12. A normal distribution has a mean of 34 and a standard deviation of 7. Find the range of
values that represent the middle 95% of the data.

F. 27 G. 20 X 48
H. 13 J. 6

The derivative formula for the function is

[tex]f'(x) = 342x^4 + 414x^3 + 342x^2 + 418x + 385[/tex]

To find the derivative of the function [tex]f(x) = (9x^2 + 11x + 7)(38x^3 + 35)[/tex], we can use the product rule of differentiation:

f(x) = u(x)v(x)

where [tex]u(x) = (9x^2 + 11x + 7)[/tex] and [tex]v(x) = (38x^3 + 35)[/tex].

The product rule states that:

f'(x) = u'(x)v(x) + u(x)v'(x)

where u'(x) and v'(x) are the derivatives of u(x) and v(x), respectively.

Taking the derivatives, we get:

u'(x) = 18x + 11

[tex]v'(x) = 114x^2[/tex]

Now, substituting everything into the product rule formula, we get:

[tex]f'(x) = (18x + 11)(38x^3 + 35) + (9x^2 + 11x + 7)(114x^2)[/tex]

Simplifying this expression gives the derivative formula for f(x):

[tex]f'(x) = 342x^4 + 414x^3 + 342x^2 + 418x + 385[/tex]

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a) equation will be 0.5x + 3 = 18.50

b)  equation will be (0.75x - 0.10x) + 3 = 21.00

a) Sydney made $18.50 selling lemonade.

she sold each cup for $0.50 and received a $3 tip from a neighbor.

let x be the cost of cup she sold.

equation will be 0.5x + 3 = 18.50

b) Daria made a profit of $21.00 selling lemonade

she sold her lemonade for $0.75 per cup, received a tip of $3 from a neighbor.

equation will be (0.75x - 0.10x) + 3 = 21.00

c) Sydney don't have to pay for the cups used while Daria paid.

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A. An equation to represent the situation in part A is 18.50 = (0.50) x + 3.

B. An equation to represent the situation in part B is 21.00 = (0.75) y - (0.10) y + 3.

C. The equations differ in the fact that one accounts for the cost per cup while the other does not.

In general, the profit is defined as the amount gained by selling a product, which should be more than the cost price of the product.

Part A: Let x be the number of cups of lemonade sold.

Then, the total amount of money Sydney made is given by:

Total money = (selling price per cup) × (number of cups sold) + tip

Substituting the given values, we get:

18.50 = (0.50) x + 3

Part B: Let y be the number of cups of lemonade sold.

Then, the total profit made by Daria is given by:

Total profit = (selling price per cup) × (number of cups sold) + tip - (cost per cup) × (number of cups sold)

Substituting the given values, we get:

21.00 = (0.75) y - (0.10) y + 3

Part C: The equation for Sydney's lemonade stand only takes into account the total amount of money she made, which includes the selling price per cup and a fixed tip.

On the other hand, the equation for Daria's lemonade stand takes into account both the total profit made (which includes the selling price per cup and a fixed tip) and the cost per cup of lemonade sold. So, the equations differ in the fact that one accounts for the cost per cup while the other does not.

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Complete question is

Part A : Sydney made $18.50 selling lemonade, by the cup, at her yard sale. She sold each cup for $0.50 and received a $3 tip from a neighbour. Write an equation to represent this situation.

Part B : Daria made a profit of $21.00 selling lemonade. She sold her lemonade for $0.75 per cup, received a tip of $3 from a neighbour, but also had to buy each plastic cup she used for $0.10 per cup. Write an equation to represent this situation.

Part C: Explain how the equation from part A and part B differ.

The coordinates of the absolute maximum point are (4, 24), and the coordinates of the absolute minimum point are (-4, -8).

To find the absolute extrema of the function[tex]f(x) = x^3 - 3x^2 - 9x + 20[/tex] on

the closed interval [-4, 4], we need to find the maximum and minimum

values of the function within the given interval.

Find the critical points of the function f(x) within the interval [-4, 4].

To find the critical points, we need to take the first derivative of the

function and set it equal to zero.

