Solve each equation

tan([tex]\frac{x}{2}[/tex] - [tex]\frac{\pi }{2}[/tex]) = [tex]\sqrt{2}[/tex]

Also what is the value of arctan([tex]\sqrt{2}[/tex])

Give in the form x=[tex]\pi[/tex]+[tex]\pi[/tex]n

Answer:

Below.

Step-by-step explanation:

Exponential growth equation is

Nt = No e^kt                 where No  is initial quantity , Nt is quantity after                       time t and k is some constant.

Here we have:

1838 = 1600 e^4k

e^4k = 1838/1600 = 1.14875

4k = ln 1.14875 = 0.138674

k = 0.034669

The solution of the system of equations is x = -5/17 and y = -16/17.

To do this, we can use a method called substitution. In this method, we solve one equation for one variable in terms of the other variable, and then substitute this expression into the other equation. This results in a new equation with only one variable, which can be solved to find the value of that variable. Once we know the value of one variable, we can substitute it into either equation to find the value of the other variable.

Let's apply this method to your system of equations:

5x - y/2 = -1 (Equation 1)

3x - 2y = 1 (Equation 2)

First, we'll solve Equation 1 for y in terms of x:

5x - y/2 = -1

y/2 = 5x + 1 (Add y/2 to both sides)

y = 10x + 2 (Multiply both sides by 2)

Now we substitute this expression for y into Equation 2:

3x - 2(10x + 2) = 1 (Substitute y = 10x + 2)

3x - 20x - 4 = 1

-17x = 5

x = -5/17

Now that we know the value of x, we can substitute it into either Equation 1 or Equation 2 to find the value of y. Let's use Equation 1:

5x - y/2 = -1

5(-5/17) - y/2 = -1

-25/17 - y/2 = -1

y/2 = -1 + 25/17

y/2 = -8/17

y = -16/17

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Complete Question:

What is the solution of the system of equations:

5x - y/2 = -1

3x - 2y = 1

Answer:

55

Step-by-step explanation:

The first we need to do is find the other angle measures. The left angle must be supplementary with 110, so it must measure 70. The right angle must be supplementary with 125, so it must measure 55. The sides of an angle add up to 180, so 180-70-55= 55

The angle of x is 55°.

There is 180° in a triangle. First, we need to calculate the angles of the other two sides. A straight line also forms 180°. Therefore the first angle will be

180 - 110= 70°

The second angle will be

180 - 125 = 55°

Angles of the triangle excluding x will be

70 + 55  =  125°

Since there is 180° in a triangle, x will be

180 - 125=  55°

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The production of biodiesel would be the least likely to impact the price of electricity. Biodiesel is not commonly used as a fuel source for electricity generation, so changes in its production would not significantly affect the overall supply and demand of electricity. On the other hand, the price of a barrel of oil, the cost of a ton of coal, and a fire in a gas pipeline could all impact the price of electricity, as these are commonly used fuel sources for electricity generation.

[tex] \sqrt{ \frac{1 - cos(150)}{2} } = \sqrt{ \frac{1}{2} - \frac{cos(150)}{2} } = \sqrt{ \frac{1}{2} - \frac{ \frac{ - \sqrt{ 3}}{2} }{2} } = \sqrt{ \frac{2}{4} + \frac{ \sqrt{3} }{4} } = \sqrt{ \frac{2 + \sqrt{3} }{4} } [/tex]

[tex]= \frac{ \sqrt{6} + \sqrt{2} }{4} [/tex]

BTW:

[tex]sin(75) = \frac{ \sqrt{6} + \sqrt{2} }{4} [/tex]

[tex] = > sin(75) = \sqrt{ \frac{1 - cos(150)}{2} } [/tex]

Ans: True

Ok done. Thank to me >:33

The value of x in the expression is

To solve for x in the the equation e^(3x+10) = 13^(x/4) we can take the natural logarithm (ln) of both sides

ln(e^(3x+10)) = ln(13^(x/4))

3x + 10 = (x/4) * ln(13)

Isolating x by collecting like terms

3x - (x/4) * ln(13) = -10

factor out x as a common factor:

x * (3 - ln(13)/4) = -10

x = -10 / (3 - ln(13)/4)

x =  -4.2395

x = -4.24 to 2 decimal place

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The most common amount of sleep: is 8 hours (6 individuals).

Outlier: Frankie (slept for 6 hours), as there is a significant difference between his sleep time and the rest of the individuals in the dataset (range of 4 hours).

The predominant resting time among the subjects is 8 hours.

Nonetheless, it's worth noting that an outlier exists within the dataset; Frankie spent a mere 6 hours sleeping - which amounts to two hours less than the sleep duration recorded by the second least rested individual (8 hours).

