Trigonometry review applications

I need help with 15-18

Answer: To complete the square, we need to add and subtract the square of half the coefficient of x:

x^2 - 4x = 2

x^2 - 4x + 4 = 2 + 4 (adding and subtracting 4)

(x - 2)^2 = 6

Now we take the square root of both sides:

x - 2 = ±√6

Adding 2 to both sides gives:

x = 2 ± √6

So the solutions are:

x = 2 + √6, 2 - √6

Step-by-step explanation:

Answer:

The linear speed of the vehicle is approximately 0.798 miles per hour.

Step-by-step explanation:

The circumference of a tire is given by the formula C = πd, where d is the diameter of the tire. So, the circumference of each tire is:

C = πd = π(32 inches) ≈ 100.53 inches

Now, to find the distance that the vehicle travels in one revolution of each tire, we use the circumference formula again:

distance traveled in one revolution = circumference of the tire = 100.53 inches

Since the tires are rotating at a rate of 800 revolutions per minute, the vehicle travels a distance of:

distance traveled in one minute = distance traveled in one revolution × number of revolutions per minute

= 100.53 inches/revolution × 800 revolutions/minute

= 80424 inches/minute

To convert this to miles per hour, we need to divide by the number of inches per mile and the number of minutes per hour:

speed = (80424 inches/minute) × (1 mile/63360 inches) × (60 minutes/1 hour)

≈ 0.798 miles/hour

Therefore, the linear speed of the vehicle is approximately 0.798 miles per hour.

Selecting the first tile (1st) and selecting the secοnd tile (2nd) are independent events.

The prοbability οf selecting an even number οn a single draw is 2/5 οr 0.4, as there are twο even numbers (2 and 4) οut οf a tοtal οf five pοssible numbers.

Since the events are independent, the prοbability οf selecting an even number οn bοth draws is the prοduct οf the prοbabilities οf selecting an even number οn each draw:

P(bοth even) = P(even οn first) × P(even οn secοnd) = 0.4 × 0.4 = 0.16

Sο the prοbability that the numbers οn bοth tiles are even is 16%.

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Answer:

1 The total distance Jane travels biking and running is the hypotenuse of a right triangle with legs 16 miles and 12 miles. Using the Pythagorean theorem, we can find this distance:

distance = sqrt(16^2 + 12^2) = sqrt(256 + 144) = sqrt(400) = 20 miles

So Jane travels a total of 20 miles.

2 The distance Jane's trainer bikes is the distance from the health club to the point where he meets Jane. Let's call this distance x. We can represent this with the following right triangle:

css

Copy code

        A (health club)

        |

        |

        | x

        |

        |

B (meeting point)

where A and B are points on a coordinate plane, and the x-axis represents the direction Jane runs.

Using the Pythagorean theorem, we can write:

x^2 + 12^2 = d^2

where d is the distance from the health club to the end of Jane's run. We know that d = 20, so we can substitute:

x^2 + 12^2 = 20^2

3. Solving for x:

x^2 + 144 = 400

x^2 = 256

x = sqrt(256) = 16

So Jane's trainer bikes 16 miles.

4 Jane travels 20 - 16 = 4 miles farther than her trainer.

Step-by-step explanation:

The value of x for measure a base angle for the isosceles trapezoid is equal to 25°.

This is a trapezoid in which the base angles are equal and therefore the left and right side lengths are also equal. The opposite angles are supplementary which implies they sum up to 180°.

angle C is equal to angle D so;

3x + 1 = 76°

3x = 76° - 1 {subtract 1 from both sides}

3x = 75°

x = 75°/3 {divide through by 3}

x = 25°

Therefore, the value of x for measure a base angle for the isosceles trapezoid is equal to 25°.

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Despite their confidence in their thinking, beliefs, and actions, European countries in power during this time frame were often wrong in their assumptions about the superiority of their culture and the legitimacy of their actions towards other nations and peoples.

European countries in power during this time frame, particularly in the 19th and early 20th centuries, believed in the concept of imperialism and the "civilizing mission" of spreading their culture and values to other parts of the world. This often led to the exploitation and oppression of colonized peoples, as well as the imposition of Western values on non-Western cultures.

However, these assumptions about the superiority of Western culture were flawed and based on a Eurocentric view of the world. Additionally, many European powers engaged in brutal and violent actions, such as the forced removal of Indigenous peoples from their lands, the use of slavery and forced labor, and the exploitation of natural resources.

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The diameter of the hemisphere to the nearest tenth of a centimeter is 62.1 cm.

What is hemisphere?