[tex]f(x) = x^3 - 3x^2 - 9x + 20[/tex]

[tex]f'(x) = 3x^2 - 6x - 9[/tex]

Setting f'(x) = 0, we get:

[tex]3x^2 - 6x - 9 = 0[/tex]

Dividing both sides by 3, we get:

[tex]x^2 - 2x - 3 = 0[/tex]

Factoring the quadratic equation, we get:

(x - 3)(x + 1) = 0

So, the critical points of the function within the interval [-4, 4] are x = -1 and x = 3.

Find the values of the function at the critical points and at the endpoints

of the interval [-4, 4].

To find the values of the function at the critical points and at the

endpoints of the interval, we evaluate the function at each of these

values.

f(-4) = -8

f(4) = 24

f(-1) = 24

f(3) = 2

Compare the values obtained in step 2 to find the maximum and minimum values of the function.

The maximum value of the function is 24, which occurs at x = -1 and x = 4.

The minimum value of the function is -8, which occurs at x = -4.

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Part III: Note reduction for payment 2 = Payment Amount - Interest Due = $1055.69 - $758.68 = $297.01.

Part I: The payment amount for payment number 2 is $1055.69 (same as the first payment).

Part II: Interest due for payment number 2 = (Unpaid Balance after payment 1) * (Monthly Interest Rate) = $189,671.31 * (4.8% / 12) = $758.68.

Part III: Note reduction for payment 2 = Payment Amount - Interest Due = $1055.69 - $758.68 = $297.01.

Part IV: Unpaid balance for payment number 2 = (Unpaid Balance after payment 1) - (Note Reduction for payment 2) = $189,671.31 - $297.01 = $189,374.30.

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a) The diameter d of the wheel has bounds:

35 cm ≤ d ≤ 45 cm

b) The circumference C has bounds, using C = πd:

π * 35 cm ≤ C ≤ π * 45 cm

The inequality representing the lower and upper bounds for the diameter d is:

35 cm ≤ d ≤ 45 cm

b) For the lower bound, we substitute the lower bound of the diameter (35 cm) into the formula:

[tex]C_l_o_w_e_r[/tex] = π * 35 cm

For the upper bound, we substitute the upper bound of the diameter (45 cm) into the formula:

[tex]C_u_p_p_e_r[/tex] = π * 45 cm

The inequality representing the lower and upper bounds for the circumference C is:

π * 35 cm ≤ C ≤ π * 45 cm

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Answer:

D) 205

Step-by-step explanation:

If the technician has a total of 2.87 oz, and can have a max of 0.014 oz in each capsule, we have to divide the total amount by the max amount per bottle.

2.87/0.014

=205

This means that the technician can fill 205 capsules with 0.014 oz of vitamin C.

Hope this helps!

(a) Indefinite integral of 5(5x^4 - 5sinx)dx is (5/3)x^5 + 5cosx + C. Definite integral over [0, π/2] is (125π/6) - 5.

We can evaluate the indefinite integral by applying the power rule and integration by substitution. The definite integral can be evaluated by substituting the limits of integration and simplifying.

(b) Indefinite integral of [(18x^2 - 1)(6x^3 - 9x^2 - 3)]^6dx is (18x^11 - 77x^9 + 126x^7 - 108x^5 + 49x^3 - 9x) / 11 + C.

To simplify the given expression, we can first expand the polynomial and then apply the power rule to integrate each term. The constant of integration can be added at the end.

(c) Definite integral of ∫tan^2(x)sec^2(x)dx over [0,π/4] is 1.

We can use the trigonometric identity sec^2(x) - 1 = tan^2(x) to simplify the integrand. Then we can apply the power rule and substitute the limits of integration to evaluate the definite integral.

(d) Indefinite integral of ∫(x+4)^2√(3x^2+4)dx is (1/15)(3x^2+4)^(3/2)(x+4) - (4/45)(3x^2+4)^(3/2) + C.

We can use substitution to simplify the integrand by setting u = 3x^2 + 4. After integrating, we can substitute back for u and simplify the constant of integration.

