if tan theta =(√2)/3, what is the value of cos theta?​

The x-intercept of the original function can be determined by setting the function equal to zero and solving for x using algebraic techniques such as factoring, the quadratic formula, or other applicable methods.

The x-intercept of a function represents the point(s) where the graph of the function intersects the x-axis. It is the value(s) of x for which the corresponding y-coordinate is zero.

To determine the x-intercept for the original function, we need to set the function equal to zero and solve for x.

In equation form, we have:

f(x) = 0

By substituting 0 for f(x) in the original function, we can solve for x. This involves manipulating the equation and solving for x using various algebraic techniques, such as factoring, using the quadratic formula, or applying other methods specific to the equation at hand.

Once we solve the equation and obtain the value(s) of x that make f(x) equal to zero, those value(s) represent the x-intercept(s) of the original function.

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The value of x and y include the following:

x = 12 units.

y = 0

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Based on the side, angle, side (SAS) similarity theorem, we can logically deduce the following congruent angles and similar sides:

2x + 7 = 3x - 5

3x - 2x = 7 + 5

x = 12 units.

For the value of y, we have:

2y - 37 = 2y - 40

2y - 2y = 40 - 37

y = 0

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The integration of the given expression 32[tex]sinx^{2}[/tex]cosx can be done by applying the formula: ∫sin²x dx = (1/2)(x - sinx cosx)which we can derive from the double-angle formula for sin 2x, as shown below:

Sin 2x = 2 sin x cos x2 sin²x = 1 - cos 2x1/2(1 - cos 2x) = sin²xIf we let u = sin x², then du/dx = 2 sin x cos x.

This is the same as the integrand, except for the factor of 32. So we have: ∫32 sin x² cos x dx= 16 ∫ 2 sin x cos x sin x² dx= 16 ∫ 2 u du= 16u² + C= 16(sin x²)² + CNote that C is the constant of integration and could be any constant.

The solution for the integral of 32[tex]sinx^{2}[/tex]cosx with respect to x is given as: 16(sin x²)² + C.The steps for the above solution are:

Let u = sin x², then du/dx = 2 sin x cos xThe integral becomes 16 ∫ 2 u du16u² + C16(sin x²)² + C where C is the constant of integration.

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A ferris wheel is 10 meters in diameter and boarded from a platform that is 2 meters above the ground. The equation for the height above the ground, h = f(t), is:

f(t) = 5 sin[(π/2) t]

The ferris wheel completes one full revolution in 2 minutes, which means it takes 2 minutes for the wheel to go from its starting position to the same position again. Since a full revolution covers 360 degrees, we can say that in 2 minutes, the angle covered by the wheel is 360 degrees.

To find the equation for the height above the ground, we can use the sine function, as the height of a point on the wheel can be represented by the vertical component of the radius.

Let's consider the position at the six o'clock as the starting point, where the height above the ground is 2 meters. At this position, the angle is 0 degrees or 0 radians.

Now, as the wheel turns, the angle increases. We can relate the angle to time by assuming a constant angular velocity since the wheel completes one revolution in 2 minutes. Therefore, the angular velocity is (360 degrees / 2 minutes) = 180 degrees per minute.

Using radians, the angular velocity is (π radians / 2 minutes) = (π/2) radians per minute.

Thus, the equation for the height above the ground, h = f(t), is given by:

h = 5 sin[(π/2) t].

In this equation, t represents time in minutes, and h represents the height above the ground. The amplitude of the sine function is 5 since the radius of the ferris wheel is 5 meters (half of the diameter). The sine function generates values between -1 and 1, so multiplying by 5 ensures that the height varies between -5 and 5, relative to the starting position at 2 meters above the ground.

Therefore, the equation for the height above the ground is f(t) = 5 sin[(π/2) t].

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a) the probability that both socks selected are blue is 25/81. b) the probability of selecting a blue sock on the second draws is 10/18

a. When selecting two socks at random, with replacement, the probability that both socks are blue can be calculated as follows:

P(Both socks are blue) = P(Blue on first draw) * P(Blue on second draw)

= (10/18) * (10/18)

= 100/324

= 25/81

Therefore, the probability that both socks selected are blue is 25/81.

b. i. Mariam's solution:

Mariam correctly uses the probability formula for independent events. The probability of selecting two blue socks is equal to the probability of selecting a blue sock on the first draw (10/18) multiplied by the probability of selecting a blue sock on the second draw (also 10/18).

ii. John's solution:

John multiplies the number of blue socks (10) by the number of remaining blue socks (7) after one blue sock has been selected. This represents the total number of favorable outcomes (blue socks on both draws) out of the total number of possible outcomes (18 socks).

c. The two solution methods in Part b are equivalent because they both correctly calculate the probability of selecting two blue socks. Mariam's solution uses the probability formula for independent events, considering the probability of each individual draw. John's solution calculates the probability based on the count of favorable outcomes (number of blue socks) and the total number of possible outcomes (total socks).

