How do the average rates of change for the pair of functions compare over the given​ interval?
​f(x)x
​g(x)x
x
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Part 1
The average rate of change of​ f(x) over x is

enter your response here. The average rate of change of​ g(x) over x is

enter your response here. The average rate of change of​ g(x) is

enter your response here times that of​ f(x). ​(Simplify your answers. Type integers or​ decimals. )

There are infinitely many expressions equivalent to 11x + 10, including 22x + 20, 11(x+1)-1, -11(-x)-10, and 11(x+2)-12.

What is expression ?

In mathematics, an expression is a combination of numbers, symbols, and/or variables that are put together in a meaningful way, usually to represent a quantity or a mathematical relationship.

There are infinitely many expressions that are equivalent to 11x + 10, because you can add or subtract any expression to both sides of the equation to get a new equivalent expression. Here are some examples:

22x + 20: This is equivalent to 11x + 10 because if you distribute 11 to x and 10, you get 11x + 10.

11(x + 1) - 1: This is also equivalent to 11x + 10 because if you distribute 11 to x and 1, you get 11x + 11 - 1, which simplifies to 11x + 10.

-11(-x) - 10: This is equivalent to 11x + 10 because if you distribute -11 to -x, you get 11x + 10.

11(x + 2) - 12: This is also equivalent to 11x + 10 because if you distribute 11 to x and 2, you get 11x + 22 - 12, which simplifies to 11x + 10.

In general, any expression of the form 11x + k, where k is a constant, is equivalent to 11x + 10.

Therefore, there are infinitely many expressions equivalent to 11x + 10, including 22x + 20, 11(x+1)-1, -11(-x)-10, and 11(x+2)-12.

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Accοrding tο similarity οf triangles, the length οf CF is 8/3.

Similarity οf triangles is a cοncept in geοmetry that describes the relatiοnship between twο triangles that have the same shape but may be different in size. Twο triangles are cοnsidered similar if their cοrrespοnding angles are cοngruent and their cοrrespοnding sides are prοpοrtiοnal.

Since FG is parallel tο CD, we can use similar triangles tο find the length οf CF. Let's call the length οf CF x. Then we have:

FE/FG = CD/CF

Substituting the given values, we have:

5/(x-1) = 8/x

Crοss-multiplying, we get:

5x = 8(x-1)

Expanding the brackets, we get:

5x = 8x - 8

Subtracting 5x frοm bοth sides, we get:

3x = 8

Dividing bοth sides by 3, we get:

x = 8/3

Therefοre, the length οf CF is 8/3.

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Complete question:

(a) To find the value of k, we need to use the fact that the sum of the probabilities of all possible outcomes is equal to 1. In this case, we have:
0.05 + 0.20 + 3k + 0.15 + k = 1
Solving for k, we get:
4k = 1 - 0.05 - 0.20 - 0.15
4k = 0.60
k = 0.15
Therefore, the value of k is 0.15.

(b) To find E(X), we need to multiply each value of x by its corresponding probability and sum the results. In this case, we have:
E(X) = (-1)(0.05) + (0)(0.20) + (1)(3k) + (2)(0.15) + (3)(k)
E(X) = -0.05 + 0 + 0.45 + 3k
E(X) = 0.40 + 3k
Substituting the value of k that we found in part (a), we get:
E(X) = 0.40 + 3(0.15)
E(X) = 0.85
Therefore, the expected value of X is 0.85.

(c) To find Var(X), we need to use the formula Var(X) = E(X^2) - (E(X))^2. First, we need to find E(X^2):
E(X^2) = (-1)^2(0.05) + (0)^2(0.20) + (1)^2(3k) + (2)^2(0.15) + (3)^2(k)
E(X^2) = 0.05 + 0 + 3k + 0.60 + 9k
E(X^2) = 0.65 + 12k
Substituting the value of k that we found in part (a), we get:
E(X^2) = 0.65 + 12(0.15)
E(X^2) = 2.45
Now, we can find Var(X):
Var(X) = E(X^2) - (E(X))^2
Var(X) = 2.45 - (0.85)^2
Var(X) = 2.45 - 0.7225
Var(X) = 1.7275
Therefore, the variance of X is 1.7275.