(e) Indefinite integral of ∫(120/(1+2x^2))dx is 60√2tan^(-1)(√2x) + C.

We can use substitution to simplify the integrand by setting u = 1 + 2x^2. After integrating, we can substitute back for u and simplify the constant of integration.

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the evaluated double integral is approximately 14.25.

To evaluate the given double integral, we need to first understand the problem properly. We have the function f(x, y) = y + xy, and the region R is described by the inequalities: 0 < x < y^2, and 1 < y < 2.

Now we can set up the double integral:

∬(y + xy) dA over the region R.

Since we are given that 0 < x < y^2 and 1 < y < 2, we can set up the integral using the given limits of integration:

∫(from y = 1 to 2) ∫(from x = 0 to y^2) (y + xy) dx dy.

Now, we can start by integrating the inner integral with respect to x:

∫(from y = 1 to 2) [(yx + (1/2)x^2*y) evaluated from x = 0 to x = y^2] dy.

After evaluating the inner integral, we have:

∫(from y = 1 to 2) (y^3 + (1/2)(y^2)^2*y) dy.

Now, we can integrate the outer integral with respect to y:

[((1/4)y^4 + (1/6)y^6) evaluated from y = 1 to y = 2].

After evaluating the outer integral, we get:

[(1/4)(2^4) + (1/6)(2^6)] - [(1/4)(1^4) + (1/6)(1^6)].

Calculating the final result:

(4 + 10.6667) - (0.25 + 0.1667) = 14.6667 - 0.4167 ≈ 14.25.

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Sal will pay to tile his entryway by calculating the area of the entryway in square feet and multiplying it by the cost of the tile per square foot.

To determine the cost of tiling Sal's entryway, we need to calculate the area of the floor plan. Since each unit length represents 1 foot, we can consider the dimensions of the floor plan in terms of feet. Let's say the length is 'L' feet and the width is 'W' feet. The area can be found by multiplying L by W, giving us the total area in square feet.

Once we have the area, we can multiply it by the cost per square foot, which is $1.15. This will give us the total cost of the tiles needed to cover the entryway.

It's important to note that rounding the final answer to the nearest cent is necessary to provide a precise cost value.

Therefore, by calculating the area of the entryway and multiplying it by the cost per square foot, we can determine the total amount Sal will pay to tile his entryway.

In conclusion, to calculate the cost, we need to find the area of the entryway by multiplying the length and width in units, convert it to square feet, and then multiply it by the tile cost per square foot.

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Answer:

Step-by-step explanation:

To prove that lines a and b are parallel lines cut by transversal f, we need to show that the alternate interior angles are congruent. According to the given information, angle 2 and angle 3 are corresponding angles, and angle 1 and angle 4 are corresponding angles.

Therefore, the set of equations that is enough information to prove that lines a and b are parallel lines cut by transversal f is:

angle 2 = angle 3 (corresponding angles)

angle 1 = angle 4 (corresponding angles)

Answer: x ≅ -0.9 or -1.8

Step-by-step explanation:

[tex]3(3x+4)^2 - 6 = 0[/tex]

[tex]3(3x+4)^2 = 6[/tex]

[tex]3(9x^2+24x+16) = 6[/tex]

[tex]9x^2+24x+16 = 2[/tex]

[tex]9x^2+24x+14 = 0[/tex]

Use the quadratric formula to get:

x ≅ -0.9 or -1.8

1) If there are 2500 tickets sold, and you buy 1 ticket, then the probability of your ticket being the winning ticket is 1/2500 or 0.04%.

2) If you buy 5 tickets, then the probability of having a winning ticket is 5/2500 or 0.2%.

3) If you and your 4 friends each buy 5 tickets, then there will be a total of 25 tickets. The probability of having a winning ticket in this scenario is 5/25 or 20%.

4) The chances of holding a winning ticket have not changed. This is because the consolation prizes are separate from the main prize, and the probability of winning the main prize is still the same.

The addition of consolation prizes does not affect the probability of winning the main prize.

