An open box is made from a 40​-cm by ​70-cm piece of tin by cutting a square from each corner and folding up the edges. The area of the resulting base is 1984cm Superscript 2. What is the length of the sides of the​ squares?

The equation of the line that passes through the points (1, 0) and (3, 6) is y = 3x - 3.

To find the equation of a line passing through two points, 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.

Given points:

Point 1: (1, 0)

Point 2: (3, 6)

Step 1: Calculate the slope (m) using the formula:

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

Substituting the coordinates:

m = (6 - 0) / (3 - 1)

m = 6 / 2

m = 3

Step 2: Substitute one of the given points and the slope into the equation y = mx + b to find the y-intercept (b).

Using Point 1 (1, 0):

0 = 3(1) + b

0 = 3 + b

b = -3

Step 3: Write the equation of the line using the slope (m) and the y-intercept (b):

y = 3x - 3

Therefore, the equation of the line that passes through the points (1, 0) and (3, 6) is y = 3x - 3.

This equation represents a line with a slope of 3, indicating that for every increase of 1 unit in the x-coordinate, the y-coordinate increases by 3 units. The y-intercept of -3 means that the line crosses the y-axis at the point (0, -3). By substituting any x-value into the equation, we can determine the corresponding y-value on the line.

Hence, the equation of the line passing through (1, 0) and (3, 6) is y = 3x - 3.

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The equations that represent a line parallel to 3x - 4y = 7 and passes through the point (-4, -2) are:

0 = 3x - 4y - 4

3x - 4y = 4

To determine which equations represent a line that is parallel to the given line and passes through the point (-4, -2), we need to find the equations that have the same slope as the given line.

The equation 3x - 4y = 7 can be rewritten in slope-intercept form as y = (3/4)x - 7/4. From this form, we can see that the slope of the given line is 3/4.

Now let's analyze the options:

Oy = -3x + 1: This equation has a slope of -3, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

0 = 3x - 4y - 4: This equation can be rewritten as 3x - 4y = 4. Comparing this to the given line, we can see that it has the same coefficients of x and y, which means it has the same slope of 3/4. Therefore, this option represents a line parallel to the given line.

4x - 3y = -3: This equation has a slope of 4/3, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

Oy - 2 = -(x - 4): This equation can be rewritten as y = -x + 2. The slope of this line is -1, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

Oy + 2 = 2(x + 4): This equation can be rewritten as y = 2x + 6. The slope of this line is 2, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

In conclusion, the equations that represent a line parallel to 3x - 4y = 7 and passes through the point (-4, -2) are:

0 = 3x - 4y - 4

3x - 4y = 4

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The values of w, x, y, z are 120°, 60°, 120° , 60° respectively.

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.

lines PQ and RS are parallel to each other and AB and CD are also parallel to each other. A pair of parallel line will serve as transversal.

Angles on parallel lines can be ;

Vertically opposite, alternate, corresponding and in each case the angles are equal.

w = 120°( vertically opposite angles)

x = 180-120 = 60( corresponding angles)

x = z = 60° ( corresponding angles)

y = 180-60 = 120°( angles on a straight line)

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Answer:

Step-by-step explanation:

According to the triangle proportionality theorem (or thales theorem), the lines are parallel if AD/BD = AE/CE

15/12= 1.25

10/8 = 1.25

Because the ratios are equal, the lines are parallel to each other.  

Answer:

a)  ∩

Step-by-step explanation:

set A : {6,8,10,12}

set B : {5,6,7,8,9}

set C : {6,8}

A ∪ B = {6,8,10,12} ∪ {5,6,7,8,9} = {5,6,7,8,9,10,12} ≠ C

A ∩ B = {6,8,10,12} ∩ {5,6,7,8,9} = {6, 8} = C

Therfore{6,8,10,12} ∩ {5,6,7,8,9} = {6, 8}

Navern Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 11.36%.

Value-added time is the time spent on activities that directly add value to the product.

Value-added time: Process time = 5 days (the process of manufacturing the elevators)

Total cycle time: Wait time + Inspection time + Process time + Move time + Queue time

= 12 days + 12 days + 5 days + 6 days + 9 days

= 44 days

MCE = (Value-added time / Total cycle time) * 100

= (5 days / 44 days) * 100

≈ 11.36%

Therefore, Navern Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 11.36%.

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Answer:

a. 0.9375, 0.46875

b. geometric sequence

c. equation: [tex] 7.5 * (\frac{1}{2})^(n-1)[/tex]

Step-by-step explanation:

a.

