what is the answer to this problem

The requried solution of the given trigonometric function is sin7π/12 is  (√3 + 1)/(2√2).

We know that 7π/12 = 9π/12 - 2π/12.

So, we can rewrite sin(7π/12) as sin(9π/12 - 2π/12).

Using the trigonmetric identity,
sin(A - B) = sinAcosB - cosAsinB, we can rewrite sin(9π/12 - 2π/12) as sin(9π/12)*cos(2π/12) - cos(9π/12)*sin(2π/12).

We know that,
sin(2π/12) = sin(π/6) = 1/2 and cos(2π/12) = cos(π/6) = √3/2.

Also, sin(9π/12) = sin(3π/4) = 1/√2 and cos(9π/12) = cos(3π/4) = -1/√2.

Substituting these values, we get:

= sin(7π/12) = sin(9π/12 - 2π/12)

= sin(9π/12)cos(2π/12) - cos(9π/12)sin(2π/12)

= (1/√2)(√3/2) - (-1/√2)(1/2)

= (√3/2√2) + (1/2√2)

= (√3 + 1)/(2√2)

Therefore, sin(7π/12) = (√3 + 1)/(2√2).

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Answer:

Method 1:

cos(60°) = 4/x

x cos(60°) = 4

x = 4/cos(60°) = 8

Method 2 (more efficient method):

Since this is a 30°-60°-90° right triangle, the length of the hypotenuse is twice the length of the shorter leg, so x = 8.

Answer:

c = 47

Step-by-step explanation:

The solution of the equation is the value of x.

Given "a" is a solution of the equation, and a = -6, to find the value of c, substitute x = -6 into the given equation and solve for c.

⇒ 4x + 1 + 7c = 6c - 3x + 6

⇒ 4(-6) + 1 + 7c = 6c - 3(-6) + 6

⇒ -24 + 1 + 7c = 6c + 18 + 6

⇒ -23 + 7c = 6c + 24

⇒ -23 + 7c - 6c = 6c + 24 - 6c

⇒ -23 + c = 24

⇒ -23 + c + 23 = 24 + 23

⇒ c = 47

Therefore, the value of c that makes a = -6 a solution of the equation is 47.

The value of 'c' for which 'a' is a solution of the given equation can be found by substituting 'a' into the equation and simplifying. In this case, 'a' is -6, and by substituting this value into the equation and simplifying, we find that when 'c' is 47, 'a' becomes a solution.

To find the value of c for which a is a solution of the equation, we have to substitute the value of a into the equation and solve for c. Here, a = -6, which means x can be considered as -6 in the equation.

Step 1: Substitute the value of a (x) into the equation. So the equation 4x + 1 + 7c = 6c - 3x + 6 becomes 4(-6) + 1 + 7c = 6c - 3(-6) + 6.

Step 2: Simplify both sides of the equation. We get -24 + 1 + 7c = 6c + 18 + 6.

Step 3: Continue simplifying to -23 + 7c = 6c + 24.

Step 4: By rearranging terms, we find c = 47.

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Answer:

[tex](x+4)^2+(y-4)^2=16[/tex]

Step-by-step explanation:

A tangent is a straight line that touches a circle at only one point.

Since the circle is tangent to both axes, and the center of the circle is in the second quadrant, the center of the circle lies on the line y = -x.

As the circle is tangent to the x-axis, the distance between the center of the circle and the x-axis is equal to the radius of the circle.

Similarly, as the circle is tangent to the y-axis, the distance between the center of the circle and the y-axis is also equal to the radius of the circle.

Therefore, the center of the circle will be 4 units to the left of the y-axis and 4 units up from the x-axis. Therefore, the coordinates of the center are (-4, 4).

[tex]\boxed{\begin{minipage}{5 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

Substitute the center (-4, 4) and the radius, r = 4, into the standard equation of a circle:

[tex]\implies (x - (-4))^2+(y-4)^2=4^2[/tex]

[tex]\implies (x+4)^2+(y-4)^2=16[/tex]

Therefore, the equation of the circle that satisfies the given conditions is:

[tex]\boxed{(x+4)^2+(y-4)^2=16}[/tex]

71*8=568

568+379=947

Step-by-step explanation:

Rounding to the nearest hundredth, the debt of Country G after 15 years would be approximately $14.89 billion dollars.