Assuming the same raffle is held every year and you and your 4 friends buy 5 tickets each year, the average net winnings would be calculated as follows:

Total cost of tickets = $1 x 5 x 5 = $25

Total prize money = $1200 + ($400 x 2) = $2000

Probability of winning = 5/2500 = 0.2%

Expected value of winning = $2000 x 0.2% = $4

Average net winnings = ($4 - $25)/year = -$21/year

This means that on average, you and your friends would lose $21 per year if you participate in the raffle every year.

However, it is important to note that this is just an average and there is a chance of winning a larger prize which would make the net winnings positive.

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There are no points of inflection.

To determine the intervals on which the function S(x) = (x - 10)(1 - x) is concave up or down and find the points of inflection, we need to find the second derivative and analyze its sign.

First, find the first derivative, S'(x):
S'(x) = (x - 10)(-1) + (1 - x)(1) = -x + 10 - 1 + x = 9

Next, find the second derivative, S''(x):
S''(x) = d(S'(x))/dx = d(9)/dx = 0

Since the second derivative S''(x) is constant and equal to 0, there is no concavity, and the function is neither concave up nor concave down. There are no points of inflection.

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Based on the elevations given, Death Valley (-282 feet) < Tallahassee, Florida (203 feet).

To compare the elevations of Death Valley and Tallahassee, we'll use the inequality symbols i.e., "<" or ">" . The symbol "<" indicate less than and ">" indicate greater than.

Death Valley, California has an elevation of -282 feet, while Tallahassee, Florida has an elevation of 203 feet. Since -282 is less than 203, we use the "<" symbol.

So, the comparison is as follows:

Death Valley (-282 feet) < Tallahassee, Florida (203 feet)

This means that the elevation of Death Valley is lower than or less than the elevation of Tallahassee.

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Answer:  First, we need to find the composite function (fg)(x):

(fg)(x) = f(g(x))

= f(-4x-3)

= (-4x-3)^2 + (-4x-3) + 9 (substituting g(x) into f(x))

= 16x^2 + 24x + 18

Therefore, (fg)(x) = 16x^2 + 24x + 18.

Next, we need to find the quotient function (f/g)(x):

(f/g)(x) = f(x) / g(x)

= (x^2 + x + 9) / (-4x - 3) (substituting f(x) and g(x))

To simplify this expression, we can use polynomial long division or synthetic division. Using synthetic division, we get:

  -4 |   1    1    9

      |_____-4__ 12

      |  1  -3   21

Therefore, (f/g)(x) = -4x + 3 - 21 / (-4x - 3)

Simplifying further, we get:

(f/g)(x) = -4x + 3 + (21/4)(1/(x + 3/4))

Therefore, (f/g)(x) = -4x + 3 + (21/4)(1/(x + 3/4)).

Step-by-step explanation:

The location of all the relative and absolute extrema is (0, 0) (local minimum); (1, 1) (local maximum); (3, 3) (absolute maximum)

To find the relative and absolute extrema of the function f(x) = 2x - x^2 on the domain (0,3), we first take the derivative:

f'(x) = 2 - 2x

Setting this equal to zero, we find the critical point:

2 - 2x = 0
x = 1

To determine the nature of the critical point, we need to examine the second derivative:

f''(x) = -2

Since the second derivative is negative at x = 1, this critical point is a local maximum. To find the absolute extrema, we also need to examine the endpoints of the domain, x = 0 and x = 3:

f(0) = 0
f(3) = 3

So the function has an absolute maximum at x = 3 and an absolute minimum at x = 0. Therefore, the location of all the relative and absolute extrema, from smallest to largest x, is:

(0, 0) (local minimum)
(1, 1) (local maximum)
(3, 3) (absolute maximum)

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The surface area of the solids are listed below:

Case 1: A = 366 mm²

Case 2: A = 448 cm²

Case 3: A = 748 m²

Case 4: A = 221.5 in²

Case 5: A = 692 in²

Case 6: A = 276 ft²

In this question we need to determine the surface area of six solids, that is, the sum of areas of all faces in each solid. The solids can include areas of rectangles and triangles, whose formulas are:

Rectangle

A = b · h

Triangle

A = 0.5 · b · h

Where:

Case 1

A = 2 · (13 mm) · (3 mm) + 2 · (13 mm) · (9 mm) + 2 · (9 mm) · (3 mm)

A = 78 mm² + 234 mm² + 54 mm²

A = 366 mm²

Case 2

A = 2 · (20 cm) · (6 cm) + 2 · (4 cm) · (6 cm) + 2 · (20 cm) · (4 cm)

A = 240 cm² + 48 cm² + 160 cm²

A = 448 cm²

Case 3

A = 2 · (5 m) · (14 m) + 2 · (16 m) · (14 m) + 2 · (5 m) · (16 m)

A = 748 m²

Case 4

A = 2 · (2 in) · (6.5 in) + 2 · (11.5 in) · (6.5 in) + 2 · (11.5 in) · (2 in)

A = 221.5 in²

Case 5

A = 2 · 0.5 · (12 in) · (7 in) + (11 in) · (19 in) + (9 in) · (19 in) + (12 in) · (19 in)

A = 692 in²

Case 6

A = 2 · 0.5 · (8 ft) · (3 ft) + 2 · (5 ft) · (14 ft) + (8 ft) · (14 ft)

A = 276 ft²

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Answer: Goofy Ahh

Step-by-step explanation:

That question is so Goofy Ahh

Weeee

The probability of choosing a red marble, then a white marble is 5/36

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

A bag contains 4 white, 3 blue, and 5 red marbles

If the marbles were replaced, then we have

P(Red) = 5/12

P(White) = 4/12

So, we have

The probability of choosing a red marble, then a white marble  is

P = 5/12 * 4/12

Evaluate

P = 5/36

Hence, the probability of choosing a red marble, then a white marble is 5/36

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Answer:

We can use the formula for simple interest to calculate the interest earned by Henry:

Simple Interest = (Principal * Rate * Time)

where,

Principal = $850 (initial amount)

Rate = 5% per year (as given)

Time = 4 years (as given)

Substituting the values, we get:

Simple Interest = (850 * 0.05 * 4) = $170

Therefore, Henry will have earned $170 in interest after 4 years of keeping his money in the savings account.

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The statement that must be true is the original prices of the refrigerator and the stove were the same. So the answer is option B.

Let x represent the original cost of the refrigerator and y represent the original cost of the stove. The refrigerator's sale price is 0.6x (40% off means paying 60% of the original price), while the stove's sale price is 0.8y (20% off means paying 80% of the original price).

To get the overall discount, multiply the total cost after the discount by the original total cost:

(0.6x + 0.8y) / (x + y)

We want this fraction to equal 0.7 (or 30% off), so we can set up the equation:

(0.6x + 0.8y) / (x + y) = 0.7

Simplifying this equation, we get:

0.6x + 0.8y = 0.7(x + y)

0.6x + 0.8y = 0.7x + 0.7y

0.1x = 0.1y

x = y

Therefore, the statement that must be true to conclude that Alfonso received a 30% overall discount on the refrigerator and stove together is: The original prices of the refrigerator and the stove were the same.

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The area of an equilateral triangle with apothem length 'a' is (1/4) * √3 * [tex]a^2[/tex].

Let's label the equilateral triangle as ABC. An apothem is a line segment from the center of the triangle to the midpoint of one of its sides, forming a right angle with that side. Let the apothem of the triangle be 'a'.

The apothem divides the equilateral triangle into two congruent 30-60-90 triangles, where the apothem is the hypotenuse of one of the triangles. The length of the apothem 'a' is also the height of each 30-60-90 triangle.

In a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg. So, the length of the shorter leg is a/2, and the length of the longer leg (which is also the length of one side of the equilateral triangle) is √3 times the length of the shorter leg. Thus, the length of one side of the equilateral triangle is:

s = √3 * (a/2) = (√3 / 2) * a

The area of the equilateral triangle can be calculated using the formula:

A = (1/2) * base * height

where the base is one side of the equilateral triangle, and the height is the length of the apothem 'a'. Substituting the values we found, we get:

A = (1/2) * s * a = (1/2) * (√3 / 2) * a * a = (1/4) * √3 *[tex]a^2[/tex]

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The correct answer is (D) 132.7 feet.

