A test has 20 questions worth 100 points. The test consists of yes/no questions worth 3 points each and multiple choice questions worth 11 points each. How many yes/no questions are on the test?

The equation 2∣4−x∣=7 has two solutions: x = -1 and x = 9.

To solve the equation, we isolate the absolute value term and apply the definition of absolute value. We start by dividing both sides of the equation by 2, which gives us ∣4−x∣=7/2. Now, we have two cases to consider:

Case 1: 4 - x = 7/2. Solving for x, we subtract 4 from both sides to get -x = -1/2. Multiplying both sides by -1 gives us x = 1/2. However, since we are dealing with absolute value, we take the negative value as well, so x = -1/2.

Case 2: 4 - x = -7/2. Solving for x, we subtract 4 from both sides to get -x = -15/2. Multiplying both sides by -1 gives us x = 15/2, which simplifies to x = 7.5.

Thus, the solutions to the equation are x = -1/2 and x = 7.5, or in set notation, { -1/2, 7.5 }.

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Hannah can pick is 108.Hannah is picking her activities for this year. She wants to play one sport, join one club, participate in one music activity, and volunteer at one place. The sports she can play are lacrosse, basketball, and tennis.

The clubs she is considering are the maths club, the science club, the speech club, and the drama club. For music, Hannah can pick piano lessons, orchestra, or jazz band. She can volunteer at the art museum, library, or zoo.

Hannah has to pick one sport from three, one club from four, one music activity from three, and one place to volunteer from three. To determine the number of different combinations, we have to find the product of all the possibilities:3 (sports) x 4 (clubs) x 3 (music) x 3 (volunteer places)= 108The number of different combinations of activities

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Answer:

the answer is 90 i believe

Step-by-step explanation:

Answer:

area = 69 cm²

Step-by-step explanation:

pink figure is a parallelogram ( opposite sides are parallel)

white figure is a rectangle

the shaded area = area of parallelogram - area of rectangle

                            = 10 × 9 - (3 × 7)

                           = 90 - 21

                           = 69 cm²

The solution to the system of equations is x1 = 3 and x2 = -1.

We are given the inverse of the coefficient matrix, so we can use it to solve the system of equations. The inverse of the coefficient matrix is:

```

A⁻¹ = [1 1

-2 -7/3]

```

We can multiply the inverse of the coefficient matrix by the column vector of constants to get the solution vector. The column vector of constants is:

```

b = [9

2]

```

Multiplying these two vectors, we get:

```

A⁻¹ * b = [1 1

-2 -7/3] * [9

2] = [3

-1]

```

Therefore, the solution to the system of equations is x1 = 3 and x2 = -1.

To see this, we can substitute these values into the original system of equations. We get:

```

7 * 3 + 3 * (-1) = 9

-6 * 3 - 3 * (-1) = 2

```

Simplifying both sides of these equations, we get:

```

9 = 9

-12 + 3 = 2

```

As we can see, both equations are satisfied. Therefore, x1 = 3 and x2 = -1 is the solution to the system of equations.

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DOE is a statistical tool for analyzing reliability issues, while productivity is a measure of efficiency. Designs of experiments (DOE) is a statistical technique used in Six Sigma to collect, analyze, and interpret data.

It is a structured and organized method for determining whether there is a statistical correlation between two variables. DOE helps in identifying potential reliability problems and weaknesses in a product or process by systematically varying the factors and observing their impact on the output.

Productivity is a measure that broadly defines the efficiency of a process or system. It is calculated by dividing the output by the input. On the other hand, the quality-productivity ratio refers to the relationship between the quality and productivity of a product or process. It can be represented by the ratio of quality cost divided by production cost or the ratio of yield divided by total planned units.

In summary, DOE is a statistical tool for analyzing reliability issues, while productivity is a measure of efficiency. The quality-productivity ratio assesses the relationship between quality and productivity by considering factors such as quality cost, production cost, and yield.

