Advanced equation solving written problem one
Solve the equation on the interval [0,2π), showing all steps of the solution process. While you are welcome to check with a solver, no credit will be given for magic answers! If it is possible to obtain an exact value solution, you must give in that form. Otherwise, use decimal radians rounded to two places for the angles. Clearly indicate reference angles and quadrants. After solving, produce a Desmos graph showing the left and right sides of the equation graphed as functions, restricted to [0,2π), and click to reveal points of intersection. Screenshot and include. Solve: 2 sin^2 x + 20 cos x = 6

Answer:

12

Step-by-step explanation:

The required inequality expression is 2 + w ≥ 360.

In mathematics, an inequality is a statement that compares two quantities or expressions using an inequality symbol such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

An inequality states that one quantity is either less than or greater than the other, but not necessarily equal to it.

Here we have

You get 100 points for playing the game. In addition, you get 50 points for each seven-letter word you make with the ten letters you receive.

Sal wants to break the record, and he needs 18,000 or more points.

Let's assume Sal does 'w' number of seven-letter words

Then total points that Sal will gain = 100 + 50(w)

To break the record Sal wants 18000 points or more

=> 100 + 50w ≥ 18000    

=> 2 + w ≥ 360

Therefore,

The required inequality expression is 2 + w ≥ 360.

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The number 240 is a composite number and can be written as the product of prime factors 2 x 2 x 2 x 2 x 3 x 5.

The number 240 is a composite number. This is because it can be divided by numbers other than 1 and itself.

To write 240 as the product of prime factors, we can use the factor tree method:

240
/  \
2   120
  /  \
 2    60
     /  \
    2    30
        /  \
       2    15
           /  \
          3    5

So, the prime factors of 240 are 2, 2, 2, 2, 3, and 5.

We can write this as:

240 = 2 x 2 x 2 x 2 x 3 x 5

Therefore, the number 240 is a composite number and can be written as the product of prime factors 2 x 2 x 2 x 2 x 3 x 5.

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Step-by-step explanation:

this represents a trigonometric triangle (right-angled) inside an imaginary circle.

imagine the triangle is turned 90° counterclockwise, so that A is the bottom left vertex, B is the bottom right vertex.

then x is the angle (40°) defining radius of the circumscribing circle.

remember, sine is the up/down leg, cosine is the left/right leg. and in a larger circle they are all multiplied by the radius.

so,

4 = cos(40) × x

x = 4 / cos(40) = 5.221629157... ≈ 5.22

The integers are 16.8 and 26.8. To find the integers, we need to use a system of equations. Let's call the first integer x and the second integer y. We know that one integer is 10 more than another, so we can write the first equation as: x = y + 10. We also know that their product is 375, so we can write the second equation as: xy = 375.

Now we can substitute the first equation into the second equation to solve for one of the integers.

y(y + 10) = 375

y^2 + 10y - 375 = 0

Using the quadratic formula, we can find the value of y:

y = (-10 ± √(10^2 - 4(1)(-375)))/2(1)

y = (-10 ± √1900)/2

y = (-10 ± 43.6)/2

y = 16.8 or y = -26.8

Since y has to be an integer, we can only use the value of 16.8.

So, y = 16.8 and x = 16.8 + 10 = 26.8.

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Subtracting -15/18 from -12/18, the answer is -3/18 in simplest form.

Subtracting two fractions with different denominators involves finding a common denominator and then subtracting the fractions.

The common denominator for (-5)/(6) and (-2)/(3) is 6*3=18.

To make the fractions have the same denominator, the numerators need to be multiplied by the denominators and then divided by the denominator.

(-5)/(6) can be rewritten as -(5*3)/(6*3)=-15/18
(-2)/(3) can be rewritten as -(2*6)/(3*6)=-12/18

Subtracting -15/18 from -12/18, the answer is -3/18 in simplest form.

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The percentage of races with a finishing time between 64.5 and 65.5 seconds is given as follows:

68%.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

The measures in this problem are within one standard deviation of the mean, as:

Hence the percentage is of 68%.

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It is simply the largest of the common factors.

We have:

12x³- 18x² + 6x

We find GCF of 12, 18 and 6:

GCF(12, 18, 6) = 6

12 = 2 · 6

18 = 3 · 6

6 = 1 · 6

and GCF of x³, x² and x:

GCF(x³, x², x) = x

x³ = x² · x

x² = x · x

x = 1 · x

Therefore:

Pablo's monthly payments are $760.76

If Pablo pays the monthly fee each month, he will pay a 3.48% interest rate.