The table can be finished as follows:

n t(n)

1 7.5

2 3.75

3. 1.875

4. 0.9375

5 0.46875

b.

The type of sequence is a geometric sequence.

A geometric sequence is a sequence of numbers where the ratio between any two consecutive terms is constant.

In this case, the ratio between any two consecutive terms is 3.75/7.5=½ ,

so the sequence is geometric.

c.

The equation for the sequence is t(n) = 7.5 * (1/2)^n.

This equation can be found by looking at the first term of the sequence (7.5) and the common ratio (1/2).

t(1) = 7.5

t(2) = 7.5 * (1/2) = 3.75

t(3) = 7.5 * (1/2)^2 = 1.875

The equation can also be found by looking at the general formula for a geometric sequence,

which is [tex]t(n) = a*r^{n-1}[/tex]

In this case,

t(n) =[tex] 7.5 * (\frac{1}{2})^{n-1}[/tex]

This is the required equation.

Answer:

[tex]\textsf{a.}\quad \begin{array}{|c|c|c|c|c|c|}\cline{1-6}\vphantom{\dfrac12} n&1&2&3&4&5\\\cline{1-6}\vphantom{\dfrac12}t(n)&7.5&3.75&1.875&0.9375&0.4687\\\cline{1-6}\end{array}[/tex]

[tex]\textsf{b.} \quad \textsf{Geometric sequence.}[/tex]

[tex]\textsf{c.} \quad t(n)=7.5(0.5)^{n-1}[/tex]

Step-by-step explanation:

Before we can complete the table, we need to determine if the sequence is arithmetic or geometric.

To determine if a sequence is arithmetic or geometric, examine the pattern of the terms in the sequence.

Calculate the difference between consecutive terms by subtracting one term from the next:

[tex]t(2)-t(1)=3.75-7.5=-3.75[/tex]

[tex]t(3)-t(2)=1.875-3.75 = -1,875[/tex]

As the difference is not common, the sequence is not arithmetic.

Calculate the ratio between consecutive terms by dividing one term by the previous term.

[tex]\dfrac{t(2)}{t(1)}=\dfrac{3.75}{7.5}=0.5[/tex]

[tex]\dfrac{t(3)}{t(2)}=\dfrac{1.875}{3.75}=0.5[/tex]

As the ratio is common, the sequence is geometric.

To complete the table, multiply the preceding term by the common ratio 0.5 to calculate the next term:

[tex]t(4)=t(3) \times 0.5=1.875 \times 0.5=0.9375[/tex]

[tex]t(5)=t(4) \times 0.5=0.9375 \times 0.5=0.46875[/tex]

Therefore, the completed table is:

[tex]\begin{array}{|c|c|c|c|c|c|}\cline{1-6}\vphantom{\dfrac12} n&1&2&3&4&5\\\cline{1-6}\vphantom{\dfrac12}t(n)&7.5&3.75&1.875&0.9375&0.4687\\\cline{1-6}\end{array}[/tex]

To find an equation for the sequence, use the general form of a geometric sequence:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

In this case, the first term is the value of t(n) when n = 1, so a = 7.5

We have already calculated the common ratio as being 0.5, so r = 0.5.

Substitute these values into the formula to create an equation for the sequence:

[tex]t(n)=7.5(0.5)^{n-1}[/tex]

Answer:

7: 63

8: 73

arithmetic sequence

y = 10x - 7

or f(n) = 10x -7

or

[tex]a_{n}[/tex] = 3 + (n-1)10

Step-by-step explanation:

the output increases by 10 every time that the input increases by 1.  That gives us our common difference or slope.  The y intercept is -7. That is the value is you worked backwards until you get to n = 0.  The initial value  is 3.  That is when n is 1.

When n is 3, f(n) is 23

When n is 2, f(n) is 13

When n is 1, f(n) is 3

When n is 0, f(n) is -7

I am not sure if this is clear.  I am assuming that you have a lot of knowledge of linear equations and how to write arithmetic sequence.  If my explanation is confusing it is me and not you.

Answer:

a. 63,73

b. Arithmetic sequence

c.t(n)=10n-7

Explanation:

a. Here is the completed table:

n | t(n)

4 | 33

5 | 43

6 | 53

7 | 63

8 | 73

b.

The type of sequence is arithmetic.

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant.

In this case, the difference between any two consecutive terms is 10.

c.

