Melanie cuts a rectangle out of a piece of scrap paper as shown. She wants to calculate the angle measure of the piece that is left over. What is the measure in degrees?

Point A (1, 0) and point B (3/5, 4/5) both lie on the unit circle with center at the origin.

A circle is a geometric shape defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance between any point on the circle and the center is called the radius.

Center: The center is a fixed point in the plane from which all points on the circle are equidistant. It is denoted by the coordinates (h, k), where h represents the x-coordinate and k represents the y-coordinate of the center.

Radius: The radius is the distance between the center of the circle and any point on the circle. It is denoted by the letter "r". The radius is constant for all points on the circle.

Diameter: The diameter is a line segment passing through the center of the circle and with endpoints on the circle. It is twice the length of the radius, and it divides the circle into two equal halves.

In the xy-plane, the unit circle with center at the origin (0, 0) is defined by all the points that are at a distance of 1 unit from the origin.

Point A: (1, 0)

Point A lies on the x-axis and is 1 unit away from the origin. It represents a point on the unit circle.

Point B: (3/5, 4/5)

Point B lies on the unit circle but is not one of the standard points (such as (1, 0), (-1, 0), (0, 1), or (0, -1)). Instead, it represents a point on the unit circle with coordinates (3/5, 4/5). The x-coordinate of B is 3/5, and the y-coordinate is 4/5. The distance between point B and the origin (0, 0) is 1 unit, which satisfies the condition of being on the unit circle.

Therefore, point A (1, 0) and point B (3/5, 4/5) both lie on the unit circle with center at the origin.

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\(\begin{align}\sf\:\text{LHS} &= \cos(A)\cos(60^\circ - A)\cos(60^\circ + A) \\&= \cos(A)\cos(60^\circ)\cos(60^\circ) - \cos(A)\sin(60^\circ)\sin(60^\circ) \\&= \frac{1}{2}\cos(A)\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) - \frac{\sqrt{3}}{2}\cos(A)\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) \\&= \frac{1}{8}\cos(A) - \frac{3}{8}\cos(A) \\ &= \frac{-2}{8}\cos(A) \\ &= -\frac{1}{4}\cos(A).\end{align} \\\)

Now, let's calculate the value of \(\sf\:\cos(3A) \\\):

\(\begin{align}\sf\:\text{RHS} &= \frac{1}{4}\cos(3A) \\&= \frac{1}{4}(4\cos^3(A) - 3\cos(A)) \\&= \cos^3(A) - \frac{3}{4}\cos(A).\end{align} \\\)

Comparing the \(\sf\:\text{LHS} \\\) and \(\text{RHS} \\\), we have:

\(\sf\:-\frac{1}{4}\cos(A) = \cos^3(A) - \frac{3}{4}\cos(A). \\\)

Adding \(\sf\:\frac{1}{4}\cos(A) \\\) to both sides, we get:

\(\sf\:0 = \cos^3(A) - \frac{2}{4}\cos(A). \\\)

Simplifying further:

\(\sf\:0 = \cos^3(A) - \frac{1}{2}\cos(A). \\\)

Factoring out a common factor of \(\sf\:\cos(A) \\\), we have:

\(\sf\:0 = \cos(A)(\cos^2(A) - \frac{1}{2}). \\\)

Using the identity \(\sf\:\cos^2(A) = 1 - \sin^2(A) \\\), we can rewrite the equation as:

\(\sf\:0 = \cos(A)(1 - \sin^2(A) - \frac{1}{2}). \\\)

Simplifying:

\(\sf\:0 = \cos(A)(1 - \frac{3}{2}\sin^2(A)). \\\)

Since \(\sf\:\cos(A) \\\) cannot be zero (as it would result in undefined values), we can divide both sides of the equation by \(\sf\:\cos(A) \\\):

\(\sf\:0 = 1 - \frac{3}{2}\sin^2(A). \\\)

Rearranging the terms:

\(\sf\:\sin^2(A) = \frac{2}{3}. \\\)

Taking the square root of both sides, we get:

\(\sf\:\sin(A) = \pm\sqrt{\frac{2}{3}}. \\\)

The solution \(\sf\:\sin(A) = \sqrt{\frac{2}{3}} \\\) corresponds to the range where \(\sf\:0° \leq A \leq 90° \\\). Therefore, the solution \(\sf\:\sin(A) = \sqrt{\frac{2}{3}} \\\) is valid.