Monthly payments refer to the amount of money that is paid on a regular basis, typically every month, to repay a debt or loan over a specified period of time. The monthly payment is usually calculated based on the total amount borrowed, the interest rate, and the repayment period

How to solve

Given that:

Loan amount = $1400

Interest rate r = 5.7%

Time n = 5 years

To solve for monthly payment

using the formula

1400 * [tex]\frac{0.0157(1 +0.0517)^6^0}{(1 + 0.0517)^6^0 -1}[/tex]

= 760.76.

Pablo's monthly payments is $760.76

The total interest to be paid:

A = P(1+rt)

1400= 760.76 + 3803.8

= 3.48

Thus, if Pablo pays the monthly fee each month, he will pay 3.48% interest rate.

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Answer:

19/3

Step-by-step explanation:

Answer:

The slope of the given line on the graph is 1/3.

Answer:

1/3

Step-by-step explanation:

The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

Let's start at 0 with x = 0 and y = 0 or (0,0)

You see the graph crosses coordinate (3,1)

so y goes from 0 to 1 => rise = 1

x goes from 0 to 3 => run = 3

then rise/run = 1/3

We can use the notation definition to show that Π does not perform too differently between two environments vj and v0 by comparing the performance of Π in both environments and calculating the difference between them.

In order to show that Π does not perform too differently between two environments vj and v0, we can use the notation definition to compare the performance of Π in both environments.

First, we can use the notation Π(vj) to represent the performance of Π in environment vj and Π(v0) to represent the performance of Π in environment v0.

Next, we can compare the two performances by calculating the difference between them using the notation |Π(vj) - Π(v0)|.

If the difference is small, then we can say that Π does not perform too differently between the two environments. However, if the difference is large, then we can say that Π performs differently in the two environments.

In conclusion, we can use the notation definition to show that Π does not perform too differently between two environments vj and v0 by comparing the performance of Π in both environments and calculating the difference between them.

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Using the given percentage we can see that there are 78 filled pages.

We know that there are 800 cards and each page holds 8 cards, then the number of pages is:

N = 800/8 = 100

And we know that 22% of the album is empty, then 78% (the complement) of the album is filled, this means that the number of filled pages is given by the formula below.

F = 100*(78%/100%) = 100*0.78 = 78

We divide the percentage by 100% to make it a decimal number, so we can use it in the operation.

There are 78 filled pages in the album.

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The exact value of Sin^-1 (cos π) can be found without using a calculator by using the properties of the unit circle and the trigonometric functions.

First, we need to find the value of cos π. On the unit circle, the point (1,0) corresponds to an angle of 0 radians or 0°. The point (-1,0) corresponds to an angle of π radians or 180°. Therefore, cos π = -1.

Next, we need to find the inverse sine of -1. The inverse sine function, Sin^-1, is the inverse of the sine function. This means that Sin^-1(sin x) = x. Therefore, we need to find an angle x such that sin x = -1.

On the unit circle, the point (0,-1) corresponds to an angle of 3π/2 radians or 270°. Therefore, sin 3π/2 = -1. This means that Sin^-1(-1) = 3π/2.

Therefore, the exact value of Sin^-1 (cos π) is 3π/2.

Answer: 3π/2.

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To simplify 7-5/6•7-7/6, we need to follow the order of operations (PEMDAS) and perform the multiplication and division before addition and subtraction.

7-5/6•7-7/6

= 7 - (5/6) * 7 - (7/6) (Multiplication first)

= 7 - (35/6) - (7/6) (Simplify the multiplication)

= (42/6) - (35/6) - (7/6) (Convert 7 to a fraction with a common denominator)

= (42 - 35 - 7) / 6 (Subtract the numerators)

= 0 / 6

= 0

Therefore, 7-5/6•7-7/6 simplifies to 0.

Answer: 0

Step-by-step explanation:

7 - 5 / 6 • 7 - 7 / 6

7 - 35 / 6 - 7 / 6

7 - 42 / 6

7 - 7

0

Answer:

half

Step-by-step explanation:

The multiplicative property of equality states that the same number can be added to or multiplied by both sides of an equation to obtain an equivalent equation. In this case, dividing both sides by 6 gives us u = 78/6 = 13.

