The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is

The expression for the area of the new square is y² + 10y + 25.

To find the expression that represents the area of the new square, we need to consider that when each side of a square is increased by 5 units, the new side length becomes y + 5. The area of the new square is then given by:

(New side length)² = (y + 5)²

Expanding the square, we get:

(y + 5)² = y² + 10y + 25

Therefore, the expression that represents the area of the new square is y² + 10y + 25.

So, the correct option is:

O y² + 10y + 25

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The length of diagonal ac is approximately 15.58 cm, and the length of diagonal bd is approximately 12.22 cm

Rhombus is a special type of parallelogram in which all four sides are congruent. The opposite angles of a rhombus are also congruent, and the diagonals bisect each other at right angles.

Now, let's consider the given rhombus abcd, where ad = 8cm and m∠ade=20°. We need to determine the length of diagonals ac and bd.

First, let's use the law of cosines to find the length of side ae. We know that ad = 8cm, and m∠ade=20°, so we can use the formula:

ae² = ad² + de² - 2ad(de)cos(m∠ade)

Substituting the values, we get:

ae² = 8² + de² - 2(8)(de)cos(20°)

Next, we can use the fact that a rhombus has all sides congruent to find the length of side de. Since abcd is a rhombus, we know that ac and bd are also congruent diagonals that bisect each other at right angles. Therefore, we can draw diagonal ac and use the Pythagorean theorem to find the length of ac:

ac² = (ae/2)² + (de/2)²

Substituting ae² from the previous equation, we get:

ac² = ((8² + de² - 2(8)(de)cos(20°))/4) + (de/2)²

Simplifying the equation and using the fact that ac and bd are congruent, we get:

bd² = ac² = (8² + de² - 2(8)(de)cos(20°))/2

Finally, we can use the Pythagorean theorem to find the length of diagonal bd:

bd² = ab² + ad²

Substituting ab = ac/2 and ad = 8cm, we get:

bd² = (ac/2)² + 8²

Substituting ac² from the previous equation, we get:

bd² = ((8² + de² - 2(8)(de)cos(20°))/8)² + 8²

Simplifying the equation, we get:

bd ≈ 12.22 cm

Similarly, we can solve for ac using the equation we derived earlier:

ac² = ((8² + de² - 2(8)(de)cos(20°))/4) + (de/2)²

Substituting de ≈ 9.84cm (which we can solve for from the equation ae² = 8² + de² - 2ad(de)cos(m∠ade)), we get:

ac ≈ 15.58 cm

Therefore, the length of diagonal ac is approximately 15.58 cm, and the length of diagonal bd is approximately 12.22 cm

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Considering the route as a curve of a quadratic function, the information should be completed as follows;

In Mathematics and Geometry, the vertex form of a quadratic function is represented by the following mathematical equation (formula):

y = a(x - h)² + k

Where:

Based on the information provided about this quadratic function, we can logically deduce that a mathematical equation which quickly reveals the vertex of the quadratic function is given by:

y = a(x - h)² + k

0 = a(2.9 - (-1))² + 4

3.9a = -4

a = -4/3.9

a = -1.03

Therefore, the required quadratic function is given by:

y = a(x - h)² + k

y = -1.03(x - (-1))² + 4

y = -1.03(x + 1)² + 4

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Purab initially had 17 rose plants in his lawn before buying the new ones.

Purab initially had a certain number of rose plants in his lawn. He bought twice that number, but had to discard 3 plants as they turned bad

After planting the new ones, there were a total of 48 plants in the garden.

To determine how many plants he had earlier, let's use a variable x to represent the initial number of plants.

Purab bought 2x plants, and after removing the 3 bad plants, he had (2x - 3) good plants.

Adding these to the initial number of plants, the equation becomes:

x + (2x - 3) = 48

Combining like terms, we get:

3x - 3 = 48

Next, we add 3 to both sides:

3x = 51

Finally, we divide by 3: x = 17

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Lee's total cost for the leaf blower was $215.50.