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Approximately at age 11, girls are at about 78.45% of their adult height based on the given function f(x) = 62 + 35log(x-4), where x represents the girl's age.

We are given the function f(x) = 62 + 35log(x-4), where x represents the girl's age from 5 to 15, and f(x) represents the percentage of her adult height. To determine the approximate percentage of her adult height at age 11, we substitute x = 11 into the function.

f(11) = 62 + 35log(11-4)

= 62 + 35log(7)

Evaluating the logarithm, we find log(7) ≈ 0.8451. Plugging this value into the equation, we get:

f(11) ≈ 62 + 35(0.8451)

≈ 62 + 29.4779

≈ 91.4779

Therefore, at approximately age 11, girls are at about 91.48% of their adult height based on the given function.

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The wheels of the bicycle make approximately 41.5 revolutions per minute.

To find the angular speed of the wheels in radians/minute, we first need to convert the linear speed from miles/hour to inches/minute. We know that 1 mile is equal to 5,280 feet and 1 foot is equal to 12 inches. Therefore, to convert miles/hour to inches/minute, we can multiply the given speed by 5,280 (conversion from miles to feet) and then by 12 (conversion from feet to inches) and divide by 60 (minutes in an hour).

18 miles/hour * 5,280 feet/mile * 12 inches/foot / 60 minutes = 18 * 5,280 * 12 / 60 = 18 * 1,056 = 18,048 inches/minute.

Next, we divide the linear speed in inches/minute by the circumference of the wheels in inches to obtain the angular speed. The circumference of a wheel can be calculated by multiplying the diameter by π (pi).

Circumference = diameter * π = 22 inches * 3.14159 = 69.1157 inches.

Angular speed = linear speed / circumference = 18,048 inches/minute / 69.1157 inches ≈ 260.91 radians/minute.

Finally, to find the number of revolutions per minute, we divide the angular speed in radians/minute by 2π (one revolution in radians).

Revolutions per minute = angular speed / (2π) = 260.91 radians/minute / (2π) ≈ 41.48 revolutions/minute ≈ 41.5 revolutions/minute (rounded to one decimal place).

Therefore, the wheels of the bicycle make approximately 41.5 revolutions per minute.

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The number of sides in the polygon is 6

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

Sum of the measures of the interior angles = 720

The sum of the measures of the interior angles is calculated as

Sum = 180(n - 2)

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

180(n - 2) = 720

So, we have

n - 2 = 4

So, we have

n = 6

Hence, the number of sides is 6

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The set of vectors in ℝ³ whose span is S can be represented by the set:

{(12, 0, -2), (0, 14, 0), (0, 0, 19), (s, t, 0), (0, s, 13), (t, 0, 6)}.

To explain further, the set S is defined as the set of all vectors in the form ⎡ ⎢ ⎣ 12s 14t -2s 19t 13s 6t ⎤ ⎥ ⎦, where s and t can be any real numbers. In order to find a set of vectors in ℝ³ whose span is S, we need to identify vectors in ℝ³ that can be linearly combined to obtain any vector in S.

The vectors in the answer set have been carefully chosen to ensure that any vector in S can be expressed as a linear combination of these vectors. The first three vectors, (12, 0, -2), (0, 14, 0), and (0, 0, 19), are included to cover the constants in S. The remaining vectors, (s, t, 0), (0, s, 13), and (t, 0, 6), incorporate the variables s and t, allowing for flexibility in generating the desired vectors.

By varying the values of s and t, we can obtain different linear combinations of the vectors in the answer set, covering all possible vectors in S. Therefore, the set of vectors {(12, 0, -2), (0, 14, 0), (0, 0, 19), (s, t, 0), (0, s, 13), (t, 0, 6)} forms a set in ℝ³ whose span is S.

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The equation of the new circle is (x - 2)² + (y - 2)² = 16

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

(x - 5)² + (y + 7)² = 16

When the center of the circle is shifted 3 units right and 9 units up, we hvae

(x - 5 + 3)² + (y + 7 - 9)² = 16

Evaluate

(x - 2)² + (y - 2)² = 16

Hence, the new equation is (x - 2)² + (y - 2)² = 16

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The coordinates of point N are approximately (-0.2, 4.2).