The multiplicative property of equality states that if two numbers are equal, then multiplying both sides of the equation by the same number will also result in an equation that is still equal. In other words, if a=b, then ac=bc. We can use this property to solve for the variable u in the equation 78=6u.

To isolate the variable on one side of the equation, we can divide both sides by 6. This will give us:

78/6 = 6u/6

Simplifying the equation gives us:

13 = u

So the solution for u is 13.

In conclusion, the multiplicative property of equality with whole numbers was used to solve for the variable u in the equation 78=6u. The solution is u=13.

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The vertex of the parabola is (5, 4). The solution has been obtained by using quadratic formula.

What is quadratic formula?

The quadratic formula is used to determine an equation's roots. This formula assists in evaluating the solutions of the quadratic equations instead of the factorization approach.

We are given equation as  y = -x² + 10x - 21.

Using quadratic formula, we know that x = -b/2a

So,

x = -10/-2

x = 5

On substituting this in the equation, we get

y = -5² + 10(5) - 21

y = -25 + 50 - 21

y = 4

Hence, the vertex of the parabola is (5, 4).

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Answer:

I believe the answer you are looking for is [tex]6b^{4}[/tex] -  [tex]8b^{2}[/tex]

Step-by-step explanation:

Hope it helps

Sorry if I'm wrong

Rounded to the nearest hundredth, the relative frequency of a student being a twelfth grader that gets their news from the internet is 0.25.

The proportion of times a particular value or event occurs within a set of data is termed as Relative frequency.

It is calculated by dividing the frequency of the event by the total number of observations. It is often expressed as a percentage or decimal value between 0 and 1.

The relative frequency of a student being a twelfth grader that gets their news from the internet can be calculated by dividing the number of twelfth graders who get their news from the internet by the total number of students surveyed:

Relative frequency = (number of twelfth graders who get news from the internet) / (total number of students surveyed)

Relative frequency = 43 / 175

Relative frequency = 0.2457 (rounded to four decimal places)

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Answer:

Area = 15.6

Perimeter = 20.6

Step-by-step explanation:

Both smaller triangles are right triangles

The left triangle is isosceles right triangle since 2 angles are 45o so AB is hypotenuse and the other 2 sides are equal

so c^2 = a^2 + b^2

c^2 = 2a^2

(3√2)^2 = 2a^2

2a^2 = 18

a^2 = 9

a = 3

For the right triangle the angle at B is 60o, hypotenuse is BC

so cos(60) = adjacent/hypotenuse

1/2 = 3/BC

=> BC = 6

sin(60) = opposite/hypotenuse

√3/2 = opp/6

opp = 3√3

so AC = 3 + 3√3 = 6√3

Area of triangle = A = 1/2bh = 1/2(6√3)(3) = 9√3 = 15.6

Perimeter = 3√2 + 6√3 + 6 = 20.6

This is the vertex form equation of the parabola with the given information.

From the information provided, we can write the vertex form equation of the parabola as follows:

First, we need to determine the value of "a" in the equation y = a(x - h)^2 + k, where (h,k) is the vertex and "a" determines the width and direction of the parabola.

The distance between the vertex and the focus is equal to 1/4a, so we can use this to find "a":

1/4a = (41/4 - 10)

1/4a = 1/4

a = 1

Now we can plug in the values for the vertex and "a" into the equation:

y = 1(x - 10)^2 - 8

This is the vertex form equation of the parabola with the given information.

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The step that Jackson made an error when solving the inequality was Step 1 because he did not multiply - 10 by 4.

When solving the inequality, Jackson had the right idea of multiplying the different sides of the equation by 4 in order to get rid of 4 as a denominator.

However, he was to multiply every figure in the equation by 4. He did not do this with the - 10 on the left side of the equation. This meant that Jackson did not convert that number to - 40 which then led to an incorrect inequality.

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From the graph of the function we determine, [tex]y = 1.(2^{\frac{1}{3}})^x[/tex].

A function is said to be an exponential function when the exponent is the independent variable.

Let, The given function be f(x) = abˣ.

Now, At x = 0,

1 = ab⁰.

a = 1.

Again at,

x = 3,

2 = a.b³.

b³ = 2.

b = [tex]2^{\frac{1}{3}[/tex].