First, let's find the selling price of the leaf blower before sales tax was added:

The leaf blower was marked up by 150%, so the selling price is:

= [tex]80 + (\frac{150}{100}) 80[/tex]

= 80 + 120

= 200

So the selling price of the leaf blower before sales tax was $200.

Next, we need to find the amount of sales tax that Lee paid. To do this, we need to multiply the selling price by the sales tax rate:

Sales tax = 7.75% ($200)

= 0.0775 ($200)

= $15.50

Finally, we can find Lee's total cost by adding the selling price and the sales tax:

Total cost = Selling price + Sales tax

= $200 + $15.50

= $215.50

Therefore, Lee's total cost for the leaf blower was $215.50.

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The solution is: 14 karat gold is 58.3333...% gold

We have given that;

75% gold and 50% gold and we need to make 200 grams of 58.3333...% gold.

Since, A percentage is a number or ratio that can be expressed as a fraction of 100. A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

Here, we have,

A) x + y = 200

B) .75x + .50y = ( (14/24) * 200)

We multiply equation B) by -1.3333... and get

B) -x -.6666...y = -155.5555... then adding A)

A) x + y = 200  we get

.3333...y = 44.4444...

y = 133.3333... grams 12 karat gold

x = 66.6666... grams  18 karat gold

Double-Checking the answer

133.3333... * .5 = 66.6666...

66.6666 * .75 = 50.0000...

Hence, Concentration of final solution = (66.6666... + 50) / 200 = 58.3333...% which is 14 karat gold

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△ABC and △DEF are similar triangles if they have corresponding sides that are proportional and the corresponding angles are all congruent. Thus, the options that are applied are B and C.

Similar shapes are enlargements or shortening of other shapes using a scale factor.

Two triangles are said to be similar if the corresponding sides are proportional and the corresponding angles are the same. There are the following similarity criteria:

1. AA or AAA where all the angles are equal

2. SSS where all the sides are proportional to the corresponding sides

3. SAS where the corresponding sides and the angle between are proportional and congruent.

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The probability that the coin will show tails at least once is 0.56.

To find the probability that the coin will show tails at least once, you can sum the probabilities of getting 1, 3, or 4 tails, as shown in the table:

Probability of 1 tail: 0.25
Probability of 3 tails: 0.25
Probability of 4 tails: 0.06

Now, add these probabilities together:

0.25 + 0.25 + 0.06 = 0.56

So, the probability that the coin will show tails at least once is 0.56.

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This is the derivative of the given function f(x) = (3x + 1)^5(3x - 1)^6 using the Generalized Power Rule.

To find the derivative of the function f(x) = (3x + 1)^5(3x - 1)^6 using the Generalized Power Rule, we will need to apply both the Product Rule and the Chain Rule.

The Product Rule states that if you have a function f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).

First, let's identify g(x) and h(x) in your function:
g(x) = (3x + 1)^5
h(x) = (3x - 1)^6

Next, we'll find the derivatives g'(x) and h'(x) using the Chain Rule, which states that if you have a function y = [u(x)]^n, then y' = n[u(x)]^(n-1) * u'(x).

For g'(x):
u(x) = 3x + 1
n = 5
u'(x) = 3
g'(x) = 5(3x + 1)^(5-1) * 3 = 15(3x + 1)^4

For h'(x):
u(x) = 3x - 1
n = 6
u'(x) = 3
h'(x) = 6(3x - 1)^(6-1) * 3 = 18(3x - 1)^5

Now, we apply the Product Rule:
f'(x) = g'(x)h(x) + g(x)h'(x) = 15(3x + 1)^4(3x - 1)^6 + (3x + 1)^5 * 18(3x - 1)^5

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The explanation on why △ACF is 30°-60°-90° triangle is given below.

With a regular hexagon, each of its sides and angles are equal in measure. Consider the centre of the encompassing circle, connected to two neighbouring vertices - labeled A and B here. This then creates a radius wherein the length of AB is basically equal to any other side, denoted as 's'. Furthermore, △ABF will be an isosceles triangle (with AB = BF).  