The equation for the arithmetic sequence is:

t(n)=a+(n-1)d

where:

For Question:

Now

equation becomes:

t(4) = a+(4-1)10

33=a+30

a=33-30

a=3

Now, the Equation becomes

t(n) = 3+(n-1)10

t(n) = 3+10n-10

t(n)=10n-7

Answer: true

Step-by-step explanation:

Answer:

Step-by-step explanation:

                                                   error

Answer:

x ≈ 75.2

Step-by-step explanation:

using the tangent ratio in the right triangle

tan62° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{40}[/tex] ( multiply both sides by 40 )

40 × tan62° = x , then

x ≈ 75.2 ( to the nearest tenth )

In 2025, both colleges will have an enrollment of 21,200 students.

To determine when the colleges will have the same enrollment, we need to set up an equation based on the projected enrollment increases and declines.

Let's assume the number of years from 2020 is represented by 'x'.

For College A, the projected enrollment can be represented by: 18,700 + 500x.

For College B, the projected enrollment can be represented by: 26,200 - 1000x.

To find when the colleges will have the same enrollment, we need to solve the equation:

18,700 + 500x = 26,200 - 1000x

Combining like terms, we get:

1500x = 7,500

Dividing both sides by 1500, we find:

x = 5

Therefore, the colleges will have the same enrollment in 5 years, or in the year 2025.

To determine the enrollment at that time, we substitute x = 5 into either equation. Let's use College A's equation:

Enrollment in College A in 2025:

18,700 + 500(5) = 18,700 + 2500 = 21,200 students

Enrollment in College B in 2025:

26,200 - 1000(5) = 26,200 - 5000 = 21,200 students

Both colleges will have an enrollment of 21,200 students in the year 2025 when they have the same enrollment.

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Avoidable costs are costs that can be eliminated if the activity that caused them is discontinued. The examples are listed.

Direct materials: Direct materials are the materials that go into a product and can be easily traced to it. For example, the wood used to make a table is a direct material. If a company decides to stop making tables, it can avoid the cost of buying wood.

Direct labor: Direct labor is the labor that is directly involved in making a product. For example, the wages paid to the workers who assemble a car are direct labor. If a company decides to stop making cars, it can avoid the cost of paying direct labor.

Variable overhead: Variable overhead is the overhead costs that vary with the number of units produced. For example, the cost of electricity used to power a factory is a variable overhead cost. If a company decides to stop producing a product, it can avoid the variable overhead costs associated with that product.

Sunk costs: Sunk costs are costs that have already been incurred and cannot be recovered. For example, the cost of research and development for a new product is a sunk cost. If the company decides not to launch the product, it cannot recover the cost of research and development.

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The notation of the transformation of the polygons is (x, y) = (x - 9, y + 2)

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

The polygons 1 and 2

In the graph, we can see that

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

(x, y) = (x - 9, y + 2)

This means that the transformation of the polygons is (x, y) = (x - 9, y + 2)

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The valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Let's solve each inequality step by step:

1. Two less than a number (x) is no more than 5:

The inequality is given as x - 2 ≤ 5. Adding 2 to both sides, we get x ≤ 7. This means that any number less than or equal to 7 satisfies the inequality.

2. Seven subtracted from 4 times a number (x) is more than 13:

The inequality is given as 4x - 7 > 13. Adding 7 to both sides, we get 4x > 20. Dividing both sides by 4, we obtain x > 5. So any number greater than 5 satisfies the inequality.

3. The sum of two times a number (x) and -2 is at least 8:

The inequality is given as 2x - 2 ≥ 8. Adding 2 to both sides, we get 2x ≥ 10. Dividing both sides by 2, we have x ≥ 5. So any number greater than or equal to 5 satisfies the inequality.

4. Four added to 3 times a number (x) is less than 19:

The inequality is given as 3x + 4 < 19. Subtracting 4 from both sides, we get 3x < 15. Dividing both sides by 3, we obtain x < 5. So any number less than 5 satisfies the inequality.

Based on the above solutions, the valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. These numbers can be represented on a number line graph by marking the appropriate points and shading the corresponding regions.

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The number of $10 tickets sold is 1210 tickets.

The number of $20 tickets sold is 1344 tickets.

The number of $30 tickets sold is 827 tickets.

In order to write a system of linear equations to describe this situation, we would assign variables to the number of each tickets sold, and then translate the word problem into an algebraic equation as follows:

Since the basketball team sold 3381 tickets overall, 134 more $20 tickets than $10 tickets and the total sales are $63,790, a system of three linear equations to model this situation is given by;

x + y + z = 3381

y - x = 134

10x + 20y + 30z = 63790  ⇒ x + 2y + 3z = 6379

Next, we would solve the system of three linear equations simultaneously as follows;

x + x + 134 + z = 3381

2x + 134 + z = 3381

z = 3247 - 2x

x + 2(x + 134) + 3(3247 - 2x) = 6379

x = 1210 tickets.