A hemisphere is a three-dimensional geometric shape that consists of a half-sphere, which is a curved surface that is shaped like the surface of a ball, and a flat circular base that lies on a plane perpendicular to the sphere's axis of symmetry. A hemisphere can be thought of as the three-dimensional analogue of a circle.

The volume of a hemisphere is given by the formula:

V = (2/3)π[tex]r^3[/tex]

where V is the volume and r is the radius of the hemisphere.

We are given the volume of the hemisphere as 62617 [tex]cm^3.[/tex]

So, we have:

62617 = (2/3)π[tex]r^3[/tex]

Multiplying both sides by 3/2π, we get:

[tex]r^3[/tex] = 62617 * (3/2π)

Simplifying, we get:

[tex]r^3[/tex] = 29897.4152

Taking the cube root of both sides, we get:

r ≈ 31.04

The diameter of the hemisphere is twice the radius, so:

d = 2r ≈ 62.08

Rounding to the nearest tenth, we get:

d ≈ 62.1

Therefore, the diameter of the hemisphere to the nearest tenth of a centimeter is 62.1 cm.

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Answer: We can use the double angle identity for tangent, which states that:

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

to simplify the expression.

Let θ = 6 degrees, then we have:

tan(2θ) = tan(12 degrees)

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

tan(12 degrees) = 2tan(6 degrees) / (1 - tan²(6 degrees))

We can use the tangent half-angle identity to find tan(6 degrees), which states that:

tan(θ/2) = sin(θ) / (1 + cos(θ))

Letting θ = 12 degrees, we get:

tan(6 degrees) = sin(12 degrees) / (1 + cos(12 degrees))

We can then use the double angle identity for sine, which states that:

sin(2θ) = 2sin(θ)cos(θ)

to simplify sin(12 degrees). Letting θ = 6 degrees, we get:

sin(12 degrees) = 2sin(6 degrees)cos(6 degrees)

We can use the half-angle identity for cosine to find cos(6 degrees), which states that:

cos(θ/2) = √((1 + cos(θ)) / 2)

Letting θ = 12 degrees, we get:

cos(6 degrees) = √((1 + cos(12 degrees)) / 2)

Substituting these values into the original expression, we get:

tan(12 degrees) = 2tan(6 degrees) / (1 - tan²(6 degrees))

tan(12 degrees) = 2(sin(12 degrees) / (1 + cos(12 degrees))) / (1 - (sin²(12 degrees) / (1 + cos(12 degrees))²))

tan(12 degrees) = 2(2sin(6 degrees)cos(6 degrees) / (1 + cos(12 degrees))) / (1 - (4sin²(6 degrees)cos²(6 degrees) / (1 + cos(12 degrees))²))

tan(12 degrees) = (4sin(6 degrees)cos(6 degrees)) / (1 + cos(12 degrees) - 4sin²(6 degrees)cos²(6 degrees))

This is the simplified expression using the double angle identity.

Step-by-step explanation:

Answer:

1. 15

2. 26

3. ≈ 7.62

4. 8

5. ≈ 23.24

6. √2

7. ≈ 19.42

8. 18

9. x≈10.29 y≈5.83

Step-by-step explanation:

Use a²+b²=c² throughout the problems and plug in the given sides and hypothenuse.

1. 9²+12²=c²

81+144=225

√225 = 15

2. 10²+24²=c²

100+576=676

√676=26

3. 7²+3²=c²

49+9=58

√58 ≈ 7.62

4. a²+6²=10²

a²+36=100

a²=64

x=8

5. a²+6²=24²

a²+36=576

a²=540

x≈ 23.24

6. 1²+1²=c²

1+1=2

x=√2

7. a²+8²=21²

a²+64=441

a²=377

x≈ 19.42

8. a²+24²=30²

a²+576=900

a²=324

x=18

9. for x:

9²+5²=x²

81+25=106

√106 ≈10.29

x ≈10.29

for y:

5²+3²=y²

25+9=34

√34 ≈5.83

y ≈5.83

The value of y for the inverse variation when x = -12 is derived to be equal to 1.

Inverse variation is a mathematical relationship between two variables, in which an increase in one variable leads to a proportional decrease in the other variable. Mathematically, inverse variation can be expressed as y = k/x, where y and x are the two variables, k is a constant of proportionality, and the product of y and x is always equal to k.

when x = 6 and y = -2, then k is derived as:

-2 = k/6

k = -12 {cross multiplication}

when x = -12, y is derived as:

y = -12/-12

y = 1

Therefore, the value of y for the inverse variation when x = -12 is derived to be equal to 1.