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Answer:

CD: 8.5

m<D: 20.6°

m<C: 69.4°

Step-by-step explanation:

CD:

The first thing it wants us to do is find the length of CD. Since the triangle shown is a right triangle, we can use Pythagorean theorem ([tex]a^{2}+b^{2}=c^{2}[/tex])  to solve for the missing length. It's important to remember that when using the Pythagorean theorem, c is the hypotenuse.

[tex]a^{2}+b^{2}=c^{2}\\3^{2}+8^{2}=c^{2}\\9 + 64 = c^{2}\\73 = c^{2}\\\sqrt{73} =c[/tex]

Since our answer is no an integer, we must turn it into a decimal.

[tex]\sqrt{73}[/tex] ≈ 8.544003745 ≈ 8.5

m<D:

Now, they want us to find the measure of <D. To do this, we will need to use trig functions (sine, cosine, tangent). To help us determine which trig function to use, we can remember the acronym SOH CAH TOA. This acronym tells us that sine is equal to opposite divided by hypotenuse, cosine is equal to adjacent divided by hypotenuse, and tangent is equal to opposite divided by adjacent. Since we do the hypotenuse and sides adjacent and opposite of <D, we can choose whichever trig function we want. For this problem, we will use tangent, so we can avoid using a rounded number, 8.5, as one of our sides.

Tan(D) = opposite / adjacent

Tan(D) = 3 / 8     [Take the tan inverse of both sides}

[tex]Tan^{-1}(Tan(D))=Tan^{-1}(3/8)[/tex]     [Simplify]

[tex]D=Tan^{-1}(3/8)[/tex]     [Solve]

D ≈ 20.55604522

D ≈ 20.6°

m<C:

Lastly, we must find the last unknown angle on the triangle. Since all angles on a triangle total 180°, if know that <C+<D+<E=180°. Let's solve this equation.

<C+<D+<E=180°

<C + 20.6 + 90 = 180     [Add]

<C + 110.6 = 180     [Subtract]

<C = 180 - 110.6     [Solve]

<C = 69.4°

If this answer helped you, please leave a thanks!

Have a GREAT day!!!

Lo siento, pero como modelo de lenguaje, no tengo la capacidad de verificar el contenido original de una pregunta. Por favor, proporciona las sucesiones específicas para que pueda ayudarte a determinar si son progresiones.

Lo siento, pero como modelo de lenguaje, no tengo la capacidad de verificar el contenido original de una pregunta. Sin embargo, puedo ayudarte a comprender qué es una progresión y darte algunos ejemplos comunes de progresiones para que puedas verificar si las sucesiones que mencionas cumplen con las características.

Una progresión es una secuencia de números en la que cada término se obtiene sumando o multiplicando un número constante al término anterior. Algunos ejemplos de progresiones son:

Progresión aritmética: 2, 5, 8, 11, 14... (se suma 3 a cada término para obtener el siguiente).

Progresión geométrica: 3, 6, 12, 24, 48... (se multiplica por 2 cada término para obtener el siguiente).

Si tienes las sucesiones específicas, por favor proporciona los números para que pueda ayudarte a determinar si son progresiones y qué tipo de progresiones son.

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Answer:

X=68m

Step-by-step explanation:

X²=40²+55²

X²=1600+3025

X²=4625

X=(4625)½

X=68m.

The manager should order 120 packages of gelatin, 240 cups of citric acid, and 480 cups of corn syrup before leaving for vacation to ensure there are enough ingredients to make 24,000 grams of gummies.

To make 24,000 grams of gummies while the manager is away, the manager needs to order the following ingredients:

Gelatin: The recipe calls for 5 packages of gelatin to make 2000 grams of gummies. Therefore, to make 24,000 grams, the manager needs to order 120 packages of gelatin (5 packages x 24).

Citric acid: The recipe requires 2 cups of citric acid to make 2000 grams of gummies. Since 24,000 grams are needed, the manager should order 240 cups of citric acid (2 cups x 120).

