A rope connects the top of a pole to the ground. The rope is 15 yd long and touches the ground 11 yd from the pole. How tall is the pole?

Answer: [tex](f \cdot g)(x) = x^4-11x^2+30.[/tex]

Step-by-step explanation:

We want to find the value of [tex]f(g(x)).[/tex] To do this, we plug in [tex]g(x)[/tex] into the expression for [tex]f(x).[/tex] This gives us:

[tex]f(g(x)) = (x^2-9)^2+7(x^2-9)+12 = x^4-18x^2+81+7x^2-63+12 = \boxed{x^4-11x^2+30}.[/tex]

Answer:

Using the Pythagorean theorem, the length of the rope can be found as follows:

c^2 = a^2 + b^2

where c is the length of the rope, a is the height of the fence (8 feet), and b is the distance from the fence to the point where the rope is attached to the ground (6 feet).

Plugging in the values, we get:

c^2 = 8^2 + 6^2

c^2 = 64 + 36

c^2 = 100

c = √100

c = 10

Therefore, the length of the rope is 10 feet.

Answer:

The length of the rope is 10 feet.

Step-by-step explanation:

To find the length of the rope attached to the top of an 8-foot-tall fence and fastened to the ground 6 feet away, we can use the Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In this case, the vertical leg is the height of the fence (8 feet) and the horizontal leg is the distance from the base of the fence to where the rope is fastened (6 feet). Therefore, the length of the rope (the hypotenuse) is given by:

hypotenuse^2 = vertical leg^2 + horizontal leg^2

Substituting the values, we get:

hypotenuse^2 = 8^2 + 6^2

hypotenuse^2 = 64 + 36

hypotenuse^2 = 100

Taking the square root of both sides, we get:

hypotenuse = 10

So the length of the rope is 10 feet.

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Answer:

60 seconds

Step-by-step explanation:

Let's start by finding how much more water needs to be added to the solution to reach a total of 10 ml of water:

10 ml - 5.8 ml = 4.2 ml

Now we can use the formula:

time = amount of water added / rate of addition

to calculate the time it will take to add 4.2 ml of water at a rate of 0.07 ml/s:

time = 4.2 ml / 0.07 ml/s

time = 60 seconds (rounded to the nearest second)

Therefore, it will take 60 seconds to add enough water to the solution to make a total of 10 ml of water.

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Answer:8%

Step-by-step explanation:

you divide 25 by 32 and round to nearest tenth

hope this helps :)

To show that the angle bisector of an equilateral triangle is perpendicular to the base​, we can assume an equilateral triangle with sides ABC. Next, we can get an angle bisector in the middle represented by BD which becomes perpendicular to the base BC.  

After getting the angle bisector, we would see that the line segment BD, divides angle BAC into two equal angles, each measuring 30 degrees. Now, we can draw a perpendicular line from point D to the base BC.

We could also draw a line E, which is the perpendicular line that intersects the base BC. We want to show that BD is perpendicular to BC, which means we need to show that angle BDE is a right angle.

Since the sum of angles BDA and ADE is 90 degrees (they form a right angle), and the sum of angles BAD, ABD, and BDA is 180 degrees, it follows that angle BDE must be a right angle.

Therefore, we have shown that the angle bisector of an equilateral triangle is perpendicular to the base.

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Answer:

Step-by-step explanation:

You want the equations for the lines shown on the graph.

The slope-intercept form of the equation for a line is ...

  y = mx + b

where m is the slope, and b is the y-intercept.

The slope of a line is the ratio of its "rise" to its "run". The rise and run are the vertical distance and horizontal distance between two points, respectively. We usually want to choose points that are where the line crosses grid intersections, as this gives the most exact value for the slope.