From these facts, we can produce △ACF which is a right angled triangle – with AC being its hypotenuse, A F and FB both equating to s/2, finally concluding that ∠AFB is equivalent to 120°/2 = 60° while establishing that ∠ACF is also a right angle constituent making △ACF essentially a 30°-60°-90° triangle.

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Answer:

Step-by-step explanation:

the point at which the lines representing the linear equations intersect

A graph of polygon GREAT and its image after a counterclockwise rotation of 90° around the origin is shown in the image attached below.

In Mathematics and Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).

By applying a rotation of 90° counterclockwise around the origin to vertices W, the coordinates of the vertices of the image ΔA'B'C' are as follows:

(x, y)                               →            (-y, x)

Ordered pair G = (-3, -2) → Ordered pair G' = (2, -3)

Ordered pair R = (-4, 4) → Ordered pair R' = (-4, -4)

Ordered pair E = (0, 1) → Ordered pair E' = (-1, 0)

Ordered pair A = (1, -2) → Ordered pair A' = (2, 1)

Ordered pair T = (-1, -3) → Ordered pair T' = (3, -1)

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Answer:

Step-by-step explanation:

The volume of the given prism is 9.936 cubic meters, To calculate the volume of a right rectangular prism, we need to multiply its length, width, and height together.

Given that the length of the prism is 4.8 meters, the width is 2.3 meters, and the height is 0.9 meters, we can calculate the volume using the formula:

Volume = length x width x height

Volume = 4.8 m x 2.3 m x 0.9 m

Volume = 9.936 m^3

Therefore, the volume of the right rectangular prism is 9.936 cubic meters.

It is important to note that when we calculate volume, we are dealing with a three-dimensional space, and the units we use must be cubed (m^3 in this case). This is because we are measuring the amount of space occupied by the object in all three dimensions.

In summary, to find the volume of a right rectangular prism, we simply multiply its length, width, and height together. In this case, the volume of the given prism is 9.936 cubic meters.

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7 is the slope of BD in Quadrilateral.

What in arithmetic is a quadrilateral?

Four sides, four vertices, and four angles make up a quadrilateral, which is a two-dimensional form. Concave and convex are the two most common forms. Additionally, there are several subgroups of convex quadrilaterals, including trapezoids, parallelograms, rectangles, rhombus, and squares.

                                 There are four closed sides to a quadrilateral. Quadrilaterals are the following figures: produced by Raphael. a quadrilateral form. The form features a single pair of parallel sides and no right angles.

points A(4, 9) and C(3, 2)

      slope = y₂ - y₁/x₂ - x₁

                =  2 - 9/3 - 4

                = - 7/-1

                = 7

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The volume of the cube-shaped speaker is 42.875 cubic inches. The volume of the package is 68 cubic inches. Subtracting the volume of the speaker from the volume of the package gives the empty space remaining, which is 25.125 cubic inches.

The volume of the cube-shaped speaker is given by V₁ = (edge length)³ = 3.5³ = 42.875 cubic inches.

The volume of the square prism-shaped package is given by V₂ = (base area) x (height) = 16 x 4.25 = 68 cubic inches.

Therefore, the empty space remaining in the package after the speaker is placed in it is V₂ - V₁ = 68 - 42.875 = 25.125 cubic inches.

So, the empty space remaining in the package is 25.125 cubic inches.

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If you paid $25,000 for your car and you sell it after owning the car for 3 years, then the worth of the car is t₃=$7,910. 15 (option c).

To find the value of your car after owning it for 3 years, we need to evaluate the function at t=3. This means we need to substitute t=3 into the function and simplify the expression.

f(3) = 0.75(3) = 2.25

The output of the function when t=3 is 2.25. But what does this number mean? It represents the fraction of the original value of the car that remains after owning it for 3 years.

To find the actual value of the car, we need to multiply this fraction by the original value of the car, which is given as $25,000.

Value of car after 3 years = 2.25 x $25,000 = $56,250

Therefore, the value of the car after owning it for 3 years is $7,910.15. This is the option (C) in the given choices.