From the given information, we can draw a diagram as follows:

       A

        /|

       / |

      /  |  37°

     /   |

    /    |

   /     |

  /      |

 /_ _ _ _|

    d     B

Where A represents the top of the cliff, B represents the ship and d represents the distance between the ship and the bottom of the cliff.

Since we know that the angle of depression is 37 degrees, then the angle CAB is also 37 degrees. We also know that AB is 100 feet, which is the height of the cliff. We want to find the distance d between the bottom of the cliff and the ship.

We can use the tangent function to find d:

tan(37°) = AB/BD

tan(37°) = 100/d

d = 100/tan(37°)

Using a calculator, we can find that:

d ≈ 132.7 feet

Therefore, the correct answer is (D) 132.7 feet.

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Answer:

Step-by-step explanation:

Let's use L to represent the number of large dolls and S to represent the number of small dolls Rachel purchased.

Based on the problem, we can write the following two equations:

1. L + S = 6     (equation 1 - represents the total number of dolls purchased)

2. 22L + 16S = 120     (equation 2 - represents the total amount spent on dolls)

To solve for L and S, we can use substitution or elimination. Let's use the substitution method:

From equation 1, we can express S in terms of L by subtracting L from both sides :

S = 6 - L

Substitute this equation for S in equation 2:

22L + 16(6-L) = 120

Simplify and solve for L:

22L + 96 - 16L = 120

6L = 24

L = 4

Now that we know L = 4, we can use equation 1 to find S:

L + S = 6

4 + S = 6

S = 2

Therefore, Rachel purchased 4 large dolls and 2 small dolls.

As you approach zero from the left on a number line, the integers become increasingly negative, but the absolute values of those integers remain positive.

A number line is a visual representation of numbers placed in order on a straight line. It is a graphical tool used to represent the real numbers, starting from negative infinity on the left side and extending to positive infinity on the right side. The number line is divided into equal intervals, and each point on the line corresponds to a specific value or number. The distance between any two points on the number line represents the numerical difference between the corresponding numbers. The number line is a fundamental tool in mathematics for understanding the order and magnitude of numbers, as well as for performing operations such as addition, subtraction, and comparison.

As you approach zero from the left on a number line, the integers become increasingly negative, but the absolute values of those integers remain positive.

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The new volume of the BigMini tablet is about 22.18 cubic inches.

The volume of a three-dimensional object, such as a tablet, is calculated by multiplying its length, width, and height.

Assuming that the scale factor of 1.25 is applied uniformly to all three dimensions of the iPad Mini 2, the new dimensions of the BigMini tablet would be:

Length: 1.25 x 7.87 inches = 9.84 inches

Width: 1.25 x 5.3 inches = 6.63 inches

Height: 1.25 x 0.29 inches = 0.36 inches

The new volume of the BigMini tablet can be calculated as:

9.84 x 6.63 x 0.36 = 22.18 cubic inches

Therefore, the new volume of the BigMini tablet would be approximately 22.18 cubic inches.

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The coordinates of triangle X'Y'Z' is X'(-2,-2), Y'(-3,4), Z'(1,2).

What are the coordinates of the triangle?

The triangle's vertices have the coordinates (x1,y1), (x2,y2), and (x3,y3). The line that connects the first two is split in the ratio l:k, and the line that runs from the division point to the opposing angular point is divided in the ratio m:k+l.

Here, we have

Given: Triangle XYZ has coordinates X(-2,2), Y(-3,-4). And Z(1,-2).

The triangle is reflected across the x-axis and we have to find the coordinates of triangle X'Y'Z'.

X(-2,2), Y(-3,-4) and Z(1,-2).

While reflecting any points across the x-axis with coordinates as (x, y) becomes, (x, -y). i.e. sign of y-coordinate changes.