This information can be extracted from analyzing the figures and noticing the conspicuous discrepancy between Frankie's slumbering span and that of the other participants in the pool of data and this is the outlier.

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Answer: ∠EFA and/or ∠CFB

Step-by-step explanation:

Adjacent: In order to classify adjacent the 2 angles need to have a common side and common vertex

Starting Angle: ∠CFE

Possible Adjacents: ∠ EFA and/or ∠CFB

Answer:

96

Step-by-step explanation:

8 times 12 = 96

The answers are A = 46% and B = 20%.

Given that, a table of data, each cell in the table shows the number of cars in that category.

A) The probability that the car's total repair cost is less than $10000 =

86 / (86+35+67) x 100 = 86/188 x 100 = 46%

B) The probability that the car's purchase was more than $40,000 =

40 / (40+86+71) x 100 = 40/197 x 100 = 20%

Hence, the answers are A = 46% and B = 20%.

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The percent rate of each function given is:

[tex]f(t) = 1.18^t:16.99\%\\\\g(t) = 2^{-2t}:-9.1\%\\\\h(t) = 1.19^{\frac{t}{10} }:8.5\%\\\\k(t) = 0.13^t:-19.5\%[/tex]

A measure known as the percent rate of change quantifies the amount of change in a quantity over time and is given as a percentage of the initial number. It is frequently used to keep tabs on population growth, shifts in the financial markets, and other occurrences.

You must first determine the amount of change in the quantity (often done by deducting the beginning value from the final value) in order to compute the percent rate of change. To calculate the percent rate of change, divide this amount of change by the starting value, multiply the result by 100, and add a percent sign.

[tex]f(t) = 1.18^t:[/tex] 16.99% is the percent rate of change per unit (t). This increase is exponential.

[tex]g(t) = 2^{-2t}:[/tex] Approximately -9.1% of a percent rate of change per unit, t. The decline is exponential.

[tex]h(t) = 1.19^{\frac{t}{10} }:[/tex] The average percent rate of change for unit t is around 8.5 percent. This increase is exponential.

[tex]k(t) = 0.13^t:[/tex] t represents the percent rate of change, which is around -19.5% per unit. The decline is exponential.

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Answer:

1. 62

Step-by-step explanation:

The graph of the quadratic function  [tex]y=-2x^2-4x+4[/tex] opens downward.

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function is [tex]y=-2x^2-4x+4[/tex]

The general form of the quadratic function is y=[tex]ax^2+bx+c[/tex]

Then, a= -2

Which is negative value.

Then the graph is open downward.

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The measure of angle x is  23⁰.

The value of arc angle y is 45.5⁰.

The value of arc angle z is 116⁰.

The measure of angle x, y and z is calculated as follows;

∠mEGF = ¹/₂ (arc EF - arc DC ) (intersecting chord theorem)

33 = ¹/₂ (89 - x )

2(33) = 89 - x

66 = 89 - x

x = 89 - 66

x = 23⁰

The value of arc angle z is calculated as follows;

89 + 132 + 23 + z = 360 (sum of angles of a circle)

244 + z = 360

z = 360 - 244

z = 116⁰

The value of angle y is calculated as follows;

∠mGEF = ¹/₂(z) (intersecting chord theorem)

∠mGEF =  ¹/₂(116)

∠mGEF = 58⁰

∠mEFG = 180 - (58 + 33) = 89

y + 89 + y = 180 (sum of angles on a straight line)

2y + 89 = 180

2y = 91

y = 91/2

y = 45.5⁰

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The measure of angle x is  23⁰.

The value of arc angle y is 45.5⁰.

The value of arc angle z is 116⁰.

The measure of angle x, y and z is calculated as follows;

∠mEGF = ¹/₂ (arc EF - arc DC ) (intersecting chord theorem)

33 = ¹/₂ (89 - x )

2(33) = 89 - x

66 = 89 - x

x = 89 - 66

x = 23⁰

The value of arc angle z is calculated as follows;

89 + 132 + 23 + z = 360 (sum of angles of a circle)

244 + z = 360

z = 360 - 244

z = 116⁰

The value of angle y is calculated as follows;

∠mGEF = ¹/₂(z) (intersecting chord theorem)

∠mGEF =  ¹/₂(116)

∠mGEF = 58⁰

∠mEFG = 180 - (58 + 33) = 89

y + 89 + y = 180 (sum of angles on a straight line)

2y + 89 = 180

2y = 91

y = 91/2

y = 45.5⁰

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Frank used a total of 2 14/15 gallons of gas on Saturday and Sunday combined.

To find the total amount of gas Frank used on both days, we need to add the amount of gas he used on Saturday and Sunday.