Corn syrup: The recipe calls for 4 cups of corn syrup to make 2000 grams of gummies. To make 24,000 grams, the manager needs to order 480 cups of corn syrup (4 cups x 120).

Water: The recipe states that 2 cups of water are required, and it remains the same regardless of the number of gummies being made. Therefore, no additional water needs to be ordered.

In summary, the manager should order 120 packages of gelatin, 240 cups of citric acid, and 480 cups of corn syrup before leaving for vacation to ensure there are enough ingredients to make 24,000 grams of gummies.

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In Layla's case, her debt to income ratio is approximately 42.24%, which is within an acceptable range.

In order to calculate Layla's DTI, we add up all her monthly debt expenses and divide them by her gross monthly income:

Total Monthly Debt Expenses = Car Loan + Credit Card + Mortgage

= $200 + $25 + $1000

= $1225

DTI = (Total Monthly Debt Expenses / Gross Monthly Income) * 100

= ($1225 / $2900) * 100

≈ 42.24%

Typically, lenders prefer a lower DTI, generally below 43% for most loan types. In Layla's case, her DTI is approximately 42.24%, which is within an acceptable range.

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The solution to the given system of equations is x = 14/3 and y = 10/3.

To eliminate the variable "y" in the given system of equations using the elimination method, we need to make the coefficients of "y" in both equations the same or multiples of each other. In this case, we can see that the coefficient of "y" in the second equation, x + y = 8, is already 1.

To make the coefficients of "y" the same or multiples of each other, we need to choose a number to multiply the second equation by. In this case, we want to eliminate "y," so we want the coefficient of "y" in the first equation, 5x + 2y = 30, to become the opposite of the coefficient in the second equation, which is -1.

To achieve this, we need to multiply the second equation by -2. Multiplying x + y = 8 by -2 yields -2x - 2y = -16. Now we have the equations:

5x + 2y = 30

-2x - 2y = -16

By adding these two equations together, the "y" terms will cancel out, allowing us to solve for "x":

(5x + 2y) + (-2x - 2y) = 30 + (-16)

3x = 14

Now we can solve for "x" by dividing both sides of the equation by 3:

3x/3 = 14/3

x = 14/3

Substituting the value of "x" back into one of the original equations, we can solve for "y." Let's use the second equation:

x + y = 8

14/3 + y = 8

Subtracting 14/3 from both sides:

y = 8 - 14/3

y = 24/3 - 14/3

y = 10/3

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The answer to your question is -2

I have done this quiz!

hope this helps :)

A sailor sails upstream in 2*1/2 hours and returns upstream in 3*3/4 hours. The bank velocity is 10 km/h, and the current velocity is 2 km/h.

Let's denote the speed of the sailor in still water as "s" km/h and the speed of the current as "c" km/h.

When the sailor is sailing upstream, they are moving against the current, so their effective speed is reduced. The time it takes to travel a certain distance is given by the formula:

Time = Distance / Speed

Given that the time to sail upstream is 21/2 hours (or 2.5 hours) and the time to sail downstream is 33/4 hours (or 3.75 hours), we can set up the following equations:

Distance Upstream / (s - c) = 2.5 (Equation 1)

Distance Downstream / (s + c) = 3.75 (Equation 2)

Since the distance traveled upstream is the same as the distance traveled downstream, we can set the left sides of both equations equal to each other:

Distance Upstream / (s - c) = Distance Downstream / (s + c)

Cross-multiplying, we get:

Distance Upstream * (s + c) = Distance Downstream * (s - c)

Since the distances are the same, we can simplify the equation to:

s + c = s - c

Simplifying further, we get:

2c = 0

This means that the current velocity is 0 km/h, which is not possible. Therefore, there must be an error in the problem statement. Please double-check the information provided to ensure accuracy.

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Answer:

introduction of a business invironment

Answer:

Step-by-step explanation:

Without that additional information, we cannot determine the specific measure of BD.

To find the measure of BD, we can use the given information that AD is 4 times the length of DC. Let's denote the length of DC as x.

Since AD is 4 times the length of DC, we have AD = 4x.

We also know that AD + DC + BD = 180 degrees, as the angles in a triangle add up to 180 degrees.

Substituting the values we have, we get:

4x + x + BD = 180

Combining like terms, we have:

5x + BD = 180

To find the value of BD, we need more information.

If we have additional information about the triangle, such as the measure of another angle or the length of another side, we can use it to solve for BD.

If you have any additional details about the figure or any relationships between the various segments or angles, please provide them.

This will enable me to assist you in finding the measure of BD.