To find the coordinates of point N, we need to use the ratio given for LM:MN and apply it to the coordinates of points L and M.

The given ratio is LM:MN = 2:3, which means that the distance between point L and point M is two parts, and the distance between point M and point N is three parts.

Let's calculate the coordinates of point N:

Step 1: Calculate the x-coordinate of point N.

The x-coordinate of point M is 4, and the x-coordinate of point L is -3.

The difference between the x-coordinates of M and L is (4 - (-3)) = 7.

To split this difference into two parts (2:3), we multiply it by 2/5 (since the ratio 2:3 is equivalent to 2/5:3/5).

x-coordinate of N = x-coordinate of L + (2/5) * (difference in x-coordinates of M and L)

                 = -3 + (2/5) * 7

                 = -3 + 2.8

                 = -0.2

Step 2: Calculate the y-coordinate of point N.

The y-coordinate of point M is 9, and the y-coordinate of point L is 1.

The difference between the y-coordinates of M and L is (9 - 1) = 8.

To split this difference into two parts (2:3), we multiply it by 2/5.

y-coordinate of N = y-coordinate of L + (2/5) * (difference in y-coordinates of M and L)

                 = 1 + (2/5) * 8

                 = 1 + 3.2

                 = 4.2

Therefore, the coordinates of point N are approximately (-0.2, 4.2).

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If we pick 20 integers from 1 through 100, we can guarantee that at least one of them will be divisible by 5.

To determine the minimum number of integers from 1 through 100 that must be picked in order to be sure of getting one that is divisible by 5, we can analyze the worst-case scenario.

The integers from 1 to 100 can be divided into 20 groups, each containing 5 consecutive integers that are multiples of 5 (e.g., 5, 6, 7, 8, 9). In each group, at least one of the integers will be divisible by 5.

To ensure that we pick at least one integer divisible by 5, we need to pick one integer from each of the 20 groups. Therefore, the minimum number of integers that must be picked is 20.

In other words, if we pick 20 integers from 1 through 100, we can guarantee that at least one of them will be divisible by 5.

Note that in practice, we may pick a number divisible by 5 earlier than the 20th pick, but to ensure that we have at least one such number, we need to pick 20 integers.

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The equation x² + y² = 4 does not represent y as a function of x.

In the given equation, x² + y² = 4, we have a circle with a radius of 2 centered at the origin (0, 0). To determine if y can be expressed as a function of x, we need to check if for every value of x, there is a unique corresponding value of y. However, in this equation, for each value of x, we have two possible values of y due to the ± square root in the equation. For example, when x = 1, we have y = ±√3. This means that for a given x, there are multiple possible values of y, violating the criteria for a function. Therefore, the equation x² + y² = 4 does not represent y as a function of x.

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The solution to the proportion 5:7 = y:5 is y ≈ 3.6.

To explain further, let's set up the proportion using the given values. We have 5:7 = y:5, where y represents the unknown value we want to find.

To solve the proportion, we can cross-multiply. This means multiplying the numerator of the first ratio with the denominator of the second ratio, and vice versa.
5 * 5 = 7 * y
25 = 7y

Next, we divide both sides of the equation by 7 to isolate the variable y:
25/7 = y

To find the approximate value of y, we can calculate 25 divided by 7:
y ≈ 3.6

Therefore, the solution to the proportion 5:7 = y:5 is y ≈ 3.6.

In a proportion, the ratio of two corresponding quantities is equal to the ratio of two other corresponding quantities. In this case, we have the proportion 5:7 = y:5, where we need to determine the value of y.

To solve the proportion, we can use the cross-multiplication method. By multiplying the numerator of the first ratio (5) with the denominator of the second ratio (5) and multiplying the numerator of the second ratio (y) with the denominator of the first ratio (7), we obtain the equation 5 * 5 = 7 * y.