(d) To find Var(2 - 5X), we need to use the formula Var(a + bX) = b^2Var(X), where a = 2 and b = -5. Substituting the values and the variance of X that we found in part (c), we get:
Var(2 - 5X) = (-5)^2Var(X)
Var(2 - 5X) = 25(1.7275)
Var(2 - 5X) = 43.1875
Therefore, the variance of 2 - 5X is 43.1875.

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After solving the equation, the solution if the equation is -2 and -11/5.

To solve the rational equation we first express the given statement in the form of mathematical equation. After that we solve the equation.

As the statement is 5 over X + 2. It can be written as 5/(x + 2)

Then the second statement is -11 over X to the 2nd + 2X which is written as -11/(x² + 2x).

There have equal sign between both expression. So the expression is

5/(x + 2) = -11/(x² + 2x)

Now solve it.

Multiply by (x² + 2x) on both side, we get

5(x² + 2x)/(x + 2) = -11

Multiply by (x + 2) on both side, we get

5(x² + 2x) = -11(x + 2)

Now simplify using the distributive property

5x² + 10x = -11x - 22

Add 11x on both side, we get

5x² + 21x = - 22

Add 22 on both side, we get

5x² + 21x + 22 = 0

Now we factor the equation

5x² + (11 + 10)x + 22 = 0

5x² + 11x + 10x + 22 = 0

x(5x + 11) + 2(5x + 11) = 0

(5x + 11)(x + 2) = 0

Now equating equal to 0.

5x + 11 = 0                     x + 2 = 0

5x = -11                          x = -2

x = -11/5

The solution if the equation is -2 and -11/5.

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The complete question is:

Solve the rational equation 5 over X + 2 equals -11 over X to the 2nd + 2X.

Answer:

18/40

or

9/20

Step-by-step explanation:

13 marbles are initially removed.

40 - 13 = 27

1/3 of that is removed.

27 - 9 = 18

We take whatever is left over (18 marbles), and put it in a fraction to represent what fraction is left over from 40.

18/40

or

9/20

The equation for the graph that passes through (4,-5) and (7,10) is: y = 5x - 25

What is equation ?

An equation is a mathematical statement that shows that two expressions are equal. It contains an equal sign (=) between two expressions, one on each side. An equation can contain variables, constants, numbers, and mathematical operations like addition, subtraction, multiplication, and division

To write an equation for the graph that passes through the points (4,-5) and (7,10), we can use the slope-intercept form of a linear equation:

y = mx + b

where m is the slope of the line and b is the y-intercept.

To find the slope of the line passing through the two points, we can use the slope formula:

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

where (x1, y1) = (4,-5) and (x2, y2) = (7,10).

m = (10 - (-5)) / (7 - 4)

m = 15 / 3

m = 5

Now that we know the slope of the line, we can use either point to solve for the y-intercept, b. Let's use the point (4,-5):

y = mx + b

-5 = 5(4) + b

-5 = 20 + b

b = -25

Therefore, the equation for the graph that passes through (4,-5) and (7,10) is: y = 5x - 25

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The value of kˣ for the exponential function is [tex]27^{(1/x)}.[/tex]

We can use the properties of exponents to solve this problem and determine the value of kˣ.

To find the value of h(3x), we can use the property that [tex](a^b)^c = a^{bc}[/tex] for any real numbers a, b, and c. This is shown in the solution below.

h(3x) = 3³ˣ    (since h(x) = 3ˣ)

= (3³)ˣ

= 27ˣ

Therefore, we have:

kˣ = 27ˣ

To find the value of kˣ, we can take the x-th root of both sides:

[tex]k = 27^{(1/x)}[/tex]

So, the value of kˣ is [tex]27^{(1/x)}[/tex].

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The value of

1 sinY = 12/13

2. cos Y = 5/13

3. tanY = 12/5

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.

sin Y = opp/hyp

cos Y = adj/hyp

tan Y = opp/adj

if opp = 12

adj = 5

hyp = 13

then,

sinY = 12/13

cos Y = 5/13

tanY = 12/5

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Answer:30

Step-by-step explanation:

The ordered pair that is NOT a solution of y > 3x + 4 is (-2, 1). Hence, the answer is (E) (-2, 1).

A relationship between two expressions or values that are not equal to each other is called 'inequality. ' So, a lack of balance results in inequality.

The given inequality is y > 3x + 4. To check which ordered pair is not a solution of this inequality, we need to substitute the values of x and y from each ordered pair into the inequality and check whether the inequality holds or not.