First, we need to make sure that both quantities are in the same form. We can convert the mixed number 2 1/3 to an improper fraction by multiplying the whole number by the denominator and adding the numerator. This gives us:

2 1/3 = (2 x 3 + 1)/3 = 7/3

Next, we can add the two quantities by finding a common denominator. The lowest common denominator for 3 and 5 is 15. So, we can write:

7/3 gallons + 3 2/5 gallons = (35/15 + 9/15) gallons

= 44/15 gallons

To simplify the answer, we can express the mixed number as a whole number and a proper fraction by dividing the numerator by the denominator. This gives us:

44/15 = 2 14/15

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Answer: 4/21

Step-by-step explanation: 1 1/3 =4/3 3 1/2x2= 7

If glass contains 320 cm³ of milk, the mass of the milk is 330 g, the density of the milk is approximately 1031 kg/m³.

Density is defined as the mass per unit volume of a substance. To find the density of milk in kg/m³, we need to convert the given volume and mass to the appropriate units.

First, we need to convert the volume from cm³ to m³. Since 1 m = 100 cm, 1 m³ = (100 cm)³ = 1,000,000 cm³. Therefore, we can convert 320 cm³ to m³ by dividing by 1,000,000:

320 cm³ ÷ 1,000,000 = 0.00032 m³

Next, we need to convert the mass from grams to kilograms. Since 1 kg = 1000 g, we can convert 330 g to kg by dividing by 1000:

330 g ÷ 1000 = 0.33 kg

Now that we have both the mass and volume in appropriate units, we can calculate the density by dividing the mass by the volume:

Density = mass ÷ volume = 0.33 kg ÷ 0.00032 m³ ≈ 1031 kg/m³

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If Limₙ→∞, aₙ ≠ 0 or does not exist, then ∑^∞_{n= 1} (aₙ) diverges.

option C.

The solution of the limit function is calculated as follows;

Limₙ→∞, aₙ ≠ 0 or does not exist, then ∑^∞_{n= 1} (aₙ),

So in this type of function,  we will conclude that the function diverges.

This is known as the divergence test, which states that if the limit of the nth term of a series does not approach zero, then the series diverges.

In other words, we can say that if the terms of the series do not approach zero, the series will not converge.

Thus, based on the given option, the correct answer is diverges.

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Answer:

Yes, point A on the graph represents a pair of possible values of m and w because it satisfies the equation w = 2.5m + 5.

To confirm this, we can substitute m = 20 into the equation to get:

w = 2.5(20) + 5

w = 50 + 5

w = 55

So the coordinates of point A are (20, 55), which represents a possible value of m and w that satisfies the equation.

The perimeter of the square garden is 36 feet, which means that each side of the square is 36/4 = 9 feet long. Therefore, the total cost of the fences is $92.

The length of the outer fence, which is 2 feet away from the garden, is equal to the perimeter of a larger square with sides that are 2 feet longer than the sides of the original garden.

So, the length of the outer fence is 4(9+2) = 44 feet.

The length of the inner fence, which is around the perimeter of the garden, is equal to the perimeter of the original square garden, which is 4(9) = 36 feet.

The total length of both fences is the sum of the lengths of the inner and outer fences:

36 + 44 = 80 feet.

The cost of the fences is the total length of both fences multiplied by the cost per foot:

80 x $1.15 = $92.

Therefore, the total cost of the fences is $92.

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If perimeter of rectangle is denoted as "18x+6" and width is denoted as "2x+5", then the expression for length of the rectangle will be .

The "Perimeter" of a rectangle is known as the "total-distance" around the outside of rectangle.

Let  length of rectangle be denoted by "L". The formula for the "peri-meter" for rectangle is : P = 2(length + width),

The "peri-meter" is = "18x + 6", and "width" is "2x + 5".

Substituting these values into the formula,

We get,

⇒ 18x + 6 = 2(L + 2x + 5),

⇒ 18x + 6 = 2L + 4x + 10,

⇒ 18x - 4x - 10 - 6 = 2L,

⇒ 14x - 16 = 2L,

⇒ 7x - 8 = L,

Therefore, the expression for the length of the rectangle is "7x - 8".

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Answer:

x < 0, x > 2

Step-by-step explanation:

|1-x| > 1

(x-1)^2 > 1

x^2 -2x + 1 > 1

x^2 - 2x > 0

x(x-2) > 0

Using the graph of y=x(x-2), x(x-2)> 0 when

x < 0, x > 2

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Any value between 5.3 cm and 11.7 cm (exclusive) could be the length of the third side of the triangle which is 6.7cm in the given options.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, for a triangle with sides a, b, and c, where c is the longest side, the following inequality holds:

a + b > c

This means that the length of any side of a triangle must be less than the sum of the lengths of the other two sides. This theorem is fundamental in geometry and is used in many proofs and applications.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, the length of the third side must be greater than the difference between the other two sides and less than their sum.

