Which region contains viable solutions to the systems of inequalities within the context of the situation? The situation is Everett wanted to create chocolate milk using whole milk and chocolate syrup. He wanted his chocolate milk to have less than 8 grams (g) of fat and less than 28 grams (g) of sugar. He drew the graph shown where x is the amount of whole milk in ounces (oz) and y is the amount of chocolate syrup in ounces (oz).

The perimeter of the shaded region is approximately 85.67 cm The area of the shaded region is approximately 91.84 cm² (rounded to the nearest tenth).

To calculate the perimeter and area of the shaded region, we first need to find the radius of each circle.

The larger circle has a diameter of 21 cm, which means its radius is 10.5 cm (half of the diameter). The smaller circle has a diameter of 10 cm, so its radius is 5 cm.

To find the perimeter of the shaded region, we need to add the circumference of both circles and subtract the overlap (the length of the shared segment). The circumference of the larger circle is 2π(10.5) ≈ 65.97 cm, and the circumference of the smaller circle is 2π(5) ≈ 31.42 cm.

To find the length of the shared segment, we can use the Pythagorean theorem. The distance between the centers of the circles is 15 cm (the sum of the radii), so we can form a right triangle with legs of 10.5 cm and 5 cm. Using the Pythagorean theorem, we get:

c² = a² + b²
c² = 10.5² + 5²
c² ≈ 137.25
c ≈ 11.72

So the length of the shared segment is approximately 11.72 cm.

Therefore, the perimeter of the shaded region is approximately 65.97 + 31.42 - 11.72 = 85.67 cm (rounded to the nearest tenth).

To find the area of the shaded region, we need to subtract the area of the smaller circle from the area of the larger circle, and then subtract the area of the overlap (the area of the shared segment).

The area of the larger circle is π(10.5)² ≈ 346.36 cm², and the area of the smaller circle is π(5)² ≈ 78.54 cm².

To find the area of the shared segment, we can use the formula for the area of a sector of a circle:

A = (θ/360)πr²

where θ is the central angle of the sector. In this case, the sector has a central angle of 2cos⁻¹(5/10.5) ≈ 105.2°, so:

A = (105.2/360)π(10.5)²
A ≈ 91.84 cm²

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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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The amount change in her bank account during that week after withdrawals and deposit is equal to $59.

Total number of withdrawals made by Sheila = 4

Amount made at the time withdrawals using ATM = $40

Amount withdraw using check card to purchase = $156

Amount deposited using paycheck = #375

Let us calculate the total amount of money Sheila withdrew from her bank account using ATM,

4 withdrawals of $40 each

= 4 x $40

= $160

So, she withdrew $160 and made a $156 purchase, meaning she spent a total amount of,

= $160 + $156

= $316

Sheila also deposited her paycheck of $375, so the total amount of money in her account changed by is equal to,

$375 - $316 = $59

Therefore, the amount in Sheila's account increased by $59 during that week.

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The above question is incomplete, the complete question is:

During one week, Sheila made several changes to her bank account. She made four withdrawals of $40 each from an ATM. She also used her check card for a $156 purchase. Then she deposited her paycheck of $375. By how much did the amount in her bank account change during that week?

1)  All the solution are,

(x, y) = (√2, 2) , (- √2, 2), (√2, -2) , (- √2, -2), (√2i, 2) , (- √2i, 2), (√2i, -2) , (- √2i, -2),

2) Solutions are,

⇒ x = 3, 3, - √3 / 2, - 5

3) All the even integers which are divisible by 5 is,

⇒ 10, 20, 30, 40, 50, ....

Given that;

1) Expression is,

⇒ x² = y, and y² = 4

2) Expression is,

⇒ (x - 3)² (2x + √3) (x + 5) = 0

Now, We can simplify as;

⇒ x² = y,

⇒ x⁴ = y²

x⁴ = 4

x⁴ - 2² = 0

(x²)² - 2² = 0

(x² - 2) (x² + 2) = 0

This gives,

x² = 2

x = ± √2

x² = - 2

x = ±√2 i

Hence, We get;

y² = 4

y = ± 2

Thus, All the solution are,

(x, y) = (√2, 2) , (- √2, 2), (√2, -2) , (- √2, -2), (√2i, 2) , (- √2i, 2), (√2i, -2) , (- √2i, -2),

Since, 2) Expression is,

⇒ (x - 3)² (2x + √3) (x + 5) = 0

Simplify as;

⇒ (x - 3)² = 0

⇒ x = 3, 3

⇒ (2x + √3) = 0

⇒ x = - √3 / 2

⇒ (x + 5) = 0

⇒ x = - 5

3) All the even integers which are divisible by 5 is,

⇒ 10, 20, 30, 40, 50, ....