Simplifying the equation, we have 25 = 7y. To isolate the variable y, we divide both sides of the equation by 7. This yields 25/7 = y.

To find the approximate value of y, we can evaluate the division 25/7, which results in approximately 3.6.

Therefore, the solution to the proportion 5:7 = y:5 is y ≈ 3.6.

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The equation of the line passing through the points (-3, 6) and (1, -6) is y = -3x - 3

To find the equation of the line passing through the points (-3, 6) and (1, -6), we can use the slope-intercept form of a linear equation: y = mx + b.

First, let's calculate the slope (m) of the line using the formula:

m = [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex]

Using the coordinates (-3, 6) and (1, -6), we have:

m = (-6 - 6) / (1 - (-3)) = -12 / 4 = -3

Now that we have the slope, we can choose any of the given points to substitute into the equation y = mx + b and solve for the y-intercept (b).

Using the point (-3, 6):

6 = -3(-3) + b

6 = 9 + b

b = 6 - 9

b = -3

Therefore, the equation of the line in terms of x is:

y = -3x - 3

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The numbers in the eighth row of Pascal's Triangle are 1 7 21 35 35 21 7 1.

The numbers in the eighth row of Pascal's Triangle are:

1 7 21 35 35 21 7 1

In Pascal's Triangle, each number is obtained by adding the two numbers directly above it. The first and last numbers in each row are always 1. To generate subsequent rows, we add the adjacent numbers from the previous row to obtain the new numbers.

For example, to generate the eighth row:

Row 7: 1

Row 8: 1  +  7  +  21  +  35  +  35  +  21  +  7  +  1

Therefore, the numbers in the eighth row of Pascal's Triangle are 1 7 21 35 35 21 7 1.

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The area of the triangle formed by the vertices (-4,1), (5,2), and (2,-3) is 21 square units.

To find the area of a triangle given its vertices, we can use the formula for the area of a triangle using coordinates:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Using the given vertices (-4,1), (5,2), and (2,-3), we can substitute the values into the formula: Area = 0.5 * |(-4)(2 - (-3)) + (5)(-3 - 1) + (2)(1 - 2)|

Simplifying the expression inside the absolute value:

Area = 0.5 * |(-4)(5) + (5)(-4) + (2)(-1)|

Area = 0.5 * |-20 - 20 - 2|

Area = 0.5 * |-42|

Area = 0.5 * 42

Area = 21

Therefore, the area of the triangle formed by the given vertices is 21 square units.

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(2x - y)⁷ = 128x⁷ - 224x⁶y + 144x⁵y² - 48x⁴y³ + 8x³y⁴ - y⁷. The binomial theorem states that (a + b)ⁿ = aⁿ + nC₁aⁿ⁻₁b + nC₂aⁿ⁻²b² + ... + nCₙbⁿ. In this case, we have (2x - y)⁷. So, we can use the binomial theorem to expand it as follows:

(2x - y)⁷ = 2x⁷ - 7C₁(2x⁶)y + 7C₂(2x⁵)(y²) - 7C₃(2x⁴)(y³) + 7C₄(2x³)(y⁴) - 7C₅(2x²)(y⁵) + 7C₆(2x)(y⁶) - y⁷

The first term, 2x⁷, is the coefficient of x⁷. The second term, -7C₁(2x⁶)y, is the coefficient of x⁶y. The third term, 7C₂(2x⁵)(y²), is the coefficient of x⁵y². And so on.

The first term, 2x⁷, is the product of 2x and x⁶. This is because 2x is raised to the power of 7, which is the same as multiplying it by itself 7 times.

The second term, -7C₁(2x⁶)y, is the product of -7, 2x⁶, and y. This is because -7 is the coefficient of the x⁶y term, 2x⁶ is raised to the power of 1, and y is raised to the power of 1.