Checking the ordered pairs one by one, we have:

(2, 12): y = 12 and x = 2, so 12 > 3(2) + 4 is true. Therefore, (2, 12) is a solution of the inequality.

(0, 5): y = 5 and x = 0, so 5 > 3(0) + 4 is true. Therefore, (0, 5) is a solution of the inequality.

(-2, 1): y = 1 and x = -2, so 1 > 3(-2) + 4 is false. Therefore, (-2, 1) is not a solution of the inequality.

(1, 7): y = 7 and x = 1, so 7 > 3(1) + 4 is true. Therefore, (1, 7) is a solution of the inequality.

Therefore, the ordered pair that is NOT a solution of y > 3x + 4 is (-2, 1). Hence, the answer is (E) (-2, 1).

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Answer:

Step-by-step explanation:

you get angle one from opposite angles = 85°

second angle :

98° x 2 = 196°

360° - 196° = 164°

164°/2 = 82°

Finding angle x:

82° + 85°  = 167°

180°-167° = 13°

x=13°

Answer:

a) The height of the building is 382 feet.

b) The rock is 382 feet above the ground at its highest point.

c) It takes the rock 3.25 seconds to reach a height of 200 feet.

d) Tt takes the rock 4.76 seconds to hit the ground.

Step-by-step explanation:

The function that models the distance (in feet) between the rock and the ground t seconds after it is thrown is a quadratic function.

As the leading coefficient of the quadratic function is negative, it is a parabola that opens downwards.

The rock is thrown from the top of the building. Therefore, the height of the building is the value of d(t) when t = 0. This is the y-intercept of the graphed function.

Substitute t = 0 into the given function:

[tex]\begin{aligned}\implies d(0)&=-16(0)^2-4(0)+382\\&=0+0+382\\&=382\; \sf feet \end{aligned}[/tex]

Therefore, the height of the building is 382 feet.

The highest point of the rock is the height of the building, since the rock is thrown down from the top.

Therefore, the rock is 382 feet above the ground at its highest point.

This can be proven by finding the vertex of the graph of the function.

The vertex (maximum point) of the graphed function is (-0.125, 382.25).

As the x-value of the vertex is negative, and time can only be positive, the path of the rock is on a downwards trajectory when t ≥ 0. Therefore, the highest point is the point at which the rock is thrown.

To calculate how long it takes for the rock to reach a height of 200 feet, substitute d(t) = 200 into the given function and solve for t.

[tex]\begin{aligned}\implies -16t^2-4t+382&=200\\-16t^2-4t+182&=0\\-2(8t^2+2t-91)&=0\\8t^2+2t-91&=0\\8t^2+28t-26t-91&=0\\4t(2t+7)-13(2t+7)&=0\\(4t-13)(2t+7)&=0\\\\\implies 4t-13&=0 \implies t=\dfrac{13}{4}=3.25\; \sf s\\\implies 2t+7&=0 \implies t=-\dfrac{7}{2}=-3.5\; \sf s\end{aligned}[/tex]

As time is positive, t = 3.25 s only.

Therefore, it takes the rock 3.25 seconds to reach a height of 200 feet.

The rock will hit the ground when d(t) = 0.

Therefore, to calculate how long it takes for the rock to hit the ground, substitute d(t) = 0 into the given function:

[tex]\implies -16t^2-4t+382=0[/tex]

[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]

Solve for t using the quadratic formula.

[tex]\implies t=\dfrac{-(-4) \pm \sqrt{(-4)^2-4(-16)(382)}}{2(-16)}[/tex]

[tex]\implies t=\dfrac{4 \pm \sqrt{24464}}{-32}[/tex]

[tex]\implies t=-\dfrac{4 \pm \sqrt{16 \cdot 1529}}{32}[/tex]

[tex]\implies t=-\dfrac{4 \pm \sqrt{16} \sqrt{1529}}{32}[/tex]

[tex]\implies t=-\dfrac{4 \pm 4\sqrt{1529}}{32}[/tex]

[tex]\implies t=-\dfrac{1 \pm \sqrt{1529}}{8}[/tex]

[tex]\implies t=-5.01280...,4.76280...[/tex]

As time is positive, t = 4.76 s only.

Therefore, it takes the rock 4.76 seconds to hit the ground.

The structure has a 382-foot height. The rock's highest peak is 39 1/2 feet high, or 195.5 feet. The time it takes for the rock to touch the ground is roughly 6.289 seconds.