How to solve the question?

The given equation for the debt of Country G is A = 18.5(0.995), where A represents the debt after one year, and the factor of 0.995 represents the decay factor. To estimate the debt after 15 years, we can use this equation iteratively.

After the first year, the debt would be A = 18.5(0.995) = 18.4 billion dollars. To find the debt after the second year, we can use the same equation, but replace the initial debt of 18.5 billion dollars with the debt after one year, which is 18.4 billion dollars. Therefore, the debt after two years would be A = 18.4(0.995) = 18.3 billion dollars.

We can repeat this process for each subsequent year, using the previous year's debt to calculate the next year's debt. After 15 years, the debt would be:

A = 18.5(0.995)¹⁵

A = 18.5(0.80424)

A = 14.8866 billion dollars

Rounding to the nearest hundredth, the debt of Country G after 15 years would be approximately $14.89 billion dollars.

It's important to note that this estimate assumes that the decay factor remains constant over the 15-year period. In reality, economic and political factors could impact the decay rate and the actual debt of Country G.

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The options that are valid conclusions about the data include:

More than half of the students are not enrolled in French or Spanish. This conclusion is valid, as 25 out of 120 students are not enrolled in either French or Spanish, and 25 is more than half of 120.

Of the students that are enrolled in French, fewer than half of them are also enrolled in Spanish. This conclusion is valid, as only 30 out of 50 students enrolled in French are also enrolled in Spanish, which is less than half.

Therefore, the valid conclusions are B and E.

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Answer: $840 left

Step-by-step explanation:

2,100 * 0.20 = 420

So, 20% of 2,100 is 420

420 * 3 = 1,260

2,100 - 1,260 = 840

Answer:

[tex]\pi \times \frac{3}{4} [/tex]

inches

OR

[tex] \frac{3}{8} \pi \: x {}^{2} [/tex]

For simplicity, let's use common denominators.

1/2 = 3/6

So, let's first figure out the fractional value of the water added. We can accomplish this by taking the resulting amount and subtracting the initial amount of water.

3/6 - 1/6 = 2/6

2 gallons = 2/6 of the tank

Next, let's figure out how many gallons will fill 1/6 of the tank.

2 gallons = 2/6 of the tank

1 gallon = 1/6 of the tank

Now that we know how many gallons 1/6 of the tank is, we can figure out how many gallons 6/6 of the tank is.

1 gallon = 1/6 of the tank

6 gallons = 6/6 of the tank

Answer: The total capacity of the aquarium tank is 6 gallons.

Hope this helps!

Answer:

[tex]P= 49.2m[/tex], [tex]A=140.4m^2[/tex]

Step-by-step explanation:

The diagonal splits the rectangle into two right triangles so we can use Pythagorean theorem to find the length of the missing side.

Pythagorean theorem states that a^2 + b^2 = c^2 where c is the hypotenuse and a and b are the other two sides

So we can label our unknown side as x

x^2 + 9^2 = 18^2

x^2 + 81 = 324

Subtract 81 from both sides to isolate the x

x^2 = 243

Find the square root of both sides

√x^2 = √243

x ≈ 15.6

(The instruction said round to the nearest 10th).

So now we know

l = 15.6m and w = 9m

P= 2l + 2w

A= lw

P= 2(15.6) + 2(9)

P= 31.2 + 18

P= 49.2m

A=15.6(9)

A=140.4m^2

The surface area of the regular pyramid is 536.40 square inches

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

Base length, l = 13 in

Base width , w = 15 in

Height, h = 10 in

The surface area of the regular pyramid is calculated as

Area = lw + 1/2w√[4h² + l²] + 1/2l√[4h² + w²]

Substitute the known values in the above equation, so, we have the following representation

Area = 13 * 15 + 1/2 * 15 *√[4 * 10² + 13²] + 1/2 * 13√[4 * 10² + 15²]

Evaluate

Area = 536.40

Hence, the surface area is 536.40 square inches

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With the numbers gotten from the abacus counting, you can compare with the complete question to see if the student is correct or not.