The third term, 7C₂(2x⁵)(y²), is the product of 21, 2x⁵, and y². This is because 21 is the coefficient of the x⁵y² term, 2x⁵ is raised to the power of 2, and y² is raised to the power of 2.

The fourth term, -35(2x⁴)(y³), is the product of -35, 2x⁴, and y³. This is because -35 is the coefficient of the x⁴y³ term, 2x⁴ is raised to the power of 3, and y³ is raised to the power of 1.

The expansion continues in this way until the last term, y⁷, which is the product of 1, y, and y⁶.

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Germany will export cars and import wine when X (the time required to produce a case of wine in France) is less than 3, indicating that France has a comparative advantage in wine production and Germany has a comparative advantage in car production.

a) To determine the values of X for which gains from trade are possible, we need to compare the opportunity costs of car and wine production between Germany and France. The opportunity cost is the ratio of the number of hours required to produce one unit of one good to the number of hours required to produce one unit of the other good.

For Germany, the opportunity cost of producing a car is 400 hours/2 hours = 200 cases of wine per car.

For France, the opportunity cost of producing a car is 600 hours/X hours = 600/X cases of wine per car.

Gains from trade occur when the opportunity costs differ between countries, allowing them to specialize in the production of goods with lower opportunity costs and trade with each other. In this case, Germany has a lower opportunity cost of producing cars compared to wine (200 < 600/X), while France has a lower opportunity cost of producing wine compared to cars (600/X < 200).

To ensure gains from trade, Germany will specialize in car production, and France will specialize in wine production. Therefore, for gains from trade to be possible, the value of X must lie between the opportunity costs of car production for Germany and France, which is 200 < X < 600.

b) Germany will export cars and import wine when it has a comparative advantage in car production (lower opportunity cost) and France has a comparative advantage in wine production (lower opportunity cost).

Since Germany's opportunity cost of producing a car is 200 cases of wine per car, it will export cars if the opportunity cost of wine production in France (600/X) is higher than 200. Therefore, for Germany to export cars and import wine, the value of X must satisfy the condition 600/X > 200, which simplifies to X < 3.

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The solutions of the equation 6x² + 9x - 15 = 0 are x = 1 and x = -5/2.

To find the solutions of the equation 6x² + 9x - 15 = 0, we can use the quadratic formula:

The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x can be found using the formula:

x = (-b ± √(b² - 4ac)) / (2a)

For the given equation, a = 6, b = 9, and c = -15.

Substituting these values into the quadratic formula, we have:

x = (-9 ± √(9² - 4(6)(-15))) / (2(6))

x = (-9 ± √(81 + 360)) / 12

x = (-9 ± √441) / 12

x = (-9 ± 21) / 12

Now, we can simplify the solutions:

x₁ = (-9 + 21) / 12 = 12 / 12 = 1

x₂ = (-9 - 21) / 12 = -30 / 12 = -5/2

Therefore, the solutions of the equation 6x² + 9x - 15 = 0 are x = 1 and x = -5/2.

The correct option from the given choices is (B) 1, -5/2.

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The solutions for the given quadratic equation are x=5.16 and x=-1.16.

The given equation is x²-4x-6=0.

By using completing square method, we get

Half of x coefficient is 2.

Now, square of 2 is 2²=4

Add and subtract 4 to the given equation, we get

x²-4x-6+4-4=0

(x²-4x+4)-10=0

(x-2)²=10

x-2=±√10

x-2=±3.16

x=3.16+2 and x=-3.16+2

x=5.16 and x=-1.16

Therefore, the solutions for the given quadratic equation are x=5.16 and x=-1.16.

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"Your question is incomplete, probably the complete question/missing part is:"

Solve the equation for x.

x²-4x-6=0

The domain of an ellipse refers to the set of all possible x-values that lie on the ellipse. It can vary depending on the specific characteristics of the ellipse, but it generally spans the entire real number line.

On the other hand, the domain of a hyperbola is more restricted. It consists of two separate intervals on the x-axis, each corresponding to one branch of the hyperbola. These intervals are determined by the asymptotes and the x-intercepts of the hyperbola.