Without any specific information about the figure or any relationships involving BD, it is not possible to determine its measure accurately.

Please provide any additional details or clarify the question further, and I will be happy to help you find the measure of BD within the given word limit.

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The pair that represents the same complex number is . 1 - i and

√2(cos(-π/4) + i sin(-π/4)). Option A.

Here are the following complex numbers and their solutions

A. √2(cos(-π/4) + i sin(-π/4))

⇒ cos(-π/4) √2 + √2 × (i) sin(-π/4))

= 1 - i

B. 5(cos(4/3) + i sin (4/3))

⇒ 5cos(4/3) + 5(i) sin (4/3)

= 1.17618787 + 4.85968951 i is not the same as 3 + 4i

C. √2(cos(π/4) + i sin(π/4))

⇒cos(π/4) √2 + √2 × (i) sin(π/4))

= 1 + 1i which is 1+i. It is not the same as 1 - i

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The probability that either event will occur is 0.33

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

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

Total = A + B + C

So, we have

Total = 8 + 7 + 30

Evaluate

Total = 45

So, we have

P(A) = 8/45

P(B) = 7/45

For either events, we have

P(A or B) = 8/45 + 7/45

P(A or B) = 15/45

Evaluate

P(A or B) = 0.33

Hence, the probability that either event will occur is 0.33

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Answer:

  (c)  1 1/8 lb

Step-by-step explanation:

You want to reading from the scale shown to the nearest 1/8 lb.

The smallest heavy markings represent 1/4 lb, and there are 4 marks between. Each of those lighter marks is 1/16 lb. The pointer seems to fill a space that is more than 1/16 lb away from 1 lb, and just slightly less than 1/16 away from 1 1/4 lb.

We judge the center of the pointer to be approximately 2/16 lb above 1 lb, so representing a weight of about 1 1/8 lb.

__

Additional comment

The width of the pointer makes it hard to judge the weight. Since 1 1/4 lb is not an answer choice, the most reasonable choice is 1 1/8 lb. Very likely an actual scale would have a pointer that would be a small fraction of the width of the space between smaller marks.

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Regression equation correctly models the data is: y = -43.98x + 1,279.93

To determine the regression equation that correctly models the data, we can use the method of linear regression. The regression equation for a straight line is generally expressed as y = mx + b, where m is the slope and b is the y-intercept.

Using the given table, we can calculate the necessary values to determine the regression equation. Let's denote the sigma notation as Σ.

The slope (m) can be calculated using the formula:

[tex]m = (Σxy - (Σx)(Σy) / n(Σx^2) - (Σx)^2)[/tex]

Plugging in the values from the table:

m =[tex](776,600 - (299)(12,400) / 6(18,109) - (299)^2)[/tex]

m = (776,600 - 3,708,800 / 6(18,109) - 89,401)

m = (-2,932,200 / 66,654)

m ≈ -43.98

The y-intercept (b) can be calculated using the formula:

b = (Σy - m(Σx)) / n

Plugging in the values from the table:

b = (12,400 - (-43.98)(299)) / 6

b ≈ 1,279.93

The correct regression equation that models the data is:

y = -43.98x + 1,279.93

Out of the given options, the correct regression equation is:

y = -43.98x + 1,279.93

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Answer: Its going down by odd numbers

Step-by-step explanation:

5-2=3, 2-(-3)= 5, then it would go on 7, 9, 11, 13...

Answer:

The difference between the numbers does not follow a common pattern; hence, the sequence is not arithmetic.

Step-by-step explanation:

Let's take the 1st two digits, 5 and 2

The difference between the numbers, 5-2 = 3

Following this pattern,

2-(-3) = 5

-3-(-10) = 7 and,

-10-(-19) = 9.

We can see that the differences are 3, 5, 7 and 9 and therefore we can prove that there is no common difference between the numbers.

Hence the sequence does not follow an arithmetic pattern as there is no common difference

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The yield to maturity of the ABC Inc. bonds is 3.63%.

To calculate the yield to maturity, we can use the formula for present value of a bond:

PV = C * (1 - (1 +[tex]r)^{(-n))}/r + F/(1 + r)^n[/tex]

Where:

PV = Present Value or current price of the bond ($1,099)

C = Coupon payment per period ($1,000 * coupon rate = $84)

r = Yield to maturity rate (unknown)

n = Number of periods (18 years * 2 semi-annual periods = 36 periods)

F = Face value or par value of the bond ($1,000)

We need to find the value of r in the formula. Since it is not possible to solve this equation algebraically, we can use an iterative method like trial and error or a financial calculator.