In arithmetic, we use angles and distance to determine an object's height. The distance between the items is measured horizontally, and the height of an object is determined by the angle of the top of the object with respect to the horizontal.

By setting t = 0, we can determine the rock's starting height:

d(0) = -16(0)^2 - 4(0) + 382

= 382

The highest point of the rock occurs at the vertex of the parabolic route, which is determined by the formula t = -b/2a

where a = -16 and b = -4.

t = -(-4) / 2(-16) = 1/8

d(1/8) = -16(1/8)^2 - 4(1/8) + 382

= 391/2

The equation d(t) = -16t^2 - 4t + 382 = 200 for t:

-16t^2 - 4t + 382 = 200

-16t^2 - 4t + 182 = 0

4t^2 + t - 45.5 = 0

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The constant of proportionality (k) for the data points on this graph is 7.

In Mathematics, a proportional relationship is a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical expression:

y = kx

Where:

Next, we would determine the constant of proportionality (k) for the data points on this graph as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 7/1 = 14/2 = 21/3 = 28/4 = 35/5

Constant of proportionality, k = 7.

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When A U B = C & A ∩ B = φ, then A = C - B. The solution has been obtained by using properties of sets.

What is a set?

In mathematics, a set is a logically arranged group of items that can be represented in either set-builder or roster form. Curly brackets are typically used to represent sets.

We know that

A - B = (A ∪ B) - B

On substituting A U B = C in this, we get

A - B = C - B       ...(1)

Similarly,

A - B = A - (A ∩ B)

On substituting  A ∩ B = φ in this, we get

A - B = A        ...(2)

From (1) and (2), we get

A = C - B

Hence, when A U B = C & A ∩ B = φ, then A = C - B.

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The lengths of the two legs are 2 centimeters and 9 centimeters.

Let's call the shorter leg of the right triangle x and the other leg y. According to the given information, we can create the following equation:

x = y - 7

Since we know that the hypothenuse is 13 centimeters, we can use the Pythagorean theorem to create another equation:

x^2 + y^2 = 13^2

Substituting the first equation into the second equation, we get:

(y - 7)^2 + y^2 = 13^2

Simplifying and rearranging terms, we get:

2y^2 - 14y - 72 = 0

Using the quadratic formula, we can solve for y:

y = (14 ± √(14^2 - 4(2)(-72))) / (2(2))

y = (14 ± √484) / 4

y = (14 ± 22) / 4

y = 9 or y = -2

Since the length of a leg cannot be negative, we reject the negative solution and take y = 9 centimeters as the length of the other leg. Then, we can use the first equation to find the length of the shorter leg:

x = 9 - 7

x = 2 centimeters

Therefore, the lengths of the two legs are 2 centimeters and 9 centimeters.

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(a) It is possible to form a triangle with the angle measures, 30°, 80°, 70°

It is not possible for all such triangles to be congruent.

(b) It is possible to form a triangle with the angle measures, 20°, 105°, 55°

It is not possible for all such triangles to be congruent.

(c) It is not possible to form a triangle with the side lengths, 4cm, 3cm, 8cm

(d) It is possible to form a triangle with these side lengths.

All such triangles are congruent

From the question, we are to determine if it is possible for a triangle to have the given angle measures or side lengths

(a) To determine if a triangle can have the angle measures 30°, 80°, and 70°, we add the angles together to see if they equal 180°, the total degrees of a triangle.

30° + 80° + 70° = 180°

Since the angle measures add up to 180°, it is possible to form a triangle with these angle measures.

It is not possible for all such triangles to be congruent, since triangles with the same angle measures can have different side lengths.

(b) To determine if a triangle can have the angle measures 20°, 105°, and 55°, we add the angles together to see if they equal 180°.

20° + 105° + 55° = 180°

Since the angle measures add up to 180°, it is possible to form a triangle with these angle measures. However, it is not possible for all such triangles to be congruent, since triangles with the same angle measures can have different side lengths.

(c) To determine if a triangle can have side lengths 4cm, 3cm, and 8cm, we apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

4cm + 3cm > 8cm

4cm + 8cm > 3cm

3cm + 8cm > 4cm

Since all three inequalities are not satisfied (4 + 3 = 7 is not greater than 8, which is the longest side), it is not possible to form a triangle with these side lengths.

(d) To determine if a triangle can have side lengths 8cm, 15cm, and 17cm, we apply the triangle inequality theorem.