For the value of y, we have:

y = x + 134

y = 1210 + 134

y = 1344 tickets.

For the z-value, we have:

z = 3247 - 2x

z = 3247 - 2(1210)

z = 827 tickets.

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Complete Question:

A basketball team sells tickets that cost $10, $20, or, for VIP seats, $30. The team has sold 3381 tickets overall. It has sold 134 more $20 tickets than $10 tickets. The total sales are $63,790. How many tickets of each kind have been sold?

How many $10 tickets were sold?

How many $20 tickets were sold?

how many $30 tickets sold?

Matching of the equations of the Ellipses to their equivalent equations in standard form are:

The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² - 36y + 9y² = 0

The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

The general form and the standard form of the ellipse are:

- The general form is: Ax² + Bxy + Cy² + Dx + Ey + F = 0

- The standard form is: (x - h)²/a² + (y - k)²/b² = 1

1) x²/3² + (y - 2)²/2² = 1

Expanding gives:

x²/9 + (y - 2)²/4 = 1

Multiply through by 36 to get:

4x² + 9(y - 2)² = 36

Expand bracket to get:

4x² + 9y² - 36y + 36 = 36

Subtract 36 from both sides to get:

4x² - 36y + 9y² = 0

The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² + 9y² - 36y = 0

2) (x + 7)²/7² + y²/6² = 1

(x + 7)²/49 + y²/36 = 1

Multiply through by 1764 to get:

36(x + 7)² + 49y² = 1764

Expand bracket to get:

36x² + 504x + 1764 + 49y² = 1764

Subtract 1764 from both sides to get:

36x² + 504x + 49y² = 0

The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

3) x²/6² + (y + 4)²/4² = 1

x²/36 + (y + 4)²/16 = 1

Multiply through by 576 to get:

16x² + 36(y + 4)² = 576

Expand bracket to get:

16x² + 36y² + 288y + 576 = 576

Subtract 576 from both sides to get:

16x² + 288y + 36y² = 0

The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

4)  (x - 3)²/3² + y²/5² = 1

(x - 3)²/9 + y²/25 = 1

Multiply through by 225 to get:

25(x - 3)² + 9y² = 225

Expand the bracket power to get:

25x² - 150x + 225 + 9y² = 225

Subtract 225 from both sides

25x² - 150x + 9y² = 0

The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

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Match the equations of ellipses to their equivalent equations in standard form.

25x² - 150x + 9y² = 0

16x² + 288y + 36y² = 0

49x² + 686 + 36y² = 0

4x² - 36y + 9y² = 0

36x² + 504x + 49y² = 0

9x² - 54x + 25y² = 0

x²/3² + (y - 2)²/2² = 1  ⇒

(x + 7)²/7² + y²/6² = 1 ⇒

x²/6² + (y + 4)²/4² = 1 ⇒

(x - 3)²/3² + y²/5² = 1 ⇒  

Answer: y = (-28)x + 14

Step-by-step explanation:

The equation for the line tangent at point (a, b) can be found using the formula y = mx + c, where m is the slope of the line and c is the y-intercept. At point (6, 14), we need to find the value of m.

We start by finding the derivative of the original function g(x). Its derivative is given by:

dg(x)/dx=7/(x^2-3x+1)

Then, substitute x = 6 into the derivative expression to obtain:

dg(6)/dx = d/dx [7 * ln|x-3| ] evaluated at x = 6 = 7/3

Next, evaluate the original function g(x) at x = 6 to get g(6) = 7 * ln |6 - 3| / (6 - 3) = 7 * ln 3.

Since we know the coordinates of the point of tangency (6, 14), we can substitute them into the general form of the linear equation y = mx + c:

14 = 7 * 6 + c

14 = 42 + c

c = -28

The final equation of the line tangent at point (6, 14) is therefore:

y = (-28)x + 14

To shift the graph left, we need to subtract 1 from x. This gives us F(x) = (x - 1)², which matches option D. Therefore, the correct answer is D.

The graph of F(x) is a parabolic graph, just like the graph of G(x)=x². F(x) is shifted to the left by 1 unit.