Using a financial calculator or Excel, we can plug in different values for r until we get the present value (PV) equal to $1,099. After performing the calculations, we find that the yield to maturity is approximately 3.63%.

1. Convert the coupon rate to a decimal: 8.40% = 0.084.

2. Calculate the semi-annual coupon payment: $1,000 * 0.084 = $84.

3. Convert the number of years to the number of semi-annual periods: 18 years * 2 = 36 periods.

4. Set up the present value equation: $1,099 = $84 * (1 - (1 + [tex]r)^{(-36[/tex]))/r + $1,000/(1 + [tex]r)^{36[/tex].

5. Use a financial calculator or Excel to solve for r by trying different values until the present value is approximately $1,099.

6. Round off the yield to maturity rate to two decimal points: 3.63%.

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Given: Deposit = $120, Annual Interest = 7%, Compounded Annually, No additional deposits or withdrawals. After 8 years we need to calculate the balance of this account. To solve this question, we use the formula for compound interest. So, the balance of the account after 8 years will be $219.11.

Compound Interest FormulaThe compound interest formula is given as follows; A = P (1 + r/n)nt, Where, A is the amount earned after time t, P is the principal amount,n is the number of times interest is compounded every year,r is the annual interest rate,t is the time the money is invested in years.

Substituting the values in the given formula; P = $120, r = 7%, n = 1 (compounded Annually), t = 8 years

A = 120(1 + 0.07/1)^(1×8)A = $219.11. Therefore, the balance of the account after 8 years will be $219.11.

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Answer: the answer is 1:16

The value of p is 1 unit.

The distance between two points A( x₁, y₁) and B(x₂, y₂) is given by:

d= [tex]\sqrt{((y_{2}-y_{1})^{2}+({x_{2}-x_{1})^{2}) }[/tex]

Given that the distance between the origin and point P(-2, p-1) is 2p units.

i.e. 2p = [tex]\sqrt{(p-1-0)^2+(-2-0)^2}[/tex]

⇒  2p = [tex]\sqrt{(p-1)^{2} +(-2)^2}[/tex]

⇒ 2p = [tex]\sqrt{p^2 + 1- 2p +4}[/tex]

⇒ 2p = [tex]\sqrt{p^2 - 2p +5}[/tex]

Squaring both sides, we get:

4p² = p² - 2p + 5

⇒ 3p² + 2p - 5 = 0

⇒ 3p² + 5p - 3p - 5 = 0

⇒ p (3p + 5) - 1 (3p + 5) = 0

⇒ (3p + 5) (p-1) = 0

p = 1 unit

Note: p ≠ -5/3 since distance cannot be negative.

Hence p = 1 unit.

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Answer:

54 m²

Step-by-step explanation:

i.

xy = 24

y = 24/x

ii.

A = (x + 2)(y + 3)

A = xy + 3x + 2y + 6

A = x(24/x) + 3x + 2(24/x) + 6

A = 24 + 3x + 48/x + 6

A = 3x + 48/x + 30

iii.

A = 3x + 48x^-1 + 30

dA/dx = 3 + (-1)(48)x^-2

dA/dx = 3 - 48/x²

3 - 48/x² = 0

48/x² = 3

x² = 16

x = 4 or x = -4

We discard the negative answer.

x = 4

y = 24/x

y = 24/4

y = 6

A = xy

A = (x + 2)(y + 3)

A = (4 + 2)(6 + 3)

A = 6(9)

A = 54

The unlabeled side length is 10 units. We don't need to round to the nearest hundredth in this case.

To find the unlabeled side length, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In the given diagram, we have a right triangle with side lengths of 8 and 6. Let's label the unlabeled side as x.

Using the Pythagorean theorem, we can write the equation:

x^2 = 8^2 + 6^2

Simplifying, we have:

[tex]x^2[/tex]= 64 + 36

x^2 = 100

To find the value of x, we take the square root of both sides:

x = √100

x = 10

Therefore, the unlabeled side length is 10.

It's important to note that since the given side lengths are whole numbers, the unlabeled side length is also a whole number. As a result, we don't need to round to the nearest hundredth in this case.

Hence, the unlabeled side length is 10 units.

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Answer:

[tex]\huge\boxed{\sf 44\ boys}[/tex]

Step-by-step explanation:

= Length × Width

Length = 32 m

Width = 11 m

Area of school room:

= Length × Width

= 32 × 11

= 352 m²

= 352 / 8

[tex]\rule[225]{225}{2}[/tex]