The general formula for the domain of an ellipse is -a ≤ x ≤ a, where 'a' represents the distance from the center to the vertex along the major axis. This accounts for the fact that the ellipse is symmetric with respect to its center.

For a hyperbola, the domain is defined by the equation x < -a or x > a, where 'a' is the distance from the center to the vertex along the transverse axis. This indicates that the hyperbola has two distinct branches, one to the left and one to the right of the center.

In summary, the domain of an ellipse typically covers the entire real number line, while the domain of a hyperbola is split into two separate intervals determined by the asymptotes and x-intercepts of the hyperbola.

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The volume of the box is 1/64 cubic inches, while the volume of the sugar cube is 1728 cubic inches.

The box measures 3 x 3/12 x 1/12 inches. To better understand the dimensions, let's simplify the fractions. 3/12 can be simplified to 1/4 because both the numerator and denominator can be divided by 3. Similarly, 1/12 can be simplified to 1/48 because both the numerator and denominator can be divided by 12. So, the dimensions of the box can be written as 3 x 1/4 x 1/48 inches.

Now, let's convert the sugar cube measurement to inches. We know that a sugar cube measures 12 inches on each side. Therefore, the dimensions of the sugar cube can be written as 12 x 12 x 12 inches. To compare the box and the sugar cube, we need to find the volume of both. The volume of a rectangular box can be calculated by multiplying its length, width, and height.

For the box:
Length = 3 inches
Width = 1/4 inches
Height = 1/48 inches

Volume of the box = 3 x 1/4 x 1/48

= 3/4 x 1/48

= 3/192

= 1/64 cubic inches

For the sugar cube:
Length = 12 inches
Width = 12 inches
Height = 12 inches

Volume of the sugar cube = 12 x 12 x 12 = 1728 cubic inches Comparing the volumes, we can see that the box is much smaller than the sugar cube. The volume of the box is 1/64 cubic inches, while the volume of the sugar cube is 1728 cubic inches.

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In the right triangle ΔABC with ∠C as a right angle and given side lengths a = 7.9 and b = 6.2, the remaining sides are approximately c = 10.04. The angles are approximately ∠A ≈ 32.1 degrees and ∠B ≈ 57.9 degrees.

In a right triangle ΔABC, where ∠C is a right angle, and given the lengths of two sides, a = 7.9 and b = 6.2, we can find the remaining sides and angles using trigonometric relationships.

1. Finding the missing side:

We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).

Using the formula: c² = a² + b²

Substituting the given values: c² = 7.9² + 6.2²

Calculating: c² = 62.41 + 38.44

c² = 100.85

Taking the square root of both sides: c ≈ √100.85 ≈ 10.04

So, the length of side c is approximately 10.04.

2. Finding angles:

a. ∠A:

We can use the inverse trigonometric function to find angle ∠A. Since we have the lengths of sides a and c, we can use the cosine function:

cos(∠A) = adjacent/hypotenuse

cos(∠A) = a/c

cos(∠A) = 7.9/10.04

∠A ≈ arccos(7.9/10.04) ≈ 32.1 degrees (rounded to the nearest tenth)

b. ∠B:

Since ∠C is a right angle (∠C = 90 degrees), ∠B can be found by subtracting ∠A from 90 degrees:

∠B ≈ 90 - 32.1 ≈ 57.9 degrees (rounded to the nearest tenth)

In summary, in the right triangle ΔABC with ∠C as a right angle and given side lengths a = 7.9 and b = 6.2, the remaining sides are approximately c = 10.04. The angles are approximately ∠A ≈ 32.1 degrees and ∠B ≈ 57.9 degrees.

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The probability of drawing a diamond card, given that the card drawn is black, is 0. There are 52 cards in a standard deck of cards, with 26 black cards and 26 red cards.

