Write an equation in slope-intercept form to represent the
table.
x: 0, 1.5, 4, 6.5, 7
y: 6.4, 4.6, 1.6, -1.4, -2

Answer:

4.28

Step-by-step explanation:

To find what the length of AC is, we can use the tangent of angle A, which is 35°.  Here's the equation:

tan(35)=3/x

multiply both sides by x

x·tan(35)=3

divide both sides by the tangent of 35

x=4.28 (rounded to the nearest hundredth)

This means that AC=4.28

Hope this helps! :)

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

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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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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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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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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Complete question:

See attached photo

The volume of the cones are;

1. 84. 78 in³

2.  564. 15 ft³

3.  4710 yd³

The formula for calculating the volume of a cone is expressed as;

V = 1/3 πr²h

Given that;

From the information given, we have;

1. Volume = 1/3 × 3.14 × 3² × 9

Multiply the values and find the square

Volume = 254. 34/3

divide the values

Volume = 84. 78 in³

2. Volume = 1/3 × 3.14 × 7² × 11

Multiply the values

Volume = 1692. 46/3

Volume = 564. 15 ft³

3. Volume = 1/3 × 3.14 × 15² × 20

Multiply the values

Volume = 4710 yd³

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Given that MNPQ is a rectangle with verticles M(3, 4), N(1, -6), and P(6, -7), to find the coordinates of point Q, we can use the fact that opposite sides of a rectangle are parallel and have equal lengths.

First, let's find the vector MN and MP:

MN = N - M = (1 - 3, -6 - 4) = (-2, -10)
MP = P - M = (6 - 3, -7 - 4) = (3, -11)

Now, let's add the vector MN to point P:

Q = P + MN = (6 + (-2), -7 + (-10)) = (4, -17)

Therefore, the coordinates of point Q that make MNPQ a rectangle are Q(4, -17).


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a. A customer would save $492 during the first year by switching from ElectroniSource to Intellivision. b. A customer who would save $207 in the second year. c. Intellivision's total cost of $1654.56.

Algebra is a branch of mathematics that deals with the manipulation and properties of variables, symbols, and equations. In algebra, variables represent unknown quantities, and equations represent relationships between those unknown quantities.

The fundamental operations in algebra are addition, subtraction, multiplication, and division. Algebraic equations involve variables and constants, which are combined using these operations to form algebraic expressions. These expressions can be simplified by applying algebraic rules and properties, such as the distributive property, associative property, and commutative property.

a. To calculate the savings during the first year, we need to find the total cost for each company for all three services during a year and compare them.

For ElectroniSource, the total cost for a year would be:

$42/month x 12 months = $504 for phone service

$35/month x 12 months = $420 for Internet service

$59/month x 12 months = $708 for cable TV service

Total cost for a year with ElectroniSource = $504 + $420 + $708 = $1632

For Intellivision, the flat monthly fee is $95, so the total cost for a year would be:

$95/month x 12 months = $1140

Savings during the first year = Cost with ElectroniSource - Cost with Intellivision

= $1632 - $1140

= $492

Therefore, a customer would save $492 during the first year by switching from ElectroniSource to Intellivision.

b. After the first year, Intellivision raises the rates by 25%. The new monthly fee would be:

$95 + 25% of $95 = $118.75

To calculate the savings for the second year, we need to find the total cost for each company for all three services during the second year and compare them.

For ElectroniSource, the total cost for the second year would still be:

$504 for phone service

$420 for Internet service

$708 for cable TV service

Total cost for the second year with ElectroniSource = $504 + $420 + $708 = $1632

For Intellivision, the total cost for the second year would be:

$118.75/month x 12 months = $1425

Savings during the second year = Cost with ElectroniSource - Cost with Intellivision

= $1632 - $1425

= $207

Therefore, a customer who switched from ElectroniSource to Intellivision would save $207 in the second year.

c. If Intellivision raises the rates by 16% for the third year compared to the second year, the new monthly fee would be:

$118.75 + 16% of $118.75 = $137.88

To compare the total cost for each company for the third year, we need to find the total cost for each company for all three services during the third year and compare them.

For ElectroniSource, the total cost for the third year would still be:

$504 for phone service

$420 for Internet service

$708 for cable TV service

Total cost for the third year with ElectroniSource = $504 + $420 + $708 = $1632

For Intellivision, the total cost for the third year would be:

$137.88/month x 12 months = $1654.56

Therefore, ElectroniSource is cheaper for the third year, with a total cost of $1632 compared to Intellivision's total cost of $1654.56.