Hence, we have proved that:

\(\sf\:\cos(A)\cos(60^\circ - A)\cos(60^\circ + A) = \frac{1}{4}\cos(3A). \\\)

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Answer:

Given:

To Prove:

cosAcos(60°-A)cos(60°+A) = 1/4 cos3A

Solution:

Here are the steps in detail:

1. Expanding cosAcos(60°-A)cos(60°+A) using the product-to-sum identities:

=cosAcos(60°-A)cos(60°+A)

=(cosA)(cos(60°-A)cos(60°+A))

=(cosA)(1/2cos(60°-2A) + 1/2cos(60°+2A))

=(cosA)(1/2cos(-A) + 1/2cos(120°))

2. Substituting cos(60°-A) = cos(60°+A) = 1/2 into the expanded expression:

= cosA(1/2cos(-A) + 1/2cos(120°))

=cosA(1/2(1/2cosA) + 1/2(-1/2))

= cosA(1/4cosA - 1/4)

= (1/4)cosAcosA - (1/4)cosA

=(1/4)cos3A

3. Simplifying the resulting expression to obtain 1/4 cos3A:

=(1/4)cosAcosA - (1/4)cosA

=(1/4)cosA(cosA - 1)

=(1/4)cos3A

Therefore, we have proven that cosAcos(60°-A)cos(60°+A) = 1/4 cos3A. Hence Proved.

Decoding Forecast starting at 17:00Z:

Wind:

24014G23KT

Visibility:

5 statute miles

Weather:

Light rain showers

Clouds:

Overcast at 1500 feet

Incomplete report.

The decoded report is:

Location:

KJAX (Jacksonville International Airport)

Date/Time: 10th at 23:20Z

Wind:

00000KT

Visibility:

More than 6 statute miles

Clouds:

Scattered at 3500 feet

Forecast starting at 11:00Z:

Wind:

00000KT

Visibility:

5 statute miles

Weather:

Mist

Clouds:

Broken at 1000 feet, Broken at 2000 feet

Forecast starting at 06:00Z:

Wind:

16003KT

Visibility:

2 statute miles

Weather:

Mist

Clouds:

Broken at 500 feet, Overcast at 1000 feet

Temporary condition between 08:00Z and 12:00Z:

Visibility:

1 statute mile

Weather:

Mist

Clouds:

Overcast at 300 feet

Forecast starting at 14:00Z:

Wind:

20010G18KT

Visibility:

More than 6 statute miles

Weather:

Vicinity showers

Clouds:

Broken at 1500 feet, Overcast at 2500 feet

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Factor the radicand: If the radicand is not already in the form of a perfect square, you can factor it into its prime factors and pull out any perfect squares from the radicand.

For example, √(36) can be simplified to √(49) = √4 * √9 = 23.

Simplify the exponent: If the radical has an exponent (such as cube root), divide the exponent by the index of the radical. Example ∛64 = ∛(4^3) = ∛4 * ∛4 * ∛4 = 4 *4 *4 =4

Combine like factors: If there are any factors in the radicand that match the exponent, you can combine them and simplify the radical further. For example, √(2235) = √(2^235) = 2√(35) = 2√15 = 2*3 = 6.

It is important to note that not all radicals can be simplified to a whole number or a fraction. In some cases, the radical will be in its simplest form when it is left as an irrational number.

For Example, √2 cannot be simplified further.