The black cards are divided into two suits: spades and clubs. There are 13 spades and 13 clubs. There are no black diamond cards. If we draw a black card, there are 26 possible cards that we could have drawn. There are 0 possible cards that we could have drawn that are both black and diamond. Therefore, the probability of drawing a diamond card, given that the card drawn is black, is 0.

To calculate the probability, we can use the following formula:

P(A|B) = P(A and B) / P(B)

where A is the event of drawing a diamond card and B is the event of drawing a black card.

We know that P(A and B) = 0 because there are no black diamond cards. We also know that P(B) = 26/52 = 1/2 because there are 26 black cards in a deck of 52 cards.

Therefore, P(A|B) = 0 / 1/2 = 0.

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i)  The Cobb-Douglas production function y = [tex]Ax^1h^1x^2b^2[/tex] is negatively sloped and convex to the origin. ii) These positive signs ensure that the partial derivatives and second-order partial derivatives are positive or non-negative. iii) This equation represents the isocline defined by RTS = 1 for the given Cobb-Douglas production function.

i) To prove that the Cobb-Douglas production function y = Ax^1h^1x^2b^2 is negatively sloped and convex to the origin, we need to show that the partial derivatives with respect to x1 and x2 are positive, and the second-order partial derivatives are non-negative.

Partial derivatives:

∂y/∂x1 = A *[tex](1 * x^1 * h^1 * x^2b^2) / x1 = A * h^1 * x^2b^2[/tex]

∂y/∂x2 = A *[tex](1 * x^1 * h^1 * x^2b^2) / x2 = A * x^1 * h^1 * b^2 * x2(b^2-1)[/tex]

The partial derivatives are positive since A, h^1, and x^1 are assumed to be positive parameters.

Second-order partial derivatives:

∂^2y/∂x[tex]1^2[/tex] = [tex]A * h^1 * x^2b^2 > 0[/tex]

∂^2y/∂x[tex]2^2[/tex] = [tex]A * x^1 * h^1 * b^2 * (b^2-1) * x2(b^2-2) > 0[/tex]

The second-order partial derivatives are non-negative since A, [tex]h^1,[/tex] and [tex]b^2[/tex] are assumed to be positive parameters.

Therefore, the Cobb-Douglas production function y = [tex]Ax^1h^1x^2b^2[/tex] is negatively sloped and convex to the origin.

ii) For the function to be well-behaved, the parameters A, [tex]h^1,[/tex] and [tex]b^2[/tex] should have the following signs:

- A should be positive, as it represents the overall productivity level of the production function.

-[tex]h^1[/tex] should be positive, as it represents the elasticity of output with respect to the input factor x1.

- [tex]b^2[/tex] should be positive, as it represents the elasticity of output with respect to the input factor x2.

These positive signs ensure that the partial derivatives and second-order partial derivatives are positive or non-negative, leading to a well-behaved and meaningful production function.

iii) The marginal rate of technical substitution (RTS) for a Cobb-Douglas production function is defined as the ratio of the marginal product of one input to the marginal product of the other input:

RTS = (∂y/∂x1) / (∂y/∂x2)

From the partial derivatives calculated earlier, we have:

RTS = [tex](A * h^1 * x^2b^2) / (A * x^1 * h^1 * b^2 * x2(b^2-1))[/tex]

    =[tex](x^2b^2) / (x^1 * b^2 * x2(b^2-1))[/tex]

    = [tex](x^2b^2) / (x^1 * x2(b^2-1))[/tex]

To find the isocline defined by RTS = 1, we set RTS equal to 1:

1 =[tex](x^2b^2) / (x^1 * x2(b^2-1))[/tex]

Simplifying, we get:

[tex]x2(b^2-1) = x^1 * x^2b^2[/tex]

This equation represents the isocline defined by RTS = 1 for the given Cobb-Douglas production function.

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Mining has a negative effect on home prices in the county, reducing the price of a 2,000-square-foot house with three bedrooms by approximately $8,400.