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The searchlight should be 1/3 feet deep at the edge of the opening. Since the paraboloid is a continuous surface, the depth will increase gradually from the edge of the opening to the vertex at (0,0,1).

Determine the depth of the searchlight shaped like a paraboloid of revolution, we need to use the equation for the standard form of a paraboloid of revolution:
z = (x^2 + y^2) / (4f)
where z is the depth, x and y are the horizontal and vertical coordinates, and f is the focal length of the paraboloid.
We know that the light source is located 1 feet from the base along the axis of symmetry, which means that the vertex of the paraboloid is at (0,0,1).
We also know that the opening is 6 feet across, which means that the horizontal distance from one side of the opening to the other is 3 feet.
Using this information, we can find the value of f:
f = (d/2)^2 / 2r
where d is the diameter of the opening (6 feet), and r is the radius of curvature at the vertex (1 foot).
f = (6/2)^2 / 2(1) = 4.5 feet
Now we can plug in the values for x, y, and f to solve for z:
z = (x^2 + y^2) / (4f)
z = (x^2 + y^2) / (4(4.5))
z = (x^2 + y^2) / 18
Since the opening is 6 feet across, we know that the maximum value of x is 3 feet. Therefore, we can use the maximum value of y (also 3 feet) to find the depth at the edge of the opening:
z = (3^2 + 3^2) / 18
z = 6/18
z = 1/3 feet
So the searchlight should be 1/3 feet deep at the edge of the opening. However, since the paraboloid is a continuous surface, the depth will increase gradually from the edge of ×the opening to the vertex at (0,0,1).

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C. Yes; the graph passes the vertical line test.

The correct answer is C. Yes; the graph passes the vertical line test.

To determine whether a graph represents a function, we apply the vertical line test. The vertical line test states that for a graph to represent a function, no vertical line should intersect the graph in more than one point.

In this case, the graph passes the vertical line test if each vertical line crosses the graph at most once. If this condition is satisfied, then each x-value corresponds to a unique y-value, indicating that the graph represents a function.

Since the question states that the graph passes the vertical line test, we can conclude that it represents a function.

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You would have to receive an approximately 8.696% increase in pay to regain your original salary after an 8% decrease.

To find the percentage increase in pay needed to regain your original salary after an 8% decrease, follow these steps:

1. Assume your original salary is 100%. After an 8% decrease, your salary becomes 100% - 8% = 92%.

2. Calculate the difference between your original salary (100%) and your current salary (92%). The difference is 100% - 92% = 8%.

3. To find the percentage increase needed to regain your original salary, divide the difference (8%) by your current salary (92%): 8% / 92% = 0.08696.

4. Multiply the result by 100 to convert it to a percentage: 0.08696 (100) = 8.696%.

So, you would have to receive an approximately 8.696% increase in pay to regain your original salary after an 8% decrease.

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the final answer is 27/125.

To evaluate [tex](3/5)^3[/tex], we simply need to multiply (3/5) by itself three times:

[tex](3/5)^3 = (3/5) * (3/5) * (3/5)[/tex]

To simplify, we can first multiply the numerators together and the denominators together:

[tex](3/5)^3 = 27/125[/tex]

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When allowing repetition, you can make 676,000 passwords, and without repetition, you can make 468,000 passwords.

To create a password that starts with two letters from the alphabet, followed by three digits (for example, AB-123), you can make a different number of passwords depending on whether you are allowed to repeat letters or numbers.

1. If you are allowed to repeat letters or numbers, you can make:
- 26 (alphabet letters) x 26 (alphabet letters) x 10 (digits 0-9) x 10 (digits 0-9) x 10 (digits 0-9) = 676,000 passwords.

2. If you don't repeat any letters or numbers, you can make:
- 26 (alphabet letters) x 25 (remaining alphabet letters) x 10 (digits 0-9) x 9 (remaining digits 0-9) x 8 (remaining digits 0-9) = 468,000 passwords.

Your answer: When allowing repetition, you can make 676,000 passwords, and without repetition, you can make 468,000 passwords.