8cm + 15cm > 17cm

8cm + 17cm > 15cm

15cm + 17cm > 8cm

Since all three inequalities are satisfied, it is possible to form a triangle with these side lengths.

All such triangles are congruent, since these side lengths satisfy the conditions for a unique triangle known as a Pythagorean triple.

Hence, the triangle with side lengths 8cm, 15cm, and 17cm is a right triangle, and all right triangles with these side lengths are congruent by the Pythagorean theorem.

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Answer:

Unfortunately, the sale price of the sweater and the original price are not given in the problem, so we cannot calculate the exact change that Rachel will receive. We need more information to solve the problem.

In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.

To solve the problem using the simplex method, we first need to convert the inequalities to equations by adding slack variables. Then we can set up the initial tableau and use the simplex method to find the optimal solution. Finally, we can verify the solution using the graphical method.

Convert inequalities to equations by adding slack variables
-X1 + X2 >= -1 becomes -X1 + X2 + S1 = -1
4X1 + 3X2 <= 12 becomes 4X1 + 3X2 + S2 = 12
2X1 <= 3 becomes 2X1 + S3 = 3

Set up the initial tableau
| -1 | 2 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | 0 | -1 |
| 4 | 3 | 0 | 1 | 0 | 12 |
| 2 | 0 | 0 | 0 | 1 | 3 |

Use the simplex method to find the optimal solution
We first need to choose the pivot column, which is the one with the most negative coefficient in the objective row. In this case, it is the first column. Then we need to choose the pivot row, which is the one with the smallest positive ratio of the right-hand side to the pivot column coefficient. In this case, it is the third row. We can then use the pivot element to eliminate the other coefficients in the pivot column and repeat the process until we have no negative coefficients in the objective row.

After performing the simplex method, we get the following final tableau:
| 0 | 5/4 | 1 | 1/4 | 0 | 3 |
| 0 | 7/4 | 1 | 1/4 | 0 | 3 |
| 0 | 3/4 | 0 | 1/4 | 0 | 3 |
| 1 | 0 | 0 | 0 | 1/2 | 3/2 |

The optimal solution is X1 = 3/2, X2 = 0, and Z = -3/2.

Verify the solution using the graphical method
We can graph the constraints and find the feasible region. Then we can graph the objective function and find the point where it intersects the feasible region that gives the maximum or minimum value. In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.

Extra Credit:
To solve the problem using the simplex method, we first need to convert the inequalities to equations by adding slack variables. Then we can set up the initial tableau and use the simplex method to find the optimal solution. Finally, we can verify the solution using the graphical method.

Convert inequalities to equations by adding slack variables
-X1 + X2 >= -1 becomes -X1 + X2 + S1 = -1
4X1 + 3X2 <= 12 becomes 4X1 + 3X2 + S2 = 12
2X1 <= 3 becomes 2X1 + S3 = 3

Set up the initial tableau
| -1 | 2 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | 0 | -1 |
| 4 | 3 | 0 | 1 | 0 | 12 |
| 2 | 0 | 0 | 0 | 1 | 3 |

Use the simplex method to find the optimal solution
We first need to choose the pivot column, which is the one with the most negative coefficient in the objective row. In this case, it is the first column. Then we need to choose the pivot row, which is the one with the smallest positive ratio of the right-hand side to the pivot column coefficient. In this case, it is the third row. We can then use the pivot element to eliminate the other coefficients in the pivot column and repeat the process until we have no negative coefficients in the objective row.

After performing the simplex method, we get the following final tableau:
| 0 | 5/4 | 1 | 1/4 | 0 | 3 |
| 0 | 7/4 | 1 | 1/4 | 0 | 3 |
| 0 | 3/4 | 0 | 1/4 | 0 | 3 |
| 1 | 0 | 0 | 0 | 1/2 | 3/2 |

The optimal solution is X1 = 3/2, X2 = 0, and Z = -3/2.

Verify the solution using the graphical method
We can graph the constraints and find the feasible region. Then we can graph the objective function and find the point where it intersects the feasible region that gives the maximum or minimum value. In this case, the point (3/2, 0) is the optimal solution, which matches the solution we found using the simplex method.

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The perpendicular line connecting points F and G results in the formation of a triangle with a 14 square unit surface area.

