to a fraction, the value now has a denominator and numerator.

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Answer:

1st page:

To determine if Jason and Arianna made a mistake in their solution, we can examine the slopes and y-intercepts of the two equations.

The first equation, 5x - 3y = -1, can be written in slope-intercept form as y = (5/3)x + 1/3, where the slope is 5/3 and the y-intercept is 1/3.

The second equation, 3x + 2y = 7, can be written in slope-intercept form as y = (-3/2)x + 7/2, where the slope is -3/2 and the y-intercept is 7/2.

To solve the system of equations, we need to find the point of intersection of the two lines. We can see from the slopes that the lines are not parallel, so they must intersect at some point. However, the slopes are not perpendicular either, so they do not intersect at a right angle.

By graphing the two equations on the same coordinate plane, we can see that the point of intersection is (2, 2), not (4.04, 7.31) as Jason and Arianna calculated. Therefore, they must have made a mistake in their solution.

In summary, we can tell that Jason and Arianna made a mistake in their solution by examining the slopes and y-intercepts of the two equations and graphing them to find the actual point of intersection.

2nd page:

To determine if Jason and Arianna made a mistake in their solution, we can graph the two linear equations on the same coordinate plane and look for the point of intersection.

We can begin by rearranging the equations into slope-intercept form:

5x - 3y = -1 → y = (5/3)x + 1/3

3x + 2y = 7 → y = (-3/2)x + 7/2

Now we can graph the two lines. We can plot two points for each line and connect them with a straight line to obtain the graphs.

For the first equation, when x = 0, we get y = 1/3. When x = 3, we get y = 6/3 = 2. Plotting these two points and connecting them, we get:

Graph of the first equation

For the second equation, when x = 0, we get y = 7/2. When x = 2, we get y = 1/2. Plotting these two points and connecting them, we get:

Graph of the second equation

We can see from the graphs that the lines intersect at the point (2, 2), not at (4.04, 7.31) as Jason and Arianna found. Therefore, they must have made a mistake in their solution.

In summary, we can tell from the graphs of the equations that Jason and Arianna must have made a mistake because the lines do not intersect at the point they found.

3rd page:

We can determine if there is a unique solution to the system of linear equations by examining the slopes of the two equations.

The slope of the first equation, 5x - 3y = -1, can be found by rearranging the equation into slope-intercept form y = (5/3)x + 1/3. We can see that the slope of the line is positive and not equal to the slope of the second equation, which is -3/2.

Similarly, the slope of the second equation, 3x + 2y = 7, can be found by rearranging the equation into slope-intercept form y = (-3/2)x + 7/2. Again, we can see that the slope of the line is negative and not equal to the slope of the first equation, which is 5/3.

Since the slopes of the two lines are not equal, they will intersect at a unique point. In other words, there is only one solution to the system of equations.

Therefore, we can conclude that there is a unique solution to the system of linear equations given by 5x - 3y = -1 and 3x + 2y = 7, based on the slopes of the graphs of the equations in the system.

Step-by-step explanation:

If x and y both increase or decrease by the same factor, then z will also increase or decrease by the same factor, keeping the ratio between the variables constant, the function model used is z = 33.33xy.

When three variables are said to vary jointly, it means that they are related in such a way that if any one of them changes, the other two also change proportionally. To model the joint variation of z with x and y when x=0.2 and y=3, we can use the following function:

z = kxy

where k is a constant of proportionality.

To find the value of k, we can substitute the given values of x, y, and z into the equation:

2 = k(0.2)(3)

Solving for k, we get:

k = 33.33

Therefore, the function that models the joint variation of z with x and y is:

z = 33.33xy

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The intersection of sets A and B

The intersection of two sets is the set of elements that are common to both sets. To find the intersection of sets A and B, we simply look for the elements that are in both sets. In this case, the intersection of A and B is {h, o, e}.

The union of two sets is the set of elements that are in either one of the sets or both. To find the union of sets A and B, we simply combine the elements from both sets, making sure to not include any duplicates. In this case, the union of A and B is {h, o, m, e, u, s}.

So, the intersection of sets A and B is {h, o, e} and the union of sets A and B is {h, o, m, e, u, s}.

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Tο depict the relatiοnship between twο numerical variables using Data visualizatiοn that is knοwn as a scatterplοt.

Using this scatterplοt, we graphed a line using linear regressiοn. This line can use as οur reference pοint and ignοre the rest οf the pοints οn the graph.

We will examine this graph and try tο find apprοximately what value οf y is given when we prοvide an x value.

Since the number οf hits is the x value, tο find hοw high the graph gοes at that x value we can lοοk at the graph tοο.

Right between 100 and 150 is 125, sο we can nοw start there and mοve up.

When we mοve up, we get a line that's between 10 and 20. We can alsο view it's in the tοp half οf that bοx. That means the value οf y fοr this will be mοre than 15.

Since it's nοt lying directly οn 20, we have the οnly οptiοn left is 18.

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A conjecture which Carlos can make about the angles formed by line t and lines l, m, and n include the following: B. angles 1, 3, and 5 are congruent.

In Mathematics, corresponding angles postulate simply refers to a theorem which states that corresponding angles are always congruent (equal) if the transversal intersects two parallel lines.

This ultimately implies that, the corresponding angles would always be congruent (equal) if a transversal intersects two (2) parallel lines.

By applying corresponding angles postulate to both lines l, m, and n, we can reasonably infer and logically deduce that the following angles are congruent:

∠1 ≅ ∠3

∠3 ≅ ∠5

∠1 ≅ ∠5

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2L of water must be evaporated from the 10L of 40% salt solution to produce a 50% solution.

To produce a 50% salt solution from a 40% salt solution, we must evaporate some water to increase the concentration of the salt. We can use the formula:

C1V1 = C2V2

Where C1 is the initial concentration, V1 is the initial volume, C2 is the final concentration, and V2 is the final volume.

Plugging in the given values:

0.40(10L) = 0.50(V2)

Simplifying:

4L = 0.50V2

Dividing both sides by 0.50:

V2 = 8L

So the final volume of the solution must be 8L in order to have a 50% concentration of salt. To find the amount of water that must be evaporated, we can subtract the final volume from the initial volume:

10L - 8L = 2L


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Answer:

420km/12km = 35

1×35l = 35lites

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Answer:

35 litres

Step-by-step explanation:

We can use a ratio to solve.

1 litre               x litres

-------------  = ----------------

12 km             420 km

Using cross products

1 * 420 = 12x

Divide each side by 12.

420/12 = x

35 =x

35 litres

The orthogonal complement of the column space of a matrix A is the set of vectors that are perpendicular to the vectors in the column space. To find a basis for the orthogonal complement of the column space of A, we can use the Gram-Schmidt process.

This process starts with the columns of A, orthogonalizes them, and then adds the orthogonalized vectors to the basis of the orthogonal complement. We can find a basis for the orthogonal complement of the column space of A by performing the Gram-Schmidt process.

The result of this process is [[1,2,-1,0], [2,3,0,1]], which is the basis of the orthogonal complement of the column space of A.

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