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Answer:

The graph on the top right

Step-by-step explanation:

The slope-intercept form is y = mx + b

m = the slope

b = y-intercept

The equation is y = 1/2x + 2

The y-intercept in this equation is 2, meaning the graph has a point (0,2) on it.  Looking at the options, the only graph that has a point (0,2) is the map on the top right, and that is the answer.

Point B is not the midpoint of line AC, because angle AOB is not half of angle  AOC.

If point B is the midpoint of line AC, then angle AOB must be equal to angle  BOC.

The value of angle AOC is calculated as follows;

let angle AOC = θ

cos θ = 100 yds / 500 yds

cos θ = 0.2

θ = cos⁻¹ (0.2)

θ = 78.5⁰

The value of length AC is calculated as follows;

AC = √ (500² - 100²)

AC = 489.9

If point B is the midpoint, then AB = BC = 489.9/2 = 244.95

The value of angle AOB is calculated as follows;

tan β = AB/AO

tan β = 244.95/100
tan β = 2.4495

β = arc tan (2.4495)

β = 67.8⁰

Half of angle AOC = 78.5⁰/2 = 39.25⁰

β  ≠ 39.25⁰

So point B is not midpoint of line AC, since angle AOB is not half of angle  AOC.

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The number of 6-card hands  in which at least one card from each suit is equal to 8,184,220.

Total number of 6-card hands that can be formed from a deck of 60 cards is,

Using combination formula,

C(60, 6) = 50,063,860

Now, subtract the number of 6-card hands that do not contain at least one card from each suit.

There are 5 ways to choose the suit that will be missing from the hand.

Once this suit is chosen, there are 48 cards remaining in the other suits.

Choose 6 cards from this set, so the number of 6-card hands that do not contain any cards from the chosen suit is,

C(48, 6) = 12,271,512

Overcounted the number of hands that are missing more than one suit.

There are C(5, 2) ways to choose 2 suits that will be missing from the hand.

Once these suits are chosen, there are 36 cards remaining in the other 3 suits.

Choose 6 cards from this set, so the number of 6-card hands that do not contain any cards from the chosen suits is,

C(36, 6) = 1,947,792

We cannot have a 6-card hand that is missing more than 2 suits.

3 suits with no cards in the hand, which is not allowed.

Number of 6-card hands that have at least one card from each suit is,

C(60, 6) - 5×C(48, 6) + C(5, 2)×C(36, 6)

=50,063,860 - 5×  12,271,512 + 10 × 1,947,792

= 50,063,860 -61,357,560 + 19,477,920

= 8,184,220

Therefore, there are 8,184,220 of 6-card hands that have at least one card from each suit.

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Answer:

Let's solve this in steps:

1. Convert mixed numbers to fractions:

```

3 1/2 = 7/2

9 3/4 = 39/4

```

2. Perform the subtraction:

```

7/2 - 39/4 = -11/4

```

3. Divide by -2.5:

```

-11/4 / -2.5 = 4.4

```

Therefore, the value of the expression is **4.4**.

To use the GCF and distributive property to find the sum of 22 + 33, we first need to identify the GCF of both numbers, which is 11.

We can then write each number as a product using the GCF as a factor: 22 = 11 x 2 and 33 = 11 x 3. Next, we can apply the distributive property by multiplying the GCF by the sum of the other factors in each number: 11 x (2 + 3).

Finally, we can simplify the expression by adding the sum of the other factors, which is 5: 11 x 5 = 55. Therefore, the sum of 22 + 33 using the GCF and distributive property is 55.

In summary, to find the sum of 22 + 33 using the GCF and distributive property, we first identify the GCF as 11 and write each number as a product using the GCF as a factor.

We then apply the distributive property by multiplying the GCF by the sum of the other factors in each number. Finally, we simplify the expression by adding the sum of the other factors and arrive at the answer of 55. This method can be helpful when working with larger numbers or more complex expressions.

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Since segment CD is the mid-segment of trapezoid WXYZ, it means that segment CD is parallel to both bases WX and YZ and it is also half the length of their sum. Therefore, we can use the mid-segment formula which states that the length of segment CD is equal to the average of the lengths of the bases WX and YZ.