Let's apply this rule to the given triangle. We have two sides of length 3.2 cm and 8.5 cm.

To be a valid triangle, the length of the third side must satisfy the following inequality:

8.5 cm - 3.2 cm < third side < 8.5 cm + 3.2 cm

5.3 cm < third side < 11.7 cm

Therefore, any value between 5.3 cm and 11.7 cm (exclusive) could be the length of the third side of the triangle. So it's 6.7cm.

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Answer:

-4.01, -4, -7,

Step-by-step explanation:

It's as simple as substituting and solving. However, this one isnt so simple. The problem tries to trick you because each of the numbers that is provided is supposed to be substituted for x. For example, one of the numbers is 4, which means the equation would be -(4)+6[tex]\geq[/tex]10. But in this problem, the 4 would become a negative four due to the negative sign next to the x. In contrast, the negative numbers in the list provided would then become positive because negative times a negative is positive.

The average rate of change of the function f(x) = x² on the interval [3, 3 + h] is given as follows:

h + 6.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function.

The function in this problem is given as follows:

f(x) = x².

Then the outputs are obtained as follows:

The change in the output is given as follows:

h² + 6h + 9 - 9 = h² + 6h.

The change in the input is given as follows:

3 + h - 3 = h.

Hence the rate is:

r = (h² + 6h)/h.

r = h(h + 6)/h

r = h + 6.

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The number of scoops of seeds needed to fill three bird feeders is 13 1/7 scoops of seeds .

To calculate the total seed quantity required to fill all three feeders, begin by computing the volume for the first two dispensers: each can hold 5 and five-twelfths cups of seeds.

Next, this text assumes an understanding that one scoop of seeds holds approximately 1 and one-sixth cups of product. Using this information, we determine how many scoops are needed to satisfy all three bird feeders' capacity.

By multiplying the first fraction's value with the reciprocal of the second fraction helps you divide fractions accurately:

(46 / 3) x (6 / 7) = (46 x 6) / (3 x 7) = 276 / 21

276 / 21 = 13 3/21

13 3/21 = 13 1/7 scoops

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The account's effective annual yield if you invest $1800 and it pays 6.3% interest compounded daily, is calculated as: 6.53%.

In order to find the effective annual yield of an account, the formula to apply is expressed as:

Effective annual yield = (1 + r/n)^n - 1

where:

r represents the annual interest rate (as a decimal)

n represents the number of times the interest is compounded per year

In this case, we are given the following:

Annual interest rate = 6.3% = 0.063

The interest is compounded daily, so n = 365 (days in a year)

Plugging in the variables we have:

Effective annual yield = (1 + 0.063/365)^365 - 1

Effective annual yield = 0.0653 or 6.53%

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A. 0.3015; fail to reject the null hypothesis

The test statistic is z = 0.52 and p-value (0.3015) is greater than the significance level (0.05)

Given data ,

Under the premise that the null hypothesis is true, the p-value is the likelihood of witnessing a test statistic that is as severe as the one computed or even more extreme. The p-value of a right-tailed test is the likelihood of seeing a test statistic that is higher than or equal to the computed test statistic.

We can compare the test statistic to the critical value or use a conventional normal distribution table to calculate the p-value given that the test statistic is z = 0.52 and the significance threshold is 0.05

Looking up the p-value corresponding to z = 0.52 in a standard normal distribution table or using a calculator, we find that the p-value is approximately 0.3015.

Since the p-value (0.3015) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Hence , 0.3015; fail to reject the null hypothesis

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It will take 14.14 seconds for the object to return to the ground.

We know that the height is modeled by the quadratic equation:

h(t)=-5t^2+120t+17

The object will return to the ground when its height is zero, so we only need to solve the quadratic equation:

0 = -5t^2+120t+17

Using the quadratic formula we will get.

[tex]t = \frac{-120 \pm \sqrt{120^2 - 4*-5*17} }{2*-5} \\\\t = \frac{-120 \pm 121.4 }{-10}[/tex]

We only care for the positive solution:

t = (-120 - 121.4)/-10 = 14.14

It will take 14.14 seconds.

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Complete question:

"The height of an object is launched into the air given by the function h(t)=-5t^2+120t+17 where t is the time in seconds. How long takes for the object to hit the ground?"

The value of the thickness of each record if the volume of the carton is 100π is,

⇒ 2.78 inches

We have to given that;

Karen can fit 25 vinyl records of diameter 12 in. inside a cylindrical carton with the same diameter.

We know that;

Volume of cylinder is,

⇒ V = πr²h

Hence, We get;

100π = π × (12/2)² × h

100 = 36h

h = 100 / 36

h = 50/18

h = 2.78 inches

Thus, The value of the thickness of each record if the volume of the carton is 100π is,

⇒ 2.78 inches

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