From the given image, we can see that we are supposed to check if the answers given by the student is correct.

However, there are no data to compare from, so I would show the total sum of each of the abacus computed.

1. For GG, the total is 19

GB = 22

2. GG= 17

GB = 15

BB= 4

From these numbers gotten from the abacus counting, you can compare with the complete question to see if the student is correct or not.

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A. Mean: This measure of central tendency signifies the average value in a dataset, derived by summing each data point and dividing by total quantity.

B. Median: An indicator of central tendency is utilized to discern the middle value for a given set. Primarily, the tabulated elements must be listed from least to greatest before arriving at the average between two centermost figures if the number of components is even.

C. Mode: As an analysis concerning where centrality lies, this technique seeks out the most regularly occurring element in a gathered dataset.

D. Range: The range serves as an assessment of variability measured via the differences between the upper and lower bounds registered within the collection. To calculate, subtract the minimum value from the maximum quota.

E. Interquartile range: Better known as IQR, this method measuring disparity quantifies dissimilarities between the 75th percentile (Q3) and 25th percentile (Q1). To ascertain, subtract Q1 from Q3. Such information is often used to pinpoint irregularities and evaluate the discrepancies among separate datasets.

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Answer:

Step-by-step explanation:

The standard error of the estimate has been given in  the table that was presented in the question as  3.4835.

Standard error is a term that is used in statistics to measure the accuracy with which a sample would be the representation of a given a population.

In other words, we can say that it shows  the variability or dispersion of a sample statistic, this could be the  the sample mean or sample proportion, that is gotten through the true population value.

The standard error is stated already in the question as Standard Error 3.483483

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The angular velocity of the wheel is 11.74 radians per second and the distance travelled is 2.81 km/min.

The angular velocity is given by the formula rpm×2π/60

Angular velocity = 2 × 3.14 × 112/60

Multiply the numerator followed by division by denominator

Angular velocity = 703.36/60

Angular velocity = 11.72 radians per second

The distance travelled will be given by the formula -

Distance = circumference × revolutions per minute

Distance = 2 × 40 × 3.14 × 112

Multiply the values on Right Hand Side

Distance = 281,227 cm/min

As, 100 cm is 1 m and 1000 m is 1 km, the distance will be 2.81 km/min.

Thus, the angular velocity is 11.72 radians per second and distance is 2.81 km/min.

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Answer:

x = √(-x) + 6 are x = 9 and x = -12

Step-by-step explanation:

To find the solution for "x" in the equation x = √(-x) + 6, we can follow these steps:

Start with the original equation: x = √(-x) + 6.

Square both sides of the equation to eliminate the square root on the right-hand side: (x)^2 = (√(-x) + 6)^2.

Simplify the right-hand side by applying the distributive property: (x)^2 = (√(-x))^2 + 2(√(-x))(6) + (6)^2.

Simplify further by noting that (√(-x))^2 = -x: (x)^2 = -x + 72 + 36.

Combine like terms: (x)^2 = -x + 108.

Move all terms to one side of the equation: (x)^2 + x - 108 = 0.

This is now a quadratic equation. We can solve for "x" by factoring, using the quadratic formula, or by graphing. Factoring or using the quadratic formula would be appropriate methods in this case.

Factoring the quadratic equation, we get: (x - 9)(x + 12) = 0.

Setting each factor equal to zero and solving for "x", we get two possible solutions: x - 9 = 0 or x + 12 = 0.

Solving for "x" in each case, we get: x = 9 or x = -12.

So, the solutions to the equation x = √(-x) + 6 are x = 9 and x = -12.

[tex]x_{1} =2\\ \\x_{2} =-3[/tex].