So, we can write:

CD = (WX + YZ)/2

Since we want to find the value of XY, we need to know its length in terms of WX and YZ.

If we draw a diagonal of the trapezoid, say diagonal WZ, it will divide the trapezoid into two triangles, namely triangle WXY and triangle ZYX.

We know that the mid-segment CD is also the median of triangle WZY, so it divides it into two equal areas.

Therefore, the area of triangle WXY is equal to the area of triangle ZYX.

We can write:

1/2 * WX * CD = 1/2 * YZ * CD

Simplifying this equation by dividing both sides by CD, we get:

1/2 * WX = 1/2 * YZ

Multiplying both sides by 2, we get:

WX = YZ

Therefore, the trapezoid WXYZ is actually an isosceles trapezoid with equal bases WX and YZ.

So, we can substitute WX for YZ in the formula for CD:

CD = (WX + WX)/2

Simplifying this equation, we get:

CD = WX

Therefore, the length of segment XY is equal to the length of the shorter base of the trapezoid, which is WX.

So, the value of XY is equal to the value of WX, and we can conclude that XY = WX.

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y=4x+3

Parallel lines have the same gradient - 4x

substitute the x and y values from the coordinates into y=mx+c

so

-5=(4×-2)+c

-5=-8+c

c=3

therefore, the answer is y=4x+3

The company should increase the quantity of capital used from 6 units to 3 units, an increase of 3 units.

The cost of capital and labor can be expressed as:

C = 44K + 11L

The company wants to limit the cost of capital and labor to $330:

44K + 11L ≤ 330

Substituting Q = aK^{0.6}L^{0.4} into the inequality, we get:

44K + 11L ≤ 330

44K + 11(Q/aK^{0.6})^{0.4} ≤ 330

44K^{1.6} + 11(Q/a)^{0.4}K ≤ 330

Solving for K, we get:

K ≤ (330 - 11(Q/a)^{0.4}) / 44K^{1.6}

Substituting K = 6, Q = aK^{0.6}L^{0.4}, and solving for L, we get:

Q = aK^{0.6}L^{0.4}

Q/K^{0.6} = aL^{0.4}

L = (Q/K^{0.6})^{2.5}/a

Substituting Q = a(6)^{0.6}(6)^{0.4} = 6a into the equation, we get:

L = (6/a)^{0.4}(6)^{2.5} = 9.585a^{0.6}

Therefore, the company is currently using 6 units each of capital and labor, and the total cost of capital and labor is:

C = 44(6) + 11(6) = 330

This means that the company is already using the maximum allowable cost. To reduce the cost, the company should use less labor or less capital.

To determine whether to use more or less labor, we can take the derivative of Q with respect to L:

∂Q/∂L = 0.4aK^{0.6}L^{-0.6}

This is a decreasing function of L, so as L increases, the quantity of product Q produced will decrease. Therefore, the company should use less labor.

To determine how much less labor to use, we can find the value of L that would reduce the cost to the maximum allowable level of $330:

44K + 11L = 330

44(6) + 11L = 330

L = 18

Therefore, the company should reduce the quantity of labor used from 6 units to 18 units, a decrease of 12 units.

To determine whether to use more or less capital, we can take the derivative of Q with respect to K:

∂Q/∂K = 0.6aK^{-0.4}L^{0.4}

This is an increasing function of K, so as K increases, the quantity of product Q produced will increase. Therefore, the company should use more capital.

To determine how much more capital to use, we can find the value of K that would reduce the cost to the maximum allowable level of $330:

44K + 11L = 330

44K + 11(18) = 330

K = 3

Therefore, the company should increase the quantity of capital used from 6 units to 3 units, an increase of 3 units.

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The equation for line P such that the system has an infinite number of solutions is given as follows:

3) 2x + 2y = -2.

The first equation for the system of equations is given as follows:

x + y = -1.

A system of equations has an infinite number of solutions when the two equations are multiples.

Multiplying the equation by 2, we have that:

2x + 2y = -2.

Meaning that equation 3 is correct.

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