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(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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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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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 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 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 equation has at least one solution in the interval (-1, 1).

To prove that the equation has at least one solution in the interval (-1, 1), we can use the Intermediate Value Theorem.

First, let's simplify the equation by finding a common denominator:

a(x^3+x-2) + b(x^3+2x^2-1) = 0

Now, let's define a new function f(x) = a(x^3+x-2) + b(x^3+2x^2-1). This function is continuous on the interval (-1, 1) because it is a sum of continuous functions.

Next, we will evaluate f(-1) and f(1) to see if the Intermediate Value Theorem can be applied.

f(-1) = a(-1^3-1-2) + b(-1^3+2(-1)^2-1) = -a-b < 0

f(1) = a(1^3+1-2) + b(1^3+2(1)^2-1) = a+3b > 0

Since f(-1) is negative and f(1) is positive, there must be at least one value of x in the interval (-1, 1) such that f(x) = 0, by the Intermediate Value Theorem.
To prove that the given equation has at least one solution in the interval (-1, 1), we can use the Intermediate Value Theorem (IVT). Let's define the function f(x) as follows:

f(x) = a/(x^3 + 2x^2 - 1) + b/(x^3 + x - 2)

Since a and b are positive numbers, we can examine the behavior of f(x) at the endpoints of the interval (-1, 1).

f(-1) = a/((-1)^3 + 2(-1)^2 - 1) + b/((-1)^3 + (-1) - 2)
f(-1) = a/(-1) + b/(-4) < 0

f(1) = a/(1^3 + 2(1)^2 - 1) + b/(1^3 + 1 - 2)
f(1) = a/(2) + b/(0) = a/2 > 0

Since f(-1) < 0 and f(1) > 0, by the Intermediate Value Theorem, there must be at least one point c within the interval (-1, 1) where f(c) = 0. This means that the given equation has at least one solution in the interval (-1, 1).

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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

The absolute maximum of the function f(x,y) = y√(x-y^2)-x+3y on the domain 0≤x≤9, 0≤y≤8 is 2√2, which occurs at the point (2,2).

The absolute minimum of the function is -8, which occurs at the point (0,2).

To find the absolute maximum and minimum of the function f(x,y) = y√(x-y^2)-x+3y on the domain 0≤x≤9, 0≤y≤8, we need to evaluate the function at the critical points and at the boundary of the domain.

The critical points of the function are the points where the partial derivatives with respect to x and y are both zero. Solving these equations, we get:∂f/∂x = -1 + y/(√(x-y^2)) = 0∂f/∂y = √(x-y^2) - 1 + 3 = 0Solving these equations, we get two critical points: (0,2) and (2,2). Evaluating the function at these points, we get:f(0,2) = -8f(2,2) = 2√2Next, we need to evaluate the function at the boundary of the domain. This includes the points (0,y), (9,y), (x,0), and (x,8).

Evaluating the function at these points, we get:f(0,y) = -x+3yf(9,y) = y√(9-y^2)-6f(x,0) = -xf(x,8) = 8√(x-64/9)-x+24Taking the maximum and minimum values of the function at the critical points and on the boundary of the domain, we see that the absolute maximum of the function is 2√2, which occurs at the point (2,2), and the absolute minimum of the function is -8, which occurs at the point (0,2).

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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) 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 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:

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.

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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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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 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: 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 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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ANSWER:

(x+7)^2=16

Step-by-step explanation:

x^2-14x+49=16

x^2-14x+49-16=0

x^2-14x+33=0

subtract -33 on both sides

x^2-14x+33-33=-33

x^2-14x=-33

Add 49 on both sides

x^2-14x+49=-33+49

x^2-14x+49=16

x^2-7x-7x+49=16

x(x-7)-7(x-7)=16

(x-7)(x-7)=16

(x-7)^2=16

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)

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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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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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 length of arc CD, given that the radius, EB = 15 cm, is 29.31 cm

First, we shall determine ∠CED. Details below:

2∠CED + 2∠BEC = 360

2∠CED + (2 × 68) = 360

2∠CED + 136 = 360

Collect like terms

2∠CED = 360 - 136

2∠CED = 224

Divide both sides by 2

∠CED = 224 / 2

∠CED = 112°

Finally, we shall determine the length of the of arc CD. Details below:

Length of arc = 2πr × (θ / 360)

Length of arc CD = (2 × 3.14 × 15) × (112 / 360)

Length of arc CD = 29.31 cm

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