[tex]x=\sqrt{-x+6}[/tex]

[tex](x)^{2} =(\sqrt{-x+6})^{2} \\ \\x^{2} =-x+6[/tex]

Standard form: [tex]ax^{2} +bx+c=0[/tex]

[tex]x^{2} +x-6=0[/tex]

a= 1 (because x² doesn't show an explicit coefficient)

b= 1 (because x doesn't show an explicit coefficient)

c= -6

Quadratic formula: [tex]\frac{-b+-\sqrt{b^{2} -4ac} }{2a}[/tex]

[tex]x_{1} =\frac{-(1)+\sqrt{(1)^{2} -4(1)(-6)} }{2(1)} =2\\ \\x_{2} =\frac{-(1)-\sqrt{(1)^{2} -4(1)(-6)} }{2(1)}=-3[/tex]

[tex]x_{1} =2\\ \\x_{2} =-3[/tex].

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The function that represents this situation is Total amount = 100(1 + 0.029)ᵗ + 15

The situation can be represented by a simple interest formula, where the balance increases every year by a fixed percentage of the principal amount. Let's assume that the initial amount is $100 and the interest rate is 2.90% per year, compounded annually. The formula for the balance after t years is:

B(t) = P(1 + r)ᵗ

Where P is the principal amount, r is the annual interest rate as a decimal, and t is the time in years.

In this case, the balance after t years is:

B(t) = 100(1 + 0.029)ᵗ

At the end of t years, when the money is withdrawn, the bank will give a free $15 gift, so the total amount received by the individual will be:

Total amount = B(t) + 15

So, the function that represents this situation is:

Total amount = 100(1 + 0.029)ᵗ + 15

where t is the number of years the money is saved.

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Answer: To place √31 on the number line, we first need to find its approximate value.

√31 is between 5 and 6 since 5²=25 and 6²=36. To find a more precise value, we can use a calculator or estimate using long division.

Using a calculator, we get:

√31 ≈ 5.56776436

Rounding to tenths, we get:

√31 ≈ 5.6

To place 5.6 on the number line, we can mark the point between 5.5 and 5.7.

Here's a rough sketch of what it might look like:

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         |         |         |         |         |         |         |         |         |         |

         |         |         |         |         |         |         |         |         |         |

         |         |         |         |         |         |         |         |         |         |

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         |         |         |         |         |         |         |         |         |         |

         |         |         |         |         |         |         |         |         |         |

Step-by-step explanation:

128

With the help of Apollonius theorem, we know that

3(sum of square of sides)=4(sum of square of medians)

3(sum of square of sides)=4×96

sum of square of sides= 4×96/3= 128

The missing step in the proof include the following: C. QU║AD and UA║DQ because the segments have the same slope.

In Mathematics and Geometry, a parallelogram simply refers to a four-sided geometrical figure (shape) and it can be defined as a type of quadrilateral and two-dimensional geometrical figure that is composed of two (2) equal and parallel opposite sides.

In Mathematics and Geometry, two (2) lines are parallel under the following conditions:

m₁ = m₂

This ultimately implies that, the slope of the two line segments are equal and the same:

Slope of line segment QU = Slope of line segment AD

-1/3 = -1/3

Slope of line segment UA = Slope of line segment DQ

-5/3 = -5/3

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The perimeter of the window in inches is 138 inches.

It will cost $110.40 to insulate the perimeter of the window.

To change the measurements to inches, we can use the fact that 1 foot is equal to 12 inches. So:

The height of the window in inches is (3 feet x 12 inches/foot) + 5 inches = 36 inches + 5 inches = 41 inches.

The width of the window in inches is (2 feet x 12 inches/foot) + 4 inches = 24 inches + 4 inches = 28 inches.

(a) The perimeter of the window in inches is the sum of the lengths of all four sides. We can use the formula for the perimeter of a rectangle to find it:

Perimeter = 2(length + width)

Plugging in the values we found:

Perimeter = 2(41 inches + 28 inches) = 138 inches

So the perimeter of the window in inches is 138 inches.

(b) We must divide the cost per inch by the perimeter in order to determine the cost of insulating the perimeter:

The following equation can be used to calculate the cost:

Cost = Perimeter x Cost per Inch

Cost = 138 inches x $0.80/inch = $110.40

So it will cost $110.40 to insulate the perimeter of the window.

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Max and Pam will be able to spend $676.1 on their trip.

The formula accrued amount in a compounded interest is expressed as;

A = P( 1 + r/n )^( n × t )

Where A is accrued amount, P is principal, r is interest rate and t is time.