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Answer:

(x + 1)² = 4(y + 10)

Step-by-step explanation:

Equation of parabola:

(x - h)² = 4a(y - k)

with vertex(h, k) and focus (h, k + a)

vertex(h, k) = (-1, -10)

⇒h = -1 and k = -10

focus (h, k + a) = (-1, -9)

⇒ k + a = -9

⇒ -10 + a = -9

⇒ a = 10 - 9

⇒ a = 1

Equation of parabola:

(x - h)² = 4a(y - k):

(x - (-1))² = 4(1)(y - (-1))

= (x + 1)² = 4(y + 10)

Answer:

$3687

Step-by-step explanation:

Hope this helps!!!!

Answer:

The inverse function can be written as:

g(x) = (x - 7 + √3) / 5

Step-by-step explanation:

Starting with the equation y = 5x + 7 - √3, we switch the roles of x and y to get x = 5y + 7 - √3.

Next, we isolate y by subtracting 7 and adding √3 to both sides, which gives x - 7 + √3 = 5y.

Finally, we divide both sides by 5 to get g(x) = (x - 7 + √3) / 5.

This is the inverse function g(x) of f(x) = 5x + 7 - √3. The inverse function is useful in solving problems where the original function is not explicitly invertible.

Answer:

13 = x

Step-by-step explanation:

\(\frac{6}{8}\) = \(\frac{(x + 2)}{(x + 7)}\)

6x + 42 = 8x + 16

42 = 2x + 16

\(\frac{26}{2}\) = \(\frac{2x}{2}\)

13 = x

Answer:

A

Step-by-step explanation:

By triangle inequality,

L < 15+6

15 < L+6

6 < L+15

Solving these, we get 9 < L < 21.

Only 10 falls within the range.

The answer is therefore 10cm i.e. A.

Answer:

The answer is "triangle ABC is not a right triangle".

Step-by-step explanation:

For a right-angle triangle:

Its square upon on longest or triangular edges is equivalent to the total of the other two squares

Its parameters indicated throughout the question are;

Square of lateral length \(A = 7 \ inch^2\)

The square of lateral length \(B = 18 \ inch^2\)

The square of lateral length \(C=27\ inch^2\)

Thus, the longest side is C, as well as the size \(inch^2\) of the squares of its two sides, is\(7 + 18 = 25 \ inch^2\), lower than square \(C = 27\ inch^2\), hence, the ABC triangle is not correct. Its longest edge is consequently C.

0.1717 is Probability that none has ever been married from selected 5 young women in age category at random.

In the question we have,

the number of females age 20 to 24 has never been married = 70.3%

the number of young women in this age category (20 to 24) at random = 5

We shall find out the probability that none has been maaried from selected group of 5 females.

Probability of female age 20 to 24 has never been married = 0.703

So, the probability that none has been married from selected group of 5 females ( p)= 0.703×0.703×0.703×0.73×0.73

= 0.1717

Therefore, The probability that none has ever been married is 0.1717.

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The number of grains on square 13 will be 4096 grains.

No. of grains on the first square = 1 = 2^0

No. of grains on the second square = 2 = 1*2 = 2¹

No. of grains on the third square = 4 = 2*2 = 2²

No. of grains on the fourth square = 8 = 2*4 = 2³

From the pattern it could be seen that amount of wheat is doubled for every next square

So, the number of grains in the nth square = 2^n-1

Therefore, No. of grains on 13th square

= 2^13 -1

= 4096

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Answer:

9

Step-by-step explanation:

no, I don't think so because 6/2=3 and in the bracket 1+2=3 and if we multiple them then the answer should be 9 (3×3=9)

The answer to your question is 9 !

the calculated t-value of -2.23 is less than the critical value of -1.699, we can reject the null hypothesis and conclude that there is enough evidence to support the claim that the average wind speed in the desert is less than 24.3 kilometers per hour.

To test whether there is enough evidence to reject the claim that the average wind speed in the desert is less than 24.3 kilometers per hour, we can use a one-sample t-test. The null hypothesis is that the population mean wind speed is 24.3 kilometers per hour or greater, while the alternative hypothesis is that the population mean wind speed is less than 24.3 kilometers per hour.

Using the given information, we can calculate the test statistic as follows:

\(t = \frac{(23 - 24.3)} { (2.24 / \sqrt{(32}} = -2.23\)

where 23 is the sample mean wind speed, 24.3 is the claimed population mean wind speed, 2.24 is the population standard deviation, and √(32) is the square root of the sample size.