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The required expression equal to [tex]$\csc\left(\frac{4\pi}{9}\right)$[/tex] using a cofunction is: [tex]\begin{equation}\frac{1}{\cos\left(\frac{5\pi}{18}\right)}\end{equation}[/tex]

Trigonometry is the study of angles and the angular relationships between planar and three-dimensional figures. The cosecant, cosine, cotangent, secant, sine, and tangent are the trigonometric functions—also known as the circular functions—that make up trigonometry.

The cofunction of sine is cosine, which means:

[tex]$\begin{equation}\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\cos\left(\frac{\pi}{2}-\theta\right)}\end{equation}[/tex]

Therefore, we can express [tex]$\csc\left(\frac{4\pi}{9}\right)$[/tex] as:

[tex]$\begin{equation}\csc\left(\frac{4\pi}{9}\right) = \frac{1}{\sin\left(\frac{4\pi}{9}\right)} = \frac{1}{\cos\left(\frac{\pi}{2}-\frac{4\pi}{9}\right)}\end{equation}[/tex]

Simplifying the argument of the cosine function:

[tex]$\begin{equation}\frac{\pi}{2}-\frac{4\pi}{9} = \frac{5\pi}{18}\end{equation}[/tex]

Substituting this back into the equation, we get:

[tex]$\begin{equation}\csc\left(\frac{4\pi}{9}\right) = \frac{1}{\cos\left(\frac{5\pi}{18}\right)}\end{equation}[/tex]

Therefore, an expression equal to [tex]$\csc\left(\frac{4\pi}{9}\right)$[/tex] using a cofunction is:

[tex]\begin{equation}\frac{1}{\cos\left(\frac{5\pi}{18}\right)}\end{equation}[/tex]

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Answer:

  [A]  Multiply the bottom equation by -3/2, then add the equations.

  [C]  Multiply the top equation by -3, then add the equations.

Step-by-step explanation:

You want to know a suitable "elimination" strategy for the two equations ...

The result of an elimination strategy is that one of the variables in the combined equations has a coefficient of 0. If we add the equations as a final step, the coefficients must be opposites for that to happen.

The ratios of variable coefficients in the top equation to those in the bottom equation are ...

A suitable elimination strategy will multiply one of the equations by a value that makes this ratio -1. Let's look at the choices.

Multiplying the bottom equation by -3/2 makes the ratio of x-coefficients be ...

  [tex]\dfrac{1}{3}\times\dfrac{1}{\left(-\dfrac{3}{2}}\right)}=-\dfrac{2}{9}\ne -1[/tex]

Multiplying the bottom equation by -3/2 makes the ratio of y-coefficients be ...

  [tex]\dfrac{3}{2}\times\dfrac{1}{\left(-\dfrac{3}{2}\right)}=-\dfrac{6}{6}=-1[/tex]

This is a suitable strategy for eliminating the y-variable.

This strategy is equivalent to multiplying the bottom equation by -3, then adding. This will make the x-coefficient ratio be ...

  [tex]\dfrac{1}{3}\times\dfrac{1}{\left(-3}\right)}=-\dfrac{1}{9}\ne -1[/tex]

Multiplying the bottom equation by -3 makes the ratio of y-coefficients be ...

  [tex]\dfrac{3}{2}\times\dfrac{1}{-3}=-\dfrac{3}{6}\ne-1[/tex]

Not a suitable strategy for elimination.

Multiplying the top equation by -3 makes the ratio of x-coefficients be ...

  [tex]\dfrac{1}{3}\times\dfrac{-3}{1}}=-\dfrac{3}{3}= -1[/tex]

This is a suitable strategy for eliminating the x-variable.

Choices A and C are suitable strategies for eliminating a variable.

When the outside temperature is 30°F, the sales are estimated to be $414.73.

The outside temperature is approximately 78.03°F, the sales are estimated to be $1,993.33.

It refers to a physical property of matter that reflects how hot or cold an object is relative to other objects or a standard scale. Temperature is typically measured in units such as Celsius (°C), Fahrenheit (°F), or Kelvin (K).

When the outside temperature is 30°F, the sales are estimated to be:

y = 32.92(30) - 572.87

y = 987.60 - 572.87

y = $414.73

Therefore, when the outside temperature is 30°F, the sales are estimated to be $414.73.

When the outside temperature is x, the sales are estimated to be $1,993.33, we can solve for x as follows:

1,993.33 = 32.92x - 572.87

2,566.20 = 32.92x

x ≈ 78.03°F

Therefore, when the outside temperature is approximately 78.03°F, the sales are estimated to be $1,993.33.