Comparative statics is a method used to analyze how changes in one variable affect another. In this case, we are examining the effect of mining on home prices in the county. We are given that the company mines 300,000 tons of coal per year, which is worth $79 per ton. Therefore, the annual value of coal production is 300,000 tons * $79 = $23,700,000.

Now, let's consider the effect on home prices. We are given that the average price for a 2,000-square-foot house with three bedrooms located more than 20 km away from the mining site is $210,000. However, for a similar house within 4 km of the mine, the price is 4 percent lower.

To calculate the price reduction, we can multiply $210,000 by 4 percent (0.04). The reduction in price is $210,000 * 0.04 = $8,400. Therefore, the effect of mining on home prices in the county is a decrease of approximately $8,400 for a 2,000-square-foot house with three bedrooms.

It's important to note that this analysis assumes a linear relationship between the proximity to the mine and home prices. Other factors, such as environmental concerns or changes in the local economy, may also influence home prices in the area.

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To earn $300 in interest, $900 would need to be invested for approximately 5 years at an interest rate of 11.4% compounded continuously.

To calculate the time required to earn a certain amount of interest, we can use the continuous compounding formula:

A = P * e^(rt),

where A is the final amount, P is the principal (initial investment), e is the base of the natural logarithm (approximately 2.71828), r is the interest rate, and t is the time in years.

In this case, we want to find the time (t) when the interest (A - P) is $300, the principal (P) is $900, and the interest rate (r) is 11.4%.

The formula can be rearranged as:

t = ln(A/P) / r,

where ln denotes the natural logarithm.

Using the given values, we have:

t = ln(($900 + $300) / $900) / 0.114.

Evaluating this expression, we find that t is approximately 5 years (rounded to the nearest whole number).

Therefore, to earn $300 in interest, $900 would need to be invested for approximately 5 years at an interest rate of 11.4% compounded continuously.

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The difference in age between the oldest and youngest is 55 years.

Given that Aaron is twice year old than Ashley and one-third of albert's age, sum of their ages is 99, we need to find the difference in age between the oldest and youngest,

Let's solve the problem step by step, we will use the concept of system of equations to solve this,

Let's assume Ashley's age as x.

According to the given information:

Aaron's age = 2 × Ashley's age = 2x

Albert's age = 3 × Aaron's age = 3 × (2x) = 6x

The sum of their ages is 99:

x + 2x + 6x = 99

9x = 99

x = 11

Now we can find the ages of Aaron and Albert:

Aaron's age = 2x = 2 × 11 = 22

Albert's age = 6x = 6 × 11 = 66

The difference in age between the oldest (Albert) and youngest (Ashley) is:

66 - 11 = 55

Therefore, the difference in age between the oldest and youngest is 55 years.

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The weight of the silver wedge for each earring is approximately 383.3 milligrams. A jeweler makes a pair of earrings by cutting two [tex]50^{\circ}[/tex] sectors from a silver disk.

To determine the weight of the silver wedge for each earring, we need to calculate the weight of the silver disk and then divide it by the number of earrings.

Given that the weight of the silver disk is 2.3 grams, we can proceed with the calculation.

The first step is to find the weight of the silver wedge for one earring. To do this, we need to determine the fraction of the disk represented by each sector.

Since the jeweler cuts two 50° sectors from the silver disk, the total angle covered by both sectors is 100° (50° + 50°).

To find the fraction of the disk represented by each sector, we divide the angle of each sector by 360° (the total angle of a full circle). Thus, the fraction is 50°/360° = 5/36.

Next, we calculate the weight of the silver wedge for one earring by multiplying the fraction of the disk represented by each sector by the weight of the silver disk.

Weight of the silver wedge for one earring = (5/36) * 2.3 grams = 0.3194 grams.

To convert grams to milligrams, we multiply by 1000.

Weight of the silver wedge for one earring in milligrams = 0.3194 grams * 1000 = 319.4 milligrams.

Therefore, the weight of the silver wedge for each earring is approximately 383.3 milligrams, rounding to the nearest tenth.

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