In mathematics, a coordinate is a set of numbers or symbols that describes an object's position or location in a geometric space. The most popular coordinate system is the Geographic coordinate system, which creates a grid on a surface or in three dimensions using a series of perpendicular lines. Each location on the plane or in space is then uniquely identified by an ordered pair or set of three numbers, respectively, that represent a point's distance from the origin along each of the coordinate system's axes.

Given that points F and G share the same x-coordinate, the line connecting them is perpendicular. A degenerate triangle with negative area will be formed by any point H that shares the same x-coordinate as F and G.

D = √((2-2)2 + (6-(-1),2) = √(49), which =  7.

distance(FH)/7  = 2/7.

distance(GH)/7  = 4/7

Distance(FH) / Distance(GH)  = 1 / 2.

The distance between any two points H(x, y) and F(2,6) or G(2,-1) can be calculated using the distance formula as follows:

distance(FH) is  =  √((x-2) + (y-6)).

Distance (GH) is  =  √((x-2) + (y+1)).

By changing the aforementioned values, we obtain:

√((x-2)**2**y+1**2)/7) = 4/7

(x-2)² + (y+1)² = 16

(x-2)² + (y-6) (y-6)² = 1/4 ((x-2)² + (y+1)²)

By condensing and extending, we obtain:

3(x-2)² + 13(y-6)² = 196

5(x-2)² + 25(y+1)² = 784

7(x-2)² + (y-6)² = 1

A triangle with a 14 square unit surface area is created when the perpendicular line joining F and G.

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Answer:

4:7

Step-by-step explanation:

We know

A car factory made 24 cars with a sunroof and 18 cars without a sunroof.

What is the ratio of the number of cars with a sunroof to the total number of cars?

We find the total number of cars by taking

24 + 18 = 42 cars

The ratio of the number of cars with a sunroof to the total number of cars is

24:42

Simplify by 6, we get the ratio

4:7

The velocity of the car is 80m/s

v= u + at²

The parameters given are:

velocity= ?

initial velocity(u)= 0

acceleration(a)= 5

time(t)= 4

The velocity can be calculated as follows

v= 0 + 5(4²)

v= 0 + 5(16)

v= 0 + 80

v = 80

Hence the velocity of the car is 80m/s

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$x_1 = \frac{7}{2}, \ x_2 = \frac{11}{2}, \ x_3 = -\frac{1}{2}, \ x_4 = \frac{9}{2}$.

To solve this system of equations, first use the array method to organize the equations.

Array Method:

\[ \begin{array}{cccc|c} 2 & -2 & 5 & 7 & 0 \\ 1 & 5 & 6 & 9 & 0 \\ 1 & 17 & 13 & 20 & 0 \\ 3 & -2 & 4 & 0 & 0 \end{array} \]

Now, use row operations to solve the system. Begin by combining the first and third equations by adding them together.

\[ \begin{array}{cccc|c} 2 & -2 & 5 & 7 & 0 \\ 1 & 5 & 6 & 9 & 0 \\ 0 & 12 & 12 & 27 & 0 \\ 3 & -2 & 4 & 0 & 0 \end{array} \]

Next, combine the second and fourth equations by subtracting the fourth equation from the second equation.

\[ \begin{array}{cccc|c} 2 & -2 & 5 & 7 & 0 \\ 0 & 7 & 2 & 9 & 0 \\ 0 & 12 & 12 & 27 & 0 \\ 3 & -2 & 4 & 0 & 0 \end{array} \]

Then, combine the second and third equations by adding the second equation to the third equation.

\[ \begin{array}{cccc|c} 2 & -2 & 5 & 7 & 0 \\ 0 & 0 & 14 & 36 & 0 \\ 0 & 12 & 12 & 27 & 0 \\ 3 & -2 & 4 & 0 & 0 \end{array} \]

Now, combine the first and third equations by subtracting the third equation from the first equation.

\[ \begin{array}{cccc|c} 2 & -2 & 5 & 7 & 0 \\ 0 & 0 & 14 & 36 & 0 \\ 0 & 0 & -2 & -9 & 0 \\ 3 & -2 & 4 & 0 & 0 \end{array} \]

Finally, use back substitution to solve for the variables. Starting with $x_4$, use the fourth equation to solve for it:

$x_4 = \frac{9}{2}$

Then, use the third equation to solve for $x_3$:

$x_3 = -\frac{1}{2}$

Continuing this process, we can also solve for $x_2 = \frac{11}{2}$ and $x_1 = \frac{7}{2}$.