Therefore, The given function is, [tex]y = 1.(2^{\frac{1}{3}})^x[/tex].

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The length of side 'g' is approximately 61.54 inches.

A triangle is a 2-dimensional geometric shape that consists of three line segments or sides and three angles. It is one of the simplest polygonal shapes and is often used in mathematics and geometry to illustrate basic concepts such as area, perimeter, angles, and congruence.

To find the length of side 'g' of ΔFGH, we can use the Law of Sines, which relates the lengths of the sides of a triangle to the sine of their opposite angles. The Law of Sines states that:

a/sin(A) = b/sin(B) = c/sin(C)

Here the lengths of the sides of a ΔABC are a, b, and c, and A, B, and C are the opposite angles.

In this case, we know that the length of side 'f' is 67 inches, and the measures of angles, G and H are 66° and 30° respectively. Let's assume the length of ∠G side is 'g'. Then, using the Sin Law, we can write:

f/sin(F) = g/sin(G) = h/sin(H)

Substituting the given values, we get:

67/sin(F) = g/sin(66) = h/sin(30)

Solving for 'g', we have:

67/sin(F) = g/sin(66°)

g = 67 * {sin(66°) / sin(F)}

To find the measure of angle F, we can use the fact that the sum of the angles in a triangle is 180 degrees. Therefore:

F = 180° - G - H

F = 180 - 66° - 30°

F = 84°

Substituting this value into the equation for 'g', we get:

g = 67 * sin(66°) / sin(84°)

putting the values from Sin table: Sin(66°) = 0.913 and Sin(84°) = 0.994

g = 61.54 inches

Therefore, the length of side 'g' is approximately 61.54 inches.

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Yes, the production manager should be concerned about production rate variations. Based on the data provided, the standard deviation of the daily production rate of fiberglass boats from Dhaka Glass is 4.6 boats, which is higher than the threshold of 3 boats a day set by the production manager. This indicates that there are significant variations in the production rate of fiberglass boats, which should be a cause for concern.

To further explain, the standard deviation measures the spread of a set of data. In this case, the data provided shows that the daily production rate of fiberglass boats can vary from 17 to 32 boats a day. The large standard deviation of 4.6 boats suggests that the daily production rate can vary significantly from the average, and this can lead to instability in production.

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Answer: I think B & C not super sure though

The value of x is 23°

A line is a straight, one-dimensional figure that extends infinitely in both directions. It is often represented graphically as a straight line on a coordinate plane, with an equation that describes its position.

A line can be defined by any two points on it, and every point on the line can be expressed as a linear combination of those two points.

Line j and k are parallel and another line crossing it means transversal line, then the corresponding angles are equal which is,

(5x+9)° = (4x+32)°

5x-4x = 32-9

x = 23°

Therefore, the value of x is 23°

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One possible way to achieve this task is by using the following method:
1. Toss the biased coin twice.
2. If the result is two heads or two tails, discard the result and toss the coin again.
3. If the result is one head and one tail, record the result as a 1 if the first toss was a head and a 0 if the first toss was a tail.

This algorithm works because the probability of getting heads followed by tails or tails followed by heads is equal to 2p(1-p), which is symmetric around p=1/2. Therefore, the probability of outputting 1 is equal to the probability of outputting 0, which is 1/2. This method of generating a random process with a desired probability distribution from another random process is known as rejection sampling.

This method will produce a random process with outputs 1's and 0's with probability 1/2, regardless of the bias of the coin. This is because the probability of getting one head and one tail in two tosses is the same for both biased and unbiased coins, and we are only recording the result when we get one head and one tail. Therefore, this method will produce an unbiased random process from a biased coin.

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The convention centre has a volume of approximately 30,085,616.8 cubic metres as a result.

As opposed to length, width, and height for a rectangle's surface, the basic formula for volumes is length, breadth, and height. It doesn't matter how you refer to the different dimensions; for example, using "depth" instead of "height" wouldn't affect the calculation.

V = base area ×height (1/3)

The pyramid's height is disclosed to be 332 meters. We multiply the base's length and width to determine the base area:

Base area = 475 m × 571 m = 271525 m²

We can now enter the values into to the formula as follows:

V = (1/3) × 271525 m² × 332 m

V = 30085616.7 m³

This is rounding up to the closest tenth, giving us:

V ≈ 30085616.7 ≈ 30085616.8 m³

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