Given the data in the question;

Principal P = $600.00

Compounded quarterly n = 4

Time t = 3  years

Interest rate r = 4%

Accrued amount A = ?

First, convert R as a percent to r as a decimal

r = R/100

r = 4/100

r = 0.04 rate per year.

Plug the given values into the above formula and solve for A.

A = P( 1 + r/n )^( n × t )

A = $600( 1 + 0.04/4 )^( 4 × 3 )

A = $600( 1 + 0.01 )^( 12 )

A = $600( 1.01 )^( 12 )

A = $676.1

Therefore, the accrued amount after 3 years is $676.1.

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Area of the triangle is 56 cm².

What is the area of triangle?

The formula for the area of a triangle is:

[tex]A = \frac{ (base \times height) }{ 2}[/tex]

where A is the area of the triangle and the base is the length of one of its sides and the height is the perpendicular distance between the base and the opposite vertex.

Alternatively for the triangle we can use Heron's formula to calculate the area of a triangle if we know the lengths of all three sides,[tex]A = \sqrt{ (s(s-a)(s-b)(s-c))}[/tex] where A is the area of the triangle, a, b and c are the lengths of the sides and s is the semi-perimeter of the triangle which is half the sum of the lengths of the sides,

[tex]s = \frac{(a + b + c)}{2}[/tex]

Here, it is clear that base of the triangle is 14 cm and height is 8 cm.

Now area of the tria is[tex] \frac{1}{2} \times 14 \times 8 = 14 \times 4 = 56 \: {cm}^{2} [/tex]

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The coordinates of vertex D are (2, -2).

Given that ABCD is a parallelogram.

Coordinates of A = (1, 3) and B = (5, 1) and C = (-2, 0).

Let the fourth coordinates of the point D be (x, y).

So, the slope of AB and CD will be equal since ABCD is parallelogram and opposite sides of parallelogram are parallel and slopes of parallel are equal.

Slope of CD = (y - 0)/(x - (-2)) = y/(x + 2)

Slope of AB = (1 -3)/(5 - 1) = -2/4 = -1/2

So, y/(x + 2) = -1/2

2y = -x - 2

x + 2y = -2 ........... (i)

again slopes of AC and BD are equal for the same reason.

Slope of AC = (3 - 0)/(1 - (-2)) = 3/3 = 1

Slope of BD = (y - 1)/(x - 5)

So, (y - 1)/(x - 5) = 1

y - 1 = x - 5

x - y = 4 ...........(ii)

equation (i) - equation (ii) we get,

(x + 2y) - (x - y) = -2 -4

3y = -6

y = -2

Putting y = -2 in equation (ii) we get,

x - (-2) = 4

x = 4 - 2 = 2

So the coordinates of fourth point are (2, -2).

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The question is in another language. The English translation of the question will be -

"Solve and explain the following problem step by step:

From a parallelogram ABCD we know A(1,3), B(5,1), C(-2,0). Find the coordinates of vertex D."

The measurements only result in one triangle.

When we apply the Law of Sines, we get:

sin(B) = (16.7/14)sin(35)

sin(B) = (16.7/14) sin(B) = 0.580sin(35)

sin(B) ≈ 0.580

We obtain either B = 35.8 degrees or B = 144.2 degrees when we take the inverse sine of both sides.

We know that C = 180 - 35 - 35.8 109.2 degrees since the total of the angles in a triangle is 180 degrees.

The length of the third side, c, can be determined using the Law of Cosines as follows: c 10.8

We have c 31.2 for B 144.2 degrees.

A legal triangle with side lengths of 14, 16, and 10 with angles of 35, 35.8, and 109.2 degrees is one where B 35.8 degrees.

As a result, the measurements only result in one triangle.

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The point estimate of the population mean is 25

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

Lower bound = 22

Upper bound = 28

The point estimate of mean is calculated as

Point estimate of mean = 1/2 * (Lower bound + Upper bound)

Substitute the known values in the above equation, so, we have the following representation

Point estimate of mean = 1/2 * (22 + 28)

Evaluate

Point estimate of mean = 25

Hence, the estimate is 25

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