Using a t-distribution table with 31 degrees of freedom (32-1), we can find the critical value for a one-tailed test with alpha = 0.05 to be -1.699. Since the calculated t-value of -2.23 is less than the critical value of -1.699, we can reject the null hypothesis and conclude that there is enough evidence to support the claim that the average wind speed in the desert is less than 24.3 kilometers per hour.

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The volume of the water in the fish tank is 135.27 cubic meters if the top 1/3 of the tank is just air.

To find the volume of the fish tank containing water, we need to find the volume of the fish tank and then subtract the volume that is occupied by the air i.e 1/3 of the volume of the tank.

So, the volume of the fish tank is :

=length x width x depth of the tank

=10.2 meters x 5.8 meters x 3.5 meters

= 203.94 cubic meters

Now we need to find the volume of the air that is occupied by the fish tank which is :

=1/3x the volume of the fish tank

=1/3x203.94 cubic meters

=68.67 cubic meters

The volume of the fish tank containing just water is:

=volume of the fish tank - the volume of the air occupied

= 203.94 cubic meters - 68.67 cubic meters

=135.27 cubic meters

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The charge for the t-shirt earlier than the cut price of 25 percent is $20.

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Given, the 25% discount, the price of a t-shirt was 15$.

Since, After 25% discount price of t-shirt = 15

Thus

Original price of t-shirt = 15 * 100/75

Original price of t-shirt = 20

Therefore, the price before the discount of 25 percent is $20.

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Therefore , the solution of the given problem of volume comes out to be

the volume of cylinder is 3801.6 cubic feet.

The amount of space a three-dimensional item occupies is measured by its volume, which is measured in cubic units. Two examples of cubic units are cm3 and in3. Mass, in contrast hand, is a metric of how much matter is present in an object. It is common to use the weight of an object to calculate mass in quantities like pounds or kilograms.

Here,

Given :

Radius = 8.7 feet

and height = 16 feet

To find the volume of cylinder :

=> πr²h

=> 22/7 * 8.7 * 8.7 * 16

=> 237.6 * 16

=> 3801.6 cubic feet.

Therefore , the solution of the given problem of volume comes out to be

the volume of cylinder is 3801.6 cubic feet.

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The complete question is:  "Find the volume of cylinder which hase Radius ; 8.7 feet; height: 16 feet ."

Answer:

6 minutes past 10

Step-by-step explanation:

lowest common multiple of 2 and 3: 6 minutes

If P(B)=0.3, P(A|B)=0.5, P(B')=0.7and P(A|B')=0.8, then the value of the probability P(B|A)= 0.2113

To find the value of P(B|A), follow these steps:

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Answer:

0.45

Step-by-step explanation:

You divided 2.25 by 5

(a) Yes, T is a linear transformation.

(b) No, T is not a linear transformation.

(c) Yes, T is a linear transformation.

(a) To determine whether T is a linear transformation, we need to check two conditions: additivity and homogeneity. In this case, T(x, y, z) = (x + 1, y + 1, z + 1) satisfies both conditions.

It preserves addition since T(x₁ + x₂, y₁ + y₂, z₁ + z₂) = (x₁ + x₂ + 1, y₁ + y₂ + 1, z₁ + z₂ + 1) = (x₁ + 1, y₁ + 1, z₁ + 1) + (x₂ + 1, y₂ + 1, z₂ + 1) = T(x₁, y₁, z₁) + T(x₂, y₂, z₂). It also preserves scalar multiplication since T(c⋅x, c⋅y, c⋅z) = (c⋅x + 1, c⋅y + 1, c⋅z + 1) = c⋅(x + 1, y + 1, z + 1) = c⋅T(x, y, z). Therefore, T is a linear transformation.