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The maximum height of Anna's golf ball is 6.25 feet.

To find the maximum height of Anna's golf ball, we need to determine the vertex of the parabolic equation y = x - 0.04x^2. The x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

In this case, the coefficients a and b are:
a = -0.04
b = 1

Substituting the values into the formula:

x = -1 / (2 * -0.04)
x = -1 / (-0.08)
x = 12.5

Now, we need to find the y-coordinate of the vertex by plugging the x-coordinate back into the equation:

y = 12.5 - 0.04(12.5)^2

y = 12.5 - 0.04(156.25)

y = 12.5 - 6.25

y = 6.25

So, the maximum height of Anna's golf ball is 6.25 feet.

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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 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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The matrices are: C = [140, 24, 2, 11, 61]           P = [160, 4, 13, 21, 656] B = [40, 5, 1, 81, 23]

To represent the nutritional values of the chicken, potato salad, and broccoli in matrices, we can use a 1x5 matrix for each food, where each column represents a different nutritional value in the following order: calories, protein, fat, calcium, and sodium.

Therefore, we have:

C = [140 24 2 11 61]
P = [160 4 13 21 656]
B = [40 5 1 81 23]

In matrix C, the values are 140 calories, 24 grams of protein, 2 grams of fat, 11 milligrams of calcium, and 61 milligrams of sodium for a 3 ounce serving of roasted skinless chicken breast. In matrix P, the values are 160 calories, 4 grams of protein, 13 grams of fat, 21 milligrams of calcium, and 656 milligrams of sodium for one half cup of potato salad. In matrix B, the values are 40 calories, 5 grams of protein, 1 gram of fat, 81 milligrams of calcium, and 23 milligrams of sodium for one broccoli spear.

These matrices can be used to perform calculations and comparisons between the nutritional values of the different foods.

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Answer: The transformation is up one and to the left 5

(-5, 1)

Step-by-step explanation:

Please Give brainliest, have a great night!

Answer:   h(x) = g(x+5)+1

Step-by-step explanation:

h(x) = g(x+5)+1

You went 5 in the negative direction x direction; take the opposite sign

Also went up 1 in the y direction so add 1 to equation

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.

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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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 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?

The radius of the circle is 2mm

It is important to note that the formula that is used for calculating the circumference of a circle is expressed with the equation.

The equation is written as;

C = 2πr

Such that the parameters are;

From the information given, we have thatt;

The radius of the circle = r

The circumference = 12.56mm

Substitute the values, we have;

12. 56 = 2×3.14r

Multiply the values

r = 12. 56/6. 28

divide the values

r = 2mm

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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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Answer:

[tex]x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n[/tex]

Step-by-step explanation:

Given equation:

[tex]\sin(2x)\sin(x)+\cos(x)=0[/tex]

Rewrite sin(2x) using the trigonometric identity sin(2x) = 2sin(x)cos(x):

[tex]\implies 2\sin(x)\cos(x)\sin(x)+\cos(x)=0[/tex]

[tex]\implies 2\sin^2(x)\cos(x)+\cos(x)=0[/tex]

Factor out cos(x):

[tex]\implies \cos(x)\left[2\sin^2(x)+1\right]=0[/tex]

Applying the zero-product property:

[tex]\textsf{Equation 1:}\quad\cos(x)=0[/tex]

[tex]\textsf{Equation 2:}\quad2\sin^2(x)+1=0[/tex]

Solve each part separately.

[tex]\underline{\sf Equation \; 1}[/tex]

[tex]\begin{aligned}\cos(x)&=0\\x&=\arccos(0)\\x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n\end{aligned}[/tex]

[tex]\underline{\sf Equation \; 2}[/tex]

[tex]\begin{aligned}2\sin^2(x)+1&=0\\\sin^2(x)&=-\dfrac{1}{2}\;\;\;\;\;\;\leftarrow\;\textsf{No solution}\end{aligned}[/tex]

Therefore, the solutions of the equation in radians are:

[tex]\boxed{x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n}[/tex]

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 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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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:

22.69 ~ = 23 feet

Step-by-step explanation:

cos 19 = adj/24

24cos 18 = adj

adj = 22.69~= 23