The equation for the graph of F(x) is F(x) = (x - 1)². This is because when a parabolic graph is shifted left or right, it affects the x-value of the vertex of the parabola.

The standard form of the equation for a parabolic graph is y = a(x - h)² + k, where (h, k) is the vertex of the graph and a determines whether the graph is facing up or down. In this case, the vertex of the graph of G(x)=x² is (0, 0).

When we shift this graph left by 1 unit, the vertex becomes (-1, 0). Therefore, the equation for F(x) is F(x) = a(x + 1)².

To determine the value of a, we can look at the y-intercept of the graph, which is (0, 1). Plugging these values into the equation, we get 1 = a(0 + 1)², which simplifies to 1 = a.

Therefore, the equation for F(x) is F(x) = (x + 1)². However, if we expand this equation, we get F(x) = x² + 2x + 1, which is not in the form given in the options.

To shift the graph left, we need to subtract 1 from x. This gives us F(x) = (x - 1)², which matches option D. Therefore, the correct answer is D.

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Answer:

The length should be 11 feet and the width should be 6 feet. 11 feet is 5 feet more than 6 feet. 11 feet times 6 feet is 66 square feet.

Step-by-step explanation:

Para calcular la población de bacterias al cabo de 4 horas, primero necesitamos determinar cuántos intervalos de 20 minutos hay en 4 horas.

4 horas son equivalentes a 240 minutos. Dividiendo 240 minutos entre 20 minutos por intervalo, obtenemos 12 intervalos.

Dado que la población de bacterias aumenta un 25% cada 20 minutos, podemos aplicar este crecimiento a cada intervalo de 20 minutos.

Para cada intervalo de 20 minutos, la población de bacterias aumentará en un 25%.

Por lo tanto, podemos calcular la población de bacterias al final de los 12 intervalos de la siguiente manera:

Población final = Población inicial * (1 + tasa de crecimiento)^número de intervalos

Población inicial = 32,200 bacterias

Tasa de crecimiento = 25% = 0.25

Número de intervalos = 12

Población final = 32,200 * (1 + 0.25)^12

Realizando los cálculos, obtenemos la población final de bacterias al cabo de 4 horas.

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The complete factorization of the polynomial x³+2x²+x+2 is(x-1)(x+1)(x+2).

To factorize the polynomial x³+2x²+x+2 completely, one may use the rational root theorem, and synthetic division method.

Therefore, first we list the factors of 2 and divide it by the factors of 1 as the leading coefficient:Factors of 2 = ±1, ±2Factors of 1 = ±1Therefore, the possible rational roots for the given polynomial equation are± 1, ±2From the given polynomial, x³+2x²+x+2,

we can easily see that there is no remainder when divided by (x-1). Hence, (x-1) is a factor of the polynomial and we can use synthetic division to factorize the polynomial.x³+2x²+x+2 = (x-1)(x²+3x+2)Now we can use the quadratic formula or factoring to find the remaining factors. We get,x²+3x+2 = (x+1)(x+2).

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Answer: 33.7

Step-by-step explanation:

Let the height of the house be h, and the distance from the foot of the rock to the house be x.