(b) For T to be a linear transformation, it should preserve both addition and scalar multiplication. However, in this case, T(A) = trace(A) = a11 + a22 + · · · + ann only satisfies the condition of preserving addition. It fails to preserve scalar multiplication because T(c⋅A) = c⋅(a11 + a22 + · · · + ann) ≠ c⋅T(A). Hence, T is not a linear transformation.

(c) Similar to part (a), we need to verify additivity and homogeneity for T to be a linear transformation.

T(x, y) = (1 + x, y) satisfies both conditions. It preserves addition since T(x₁ + x₂, y₁ + y₂) = (1 + (x₁ + x₂), y₁ + y₂) = (1 + x₁, y₁) + (1 + x₂, y₂) = T(x₁, y₁) + T(x₂, y₂). It also preserves scalar multiplication since T(c⋅x, c⋅y) = (1 + c⋅x, c⋅y) = c⋅(1 + x, y) = c⋅T(x, y). Therefore, T is a linear transformation.

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Answer:

x^4 n^4

Step-by-step explanation:

(-x)^4*n^4

4 is an even power and the minus sign disappears. 4 minus signs make a plus.

x^4 n^4

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\(( { - xn})^{4} = ({ - x})^{4} \times ({n})^{4} = \)

\(( { - 1})^{4} \times {x}^{4} \times {n}^{4} = {x}^{4} {n}^{4} \)

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Answer:

y = -3/4x + 9

Step-by-step explanation:

first, find the slope by taking the difference of the y-values / difference of the x-values

(-3-12) / (16-(-4)) which equals -15/20 or -3/4

now use that value to represent the 'm' in the formula in order to find 'b'

y = mx + b

-3 = -3/4(16) + b

-3 = -12 + b

9 = b

y = -3/4x + 9

Answer:

28

Step-by-step explanation:

\( - (7) \times ( - 4) \\ - ( - 28) \\ - \times - = + \\ = 28\)

Answer:

The expressions 10³÷10⁷, 10¹²/10¹⁶, and 10⁻²/10² are all equivalent to 10⁻⁴.

Step-by-step explanation:

When dividing exponents with the same base you subtract the exponents.

For visuals, in the equation 10³÷10⁷ it would be 10³⁻⁷ which would be 10⁻⁴ since 3 minus 7 is negative 4.

It is the same thing for 10¹²/10¹⁶, it would be 10¹²⁻¹⁶ which is equal to 10⁻⁴.

The same method can be used for 10⁻²/10² (10⁻²⁻²).

Step-by-step explanation:

a+b+c=22

abc=?

2+3+17=22

2×3×17=102

Answer:

The numbers are 2, 7 and 13

Step-by-step explanation:

2+7+13 = 22

2X7X13 = 182

Answer:

3×6 + 3×6 =36

Step-by-step explanation:

3 multiply 3+3 = 3*6 =18×2 =36

Answer:

Step-by-step explanation:

=3(3+3)+3(3+3)

=3x6+3x6

=18+18

=36

I hope this helps

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Your graph will actually look like an oscillation (wave), it will be more compressed horizontally. As the value of x increases withe very whole number, your graph will compress horizontally. More like in the image below.

Mr. Krumm paid $9.27 for a tie that had a 3% sales tax added on. So, the tie cost $231.75 before the tax was included.

To find out how much the tie cost before the tax was included, we need to first calculate how much of the total price was due to the tax. We know that the total price Mr. Krumm paid was $9.27, and that this price included a $3% sales tax.

To calculate the amount of tax that was included in the price, we can start by setting up an equation:

0.03x = 9.27 - x

Here, x represents the cost of the tie before the tax was included. We know that the tax is 3% of this cost, which is why we're multiplying it by 0.03.. We're also subtracting x from 9.27 to get the amount of tax that was added on.

Simplifying this equation, we get:

0.04x = 9.27

Dividing both sides by 0.04, we get:

x = 231.75

So the tie cost $231.75 before the tax was included.

In summary, Mr. Krumm paid $9.27 for a tie that had a 3% sales tax added on. To find out how much the tie cost before the tax was included, we set up an equation and solved for the cost of the tie (x) before the tax was added. The answer is that the tie cost $231.75 before the tax was included.

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