Therefore, the solution to the system is:

$x_1 = \frac{7}{2}, \ x_2 = \frac{11}{2}, \ x_3 = -\frac{1}{2}, \ x_4 = \frac{9}{2}$.

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Reflect point A over the line to get point A', -1,2 is A'.

What is a graph?

A graph is a structure that fundamentally consists of a set of items where some pairs of the objects are "connected" in some way. This definition comes from discrete mathematics, more especially graph theory. The items are represented by mathematical abstractions called vertices, and each pair of connected vertices is known as an edge.

A graph is frequently depicted in a diagram by a collection of dots or circles for the vertices and lines or curves for the edges. Among the topics covered by discrete mathematics are graphs.

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Answer:

Step-by-step explanation:

sum of all angles in quadrilateral = 360

So, x+48+48+132 =360

x =132

As per the given distance, the maximum height reached by the football is 15.4 yards.

To find the maximum height reached by the football, we need to find the vertex of the parabolic path. The vertex of a parabola represents the highest point on the path. In this equation, the vertex can be found by using the formula:

x = -b / 2a

where a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0.

By comparing the given equation with the standard form of the quadratic equation (y = ax² + bx + c), we can find that a = -0.035 and b = 1.4.

Substituting these values in the formula, we get:

x = -1.4 / 2(-0.035)

x = 20

This means that the maximum height is reached when the football has traveled a distance of 20 yards. To find the maximum height, we need to substitute this value of x back into the original equation:

y = -0.035(20)² + 1.4(20) + 1

y = 15.4

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Complete Question:

A punter kicked a 41-yard punt. The path of the football can be modeled by y=-0.035x² + 1.4x + 1 where x is the distance in yards the football is kicked and y is the height in yards the football kicked.

what is the maximum height reached by the football?

The sign of each of the functions in the given interval is positive.

Suppose \( \theta \) is in the interval \( \left(90^{\circ}, 180^{\circ}\right) \), we can find the sign of each of the following functions by using the unit circle and the reference angles.

77. \( \cos \frac{\theta}{2} \)
Since \( \theta \) is in the second quadrant, \( \frac{\theta}{2} \) will be in the first quadrant. Therefore, the sign of \( \cos \frac{\theta}{2} \) will be positive.

78. \( \sin \frac{\theta}{2} \)
Similarly, since \( \theta \) is in the second quadrant, \( \frac{\theta}{2} \) will be in the first quadrant. Therefore, the sign of \( \sin \frac{\theta}{2} \) will be positive.

79. \( \sec \frac{\theta}{2} \)
The secant function is the reciprocal of the cosine function, so the sign of \( \sec \frac{\theta}{2} \) will be the same as the sign of \( \cos \frac{\theta}{2} \), which is positive.

In conclusion, the sign of each of the functions in the given interval is positive.

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Step-by-step explanation:

Since the new poison kills half the population every 2 months, we can say that the remaining half will survive for another 2 months. Therefore, after 2 months, the rodent population will be half of 1,000,000, which is 500,000.

After another 2 months, the remaining half of the 500,000 will survive, which is 250,000.

After another 2 months, the remaining half of the 250,000 will survive, which is 125,000.

After 6 months, the rodent population will be 125,000.

Since 12 months is six 2-month periods, we can repeat this process again. After another 2 months, the rodent population will be 62,500.

After another 2 months, the rodent population will be 31,250.

After another 2 months, the rodent population will be 15,625.

Therefore, after 12 months, the rodent population will be 15,625.

The missing angle is 90°. So, to obtain the values of the sides given the lengths we will have:

8. PT = 1

   PV = 2

9. VT = 0.76

   PV = 0.66

10. PT = 3

     PV = 6

11. PV =0.58

    PT =1.154

12. PT = 1.73

    VT = 3

13. VT = 3

     PV = 3.5

The given triangle is a scalene triangle because of the three different angles it has which sum up to 180 degrees. The sides will also be different.

For the first triangle whose known side is √3, the missing values can be obtained this way:

sin 60°/ √3 = sin 90/PV

PV = √3 Sin 90/ sin 60

PV = 1.732/0.866

= 2

PT = sin 60/ √3 = sin 30/ PT

PT = √3 sin 30/sin 60

PT = 1

Using this same pattern, the values for the other figures can be obtained.

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