The rule is (x, y) → (x, -y)

After applying the rule, we get

X(-2,2) ⇒ X'(-2,-2)

Y(-3,-4) ⇒ Y'(-3,4)

Z(1,-2) ⇒ Z'(1,2)

Hence, the coordinates of triangle X'Y'Z' is X'(-2,-2), Y'(-3,4), Z'(1,2).

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Answer: 80 trees

Step-by-step explanation:

YIELD = (NUMBER OF TREES)*(NUMBER OF ORANGES PER TREE)

Let's assume NUMBER OF TREES = 60 + x, where x is the number of additional trees above 60

The NUMBER OF ORANGES PER TREE will = (400-4x).  Hence:

YIELD = (60+x)*(400-4x) = 24000-240x+400x-4x2 = -4x2 + 160x + 24,000

To find the maximum YIELD, take the derivative of YIELD wrt x, set it to zero, and solve for x:

d(YIELD)/dx = -8x + 160

0 = -8x +160

8x = 160

x = 20

The grower should grow 60 + 20 = 80 trees to maximize yield.

Answer: 80 trees

Step-by-step explanation: just bc it is

The maximum and minimum values for given function f(x, y) = 5x² + 5y² subject to xy = 1 are both 10. The extreme values of f(x, y, z) = x + 2y; x + y + z = 6, y² + z² = 4 subject to both constraints are 7 and -4.

We can use Lagrange multipliers to find the maximum and minimum values of f(x, y) subject to the constraint xy = 1.

First, we set up the Lagrange function

L(x, y, λ) = 5x² + 5y² + λ(xy - 1)

Then, we take partial derivatives of L with respect to x, y, and λ and set them equal to 0

∂L/∂x = 10x + λy = 0

∂L/∂y = 10y + λx = 0

∂L/∂λ = xy - 1 = 0

Solving these equations simultaneously, we get

x = ±√2, y = ±√2, λ = ±5/2√2

We also need to check the boundary points where xy = 1, which are (1, 1) and (-1, -1). We evaluate f at these points and compare them to the values we get from the Lagrange multipliers.

f(√2, √2) = 10, f(-√2, -√2) = 10

f(1, 1) = 10, f(-1, -1) = 10

So the maximum and minimum values of f(x, y) subject to xy = 1 are both 10.

We can use Lagrange multipliers to find the extreme values of f(x, y, z) subject to both constraints.

First, we set up the Lagrange function

L(x, y, z, λ, μ) = x + 2y + λ(x + y + z - 6) + μ(y² + z² - 4)

Then, we take partial derivatives of L with respect to x, y, z, λ, and μ and set them equal to 0

∂L/∂x = 1 + λ = 0

∂L/∂y = 2 + λ + 2μy = 0

∂L/∂z = λ + 2μz = 0

∂L/∂λ = x + y + z - 6 = 0

∂L/∂μ = y² + z² - 4 = 0

Solving these equations simultaneously, we get

x = -1, y = 2, z = 3, λ = -1, μ = -1/2

x = 3, y = -2, z = -1, λ = -1, μ = -1/2

We also need to check the boundary points where either x + y + z = 6 or y² + z² = 4. These points are (0, 2, 2), (0, -2, -2), (4, 1, 1), and (4, -1, -1). We evaluate f at these points and compare them to the values we get from the Lagrange multipliers.

f(-1, 2, 3) = 7, f(3, -2, -1) = -1

f(0, 2, 2) = 4, f(0, -2, -2) = -4

f(4, 1, 1) = 6, f(4, -1, -1) = 2

So the maximum value of f subject to both constraints is 7, which occurs at (-1, 2, 3), and the minimum value of f subject to both constraints is -4, which occurs at (0, -2, -2).

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The best implied thesis statement for the given prompt "Write an essay about something you do for your well-being" is: "Managing stress is important for overall well-being because the physical and emotional effects of stress are harmful." (Option D).

A thesis statement is a concise statement that summarizes the main point or argument of an essay, research paper, or other academic work. It usually appears at the end of the introduction and guides the reader in understanding what the paper is going to be about.

The thesis statement should be specific, clear, and debatable, and it should provide the main focus of the paper. It is an essential component of any academic writing, as it helps to structure the paper and gives it direction.

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