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Answer:

A.57.99 B.70.39 C 45.96

Step-by-step explanation:12.4 x 4 + 8.39 = 57.99 for Part A For B, this is 12.4x5+8.39=70.39. C is 12.4+8.39x4=45.96 12.4 + 8.39 x 5 =54.35

I'm not sure whether I should group the responses because there isn't much context, but I hope this helps.

Answer: 30 inches

Step-by-step explanation:

Length x Width x Height

Your length is 2, your height is 3 and your width is 5. So this means 2x3x5.

2 times 5 equals 10. 10 times 3 equals 30. Your answer is 30! Hope this helps you!

Answer: $939 and $689

Step-by-step explanation:

     We can create a system of equations to help us solve. Let x be one and y be the other paycheck.

           y = x - $250

           y + x = $1,628

     Next, we can solve the system of equations by graphing. The point of intersection is our solution. See attached. This gives us (939, 689), so one check was written for $939 and one was written for $689.

The future value of Ted's investment after 30 years would be approximately $13,711.86.

What is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.

Ted can use the formula for the future value of an annuity to calculate the future value of his investment:

FV = Pmt x (([tex](1 + r)^{n}[/tex]) - 1) / r

Where:

FV = Future Value

Pmt = Payment per period (monthly in this case)

r = Interest rate per period (monthly in this case)

n = Number of periods (in this case, 30 x 12 = 360 months)

Substituting the given values, we get:

FV = $250 x (([tex](1 + 0.062/12)^{360}[/tex] - 1) / (0.062/12)

FV = $250 x (283.8576) / 0.0051667

FV = $13,711.86

Therefore, the future value of Ted's investment after 30 years would be approximately $13,711.86.

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The company's total expenses and revenues will both be $196 when 4 attendees register.

What is Algebraic expression ?

In mathematics, an algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, that represents a quantity or a relationship between quantities.

Let's assume that the company's goal is to break even, which means that their total revenue should be equal to their total expenses.

Let's represent the number of attendees by "x." For the company to break even, the total revenue earned from x attendees must be equal to the total expenses incurred.

Total Expenses = Fixed Cost + Variable Cost

Fixed cost = Facility rent fee = $164

Variable cost = Books and supplies per attendee = $8

Total Expenses = $164 + $8x

Total Revenue = Price per attendee x Number of attendees = $49x

To find the break-even point, we set the total revenue equal to the total expenses and solve for x:

$49x = $164 + $8x

$41x = $164

x = 4

Therefore, the company needs to have 4 attendees to break even.

To calculate the total expenses and revenues, we can substitute x = 4 into the equations we derived above:

Total Expenses = $164 + $8(4) = $196

Total Revenue = $49(4) = $196

Therefore, the company's total expenses and revenues will both be $196 when 4 attendees register.

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Answer:

VERTEX: (8,1)

GRAPH BY DESMOS IS ATTACHED TO THIS POST

Y-INTERCEPT: (5,0)

X-INTERCEPT: NONE

Step-by-step explanation:

To find the vertex, we usually use the formula -b/2a to find the x-vertex. So, the x-vertex would be:

-(-4)/2

4/2

2

The y-vertex can be found by substituting the x-vertex back into the equation. So:

y=(2)² -4(2) + 5

y= 4-8+5

y= -4 + 5

y= 1

So, we found that our vertex is (8,1)

Because we see that our "a" value from our quadratic equation is positive, this tells us that the graph opens upwards. We know our vertex is higher than our x-axis line, so there will be no x-intercepts.

The y-intercept is found when the x-value equals to 0. So, inserting this into the equation, we get :

y=(0)² - 0 +5

y= 5

So, we have found our y-intercept as (0, 5)

The graph you could've made would've been using the vertex and the y-intercept and connecting those points for one side of the quadratic graph. The other side would require an integer point for more accuracies, and I believe that the point (3,2) also works.

So, using the points (3,2), (0,5), and (8,1), we can make a graph!

I hope this helped :P

Answer:

the diver will reach the surface of the water after 1/4 seconds.

Step-by-step explanation:

To find when the swimmer reaches the surface of the water, we need to set h(t) to zero since the surface of the water is at a height of zero feet.

0 = t - 4t^2

Simplifying the equation, we get:

4t^2 = t

4t^2 - t = 0

t(4t - 1) = 0

So, either t = 0 (which corresponds to the initial time when the swimmer is at the surface) or 4t - 1 = 0. Solving for t, we get:

4t - 1 = 0

4t = 1

t = 1/4

Therefore, the diver will reach the surface of the water after 1/4 seconds.

This information can be used to understand the performance of the students and Evaluate their scores in relation to the overall group.

Suppose that the distribution of scores of 1,000 students who take a standardized intelligence test is a normal distribution. In this scenario, the distribution's mean is 470 and its standard deviation is 10.