From the given information, we have the following diagram:

```

*

/ \

/ \

/ 63° \

/ \

/θ \

A ----------------------- B

70m x

Angle A = 60° (complementary to the angle of elevation of the foot of the house)

Angle B = 63° (angle of elevation of the top of the house)

Using trigonometry, we have:

tan(63°) = h/x ----(1) (for triangle AOB)

tan(60°) = h/(x + 70) ----(2) (for triangle ABD)

Solving equations (1) and (2) simultaneously, we get:

h = (70 tan(63°) - 70 tan(60°)) meters

h ≈ 33.7 meters (rounded to one decimal place)

Therefore, the height of the house is approximately 33.7 meters.

a) The triangles in this problem are congruent.

b) The reason is the SAS congruence theorem.

The Side-Angle-Side (SAS) congruence theorem states that if two sides of two similar triangles form a proportional relationship, and the angle measure between these two triangles is the same, then the two triangles are congruent.

The equal sides for this problem are given as follows:

Hypotenuses and PS.

As angle P is bisected, we have <1 = <2, which are the angles between the equal sides, hence the SAS theorem holds true for this problem.

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Nave Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 27.27%.

To calculate the manufacturing cycle efficiency (MCE) for Nave Corporation's elevators, we need to determine the ratio of value-added time to the total lead time.

Value-added time refers to the time spent on activities that directly contribute to the production or customization of the elevators, while lead time refers to the total time from order placement to installation.

Given the breakdown of time:

12 days for manufacturing and customization

5 days for waiting or non-value-added time

The value-added time is 12 days, and the total lead time is 44 days.

To calculate MCE, we divide the value-added time by the total lead time and multiply by 100 to express it as a percentage:

MCE = (Value-added time / Total lead time) x 100

MCE = (12 / 44) x 100

MCE = 27.27%

Therefore, Nave Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 27.27%.

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Therefore, it would take approximately 36 minutes for 14 workers to complete the job.

To solve this inverse variation problem, we'll use the formula: t = k/w, where t represents the time needed, w represents the number of workers, and k is the constant of variation.

We can find the value of k by plugging in the given values of 9 workers and 56 minutes into the formula:

56 = k/9

To find the value of k, we multiply both sides of the equation by 9:

k = 504

Now that we know the constant of variation, we can determine the time it would take for 14 workers to complete the job. Plugging in the values into the formula:

t = 504/14

t ≈ 36

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If Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).

To show that x is also an eigenvector of QT, we need to demonstrate that QT * x is a scalar multiple of x.

Given that Q is an orthogonal matrix, we know that QT * Q = I, where I is the identity matrix. This implies that Q * QT = I as well.

Let's denote x as the eigenvector corresponding to the eigenvalue λ1 This means that Q * x = λ1 * x.

Now, let's consider QT * x. We can multiply both sides of the equation Q * x = λ1 * x by QT:

QT * (Q * x) = QT * (λ1 * x)

Applying the associative property of matrix multiplication, we have:

(QT * Q) * x = λ1 * (QT * x)

Using the fact that Q * QT = I, we can simplify further:

I * x = λ1 * (QT * x)

Since I * x equals x, we have:

x = λ1 * (QT * x)

Now, notice that λ1 * (QT * x) is a scalar multiple of x, where the scalar is λ1. Therefore, we can rewrite the equation as:

x = λ2 * x

where λ2 = λ1 * (QT * x).

This shows that x is indeed an eigenvector of QT, with the eigenvalue λ2 = λ1 * (QT * x).

In conclusion, if Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).

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State the null and alternative hypotheses.

c. H0: µ = 0, Ha: µ > 3

The null hypothesis (H0) states that the population mean rating is equal to 3, while the alternative hypothesis (Ha) suggests that the population mean rating is greater than 3.

Specify the rejection region for α = 0.01. Reject H0 if

a. z > 2.33

The rejection region for a one-tailed test with a significance level of 0.01 (α) is in the upper tail of the distribution. In this case, we reject the null hypothesis if the test statistic (z-score) is greater than 2.33.

Calculate the test statistic

c. 0.44

To calculate the test statistic, we use the formula:

z = (sample mean - population mean) / (sample standard deviation / √n)

Plugging in the given values, we get:

z = (3.31 - 3) / (0.70 / √172) ≈ 0.44

What is your conclusion?

b. Fail to Reject H0

Since the calculated test statistic (0.44) does not exceed the critical value (2.33) in the rejection region, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the average rating is significantly greater than 3 in the population.

Obtain the lower bound of a 99% confidence interval for the mean rating.

d. 3.17

To obtain the lower bound of a 99% confidence interval, we subtract the margin of error from the sample mean. The margin of error can be calculated by multiplying the critical value (obtained from the z-table for a 99% confidence level) with the standard error (sample standard deviation divided by the square root of the sample size).

The lower bound is given by:

3.31 - (2.33 * (0.70 / √172)) ≈ 3.17

Obtain the upper bound of a 99% confidence interval for the mean rating.

c. 3.44

To obtain the upper bound of a 99% confidence interval, we add the margin of error to the sample mean. Using the same calculation as above, the upper bound is given by:

3.31 + (2.33 * (0.70 / √172)) ≈ 3.44

What assumption(s) do you need to make in order to answer the above questions?

b. The population distribution is assumed to be normal.

To perform hypothesis testing and construct confidence intervals, it is typically assumed that the population distribution is approximately normal. Additionally, assumptions such as random sampling and independence of observations are generally made.

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The scaled triangle will be larger than the initial size by a factor 2.

The scaled square will be smaller than the initial side by a factor 4

Dilation refers to a transformation that changes the size of a geometric figure without altering its shape.

Dilation involves scaling an object by a certain factor, that might result in  enlarging or reducing its dimensions uniformly in all directions.

Based on the given diagram, the new length and size of the object is calculated as follows;

For the triangle, (measure the length with ruler)

For the square; (measure the length with ruler)

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