A normal distribution, also known as a Gaussian distribution or bell curve, is a symmetric distribution where the majority of the data falls around the mean value. In this case, the mean score is 470, which represents the average intelligence score for the 1,000 students. The standard deviation, which is 10 in this case, measures the dispersion or spread of the data around the mean.

To interpret the scores, we can use the empirical rule, which states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Applying this rule:
- About 68% of the students scored between 460 (470-10) and 480 (470+10).
- Approximately 95% of the students scored between 450 (470-2*10) and 490 (470+2*10).
- Around 99.7% of the students scored between 440 (470-3*10) and 500 (470+3*10).

This information can be used to understand the performance of the students and evaluate their scores in relation to the overall group.

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Answer:

m =  [tex]\frac{6}{5}[/tex]

Step-by-step explanation:

First, we must rewrite 5x + 6y = - 18 to the form y = mx + b.

5x + 6y = -18

6y = -5x - 18

y = [tex]\frac{-5}{ 6}[/tex]x - 3

The equation of a perpendicular line to y = [tex]\frac{-5}{ 6}[/tex]x - 3 must have a slope that is the negative reciprocal of the original slope.

So, the slope of the perpendicular line is

m = [tex]\frac{6}{5}[/tex]

Answer: 6/5

Step-by-step explanation:

trust me

Answer:

x = 47

Step-by-step explanation:

All angles in a triangle ALWAYS add up to 180 degrees, so to find a missing angle, subtract what is already given. It will be 180 - (102+31), which is the same as 180-133 which equals 47. The basic rule is to subtract given angels from 180 to find the missing side.

Answer:

The expression 4 in x + in 3 - in x can be simplified using logarithmic rules: 4 in x + in 3 - in x = in x^4 + in 3 - in x = in (x^4 * 3 / x) = in (3x^3) Therefore, the expression is equivalent to in (3x^3).

Answer:

(i) log 1 base 2 = log_{2} 1 =1.30103

Step-by-step explanation:

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The simplified expression is: √602755 = 776.372977

What do you mean by term Square root ?

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol √, which is called the radical symbol. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. The square root of a non-negative number is always a non-negative number.

Assuming

(890 / 5621) / 56 + 8987 x 67 - sqrt(x)

where x is some unknown value, we can simplify the expression as follows:

First, we can simplify the division:

(890 / 5621) / 56 = 0.000281

Then we can substitute this value into the expression and simplify the multiplication:

0.000281 + 8987 x 67 - sqrt(x) = 602755 - sqrt(x)

At this point, we cannot simplify the square root further without knowing the value of x.

Therefore, the simplified expression is:

√602755 - 776.372977

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Allison can make 1648 bookmarks with 412 feet of fabric.

First of all, the total length of fabric, and the length of each bookmark have different units.

Therefore, we convert fabric length into inches.

∵ 1 foot = 12 inches

∴ 412 feet = 412 × 12 inches

                 = 4944 inches.

Now,

length of each bookmark = 3 inches (given)

Total length of fabric in inches = 4944 inches

∴ Total number of bookmarks = 4944 ÷ 3

                                                  = 1648

Hence we find that the total number of bookmarks that Allison can make from 412 feet of fabric is 1648.

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A semicircle has been cut out of a rectangular paperboard that is 20 inches long and 12 inches broad, as seen below. After the semicircle is taken out, the paperboard's remaining perimeter is 34.58 inches.

The paperboard has a length of 20 inches and a width of 12 inches. A semicircle is cut out of it, which means we need to find the perimeter of the remaining part.

The diameter of the semicircle is equal to the width of the paperboard, which is 12 inches. So, the radius of the semicircle is half of the diameter, which is 6 inches.

The perimeter of the remaining part will be the sum of the length of the paperboard and the two straight sides of the semicircle.

The length of the paperboard is 20 inches, and the two straight sides of the semicircle are equal to the diameter of the semicircle, which is 12 inches. So, the perimeter of the remaining part is:

P = 20 + 12 + 12 = 44 inches

However, we also need to subtract the length of the curved part of the semicircle from the perimeter. The length of the semicircle can be found using the formula:

C = πr

where C is the circumference of the semicircle and r is the radius.

Since we have a semicircle, we need to divide the circumference by 2. So, the length of the curved part of the semicircle is:

C/2 = (π x 6) / 2 = 9.42 inches (rounded to two decimal places)

Therefore, the perimeter of the remaining part is:

P = 44 - 9.42 = 34.58 inches (rounded to two decimal places)

So, the perimeter of the remaining paperboard is 34.58 inches.

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