Red line: There is no rise for any value of run. The slope is ...

  m = rise/run = 0/1 = 0

Green line: The green line crosses the y-axis at y = 1, and crosses the next grid intersection to the right at (1, -1). The rise between those points is -2 (2 grid squares down), and the run is 1 (1 grid square to the right). The slope is ...

  m = -2/1 = -2

Blue line: The blue line crosses the y-axis at y = -1, and crosses the next grid intersection to the right at (1, 1). The rise between those points is +2 (2 grid squares up), and the run is 1 (1 grid square to the right). The slope is ...

  m = 2/1 = 2

Yellow line: The yellow line crosses the y-axis at y = -1, and crosses the next grid intersection to the right at (2, -2). The rise between those points is -1 (1 grid square down), and the run is 2 (2 grid squares to the right). The slope is ...

  m = -1/2

Y-Intercept.

The y-intercept is the value of y where the line crosses the y-axis. In the slope-intercept form equation (y=mx+b), this is the value of 'b'. In the previous section, we used those crossings as one of the grid intersections for finding the slope. They are ...

Using the slope and y-intercept for each line, we can now write the equations:

__

Additional comment

The slope-intercept form of the equation is not the only possible way to write an equation for a line. There are more than half a dozen other ways an equation for a line can be written. Each will have its use.

Other forms include ...

  ax +by = c . . . . . . . standard form

  ax +by -c = 0 . . . . . general form

  x/a +y/b = 1 . . . . . . . intercept form

  y -k = m(x -h) . . . . . point-slope form

Intercept forms don't work well when one of the intercepts is missing. For the red line, the x-intercept is missing. Essentially, the x-terms disappear from the standard, general, and intercept form equations. In the point-slope form, the equation of the red line is y-5=0, since the slope is 0.

Using the power reducing formulas the expression [tex]5cos^4(x) = 5(1 + cos(2x))^{2/4}[/tex].

We can define higher powers of a trigonometric function in terms of lower powers of the same function using the power-reducing formulas, which are trigonometric identities. Higher powers of cosine can be expressed using this formula in terms of sine and cosine's initial powers. For the other trigonometric functions, such as sine and tangent, there are equivalent formulas. These formulas can be used to solve trigonometric equations and to simplify trigonometric expressions.

The given function is 5cos^4(x).

Now, the power reducing formulas are given by:

[tex]cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1\\cos^2(x) = (1 + cos(2x))/2[/tex]

Substituting the value of second equation in 1 we have:

cos(2x) = 2(1 + cos(2x))/4 - 1

[tex]cos(2x) = (2cos^2(x) - 1) = cos^2(x) - sin^2(x)[/tex]

Now we have:

[tex]cos^2(x) = (1 + cos(2x))/2[/tex]

Thus,

[tex]cos^4(x) = (1 + cos(2x))/2)^2[/tex]

Now multiplying by 5:

[tex]5cos^4(x) = 5(1 + cos(2x))^{2/4}[/tex]

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A sampling distribution is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population.

To get the possible samples, use the formula "N choose n", where N is the population size, and n is the sample size to be taken. The total probability of the sample mean must be equal to 1.

Since the mean of all possible samples must be equal to the population mean. The sampling distribution is an important concept in statistics as it allows us to make inferences about the population based on the characteristics of the sample, and helps us estimate parameters and test hypotheses.

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The formula for the nth term of the arithmetic sequence is an = 4n + 2

Let d be the common difference of the arithmetic sequence, and let a1 be the first term. Then we have:

a4 = a1 + 3d = 10

a6 = a1 + 5d = 18

We can solve this system of equations to find a1 and d. Subtracting the first equation from the second, we get:

2d = 8

d = 4

Substituting this into either of the original equations, we get:

a1 + 12 = 18

a1 = 6.

So the arithmetic sequence is:

6, 10, 14, 18, ...

To find the nth term of this sequence, we can use the formula:

an = a1 + (n - 1)d

Substituting in the values of a1 and d, we get:

an = 6 + 4(n - 1)

Simplifying, we get:

an = 4n + 2

Therefore, the formula for the nth term of the arithmetic sequence whose fourth term is 10 and sixth term is 18 is:

an = 4n + 2

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Answer:

∠ A = 76°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180°

sum the 3 angles and equate to 180

9x - 5 - 5 + 6x + 55 = 180

15x + 45 = 180 ( subtract 45 from both sides )

15x = 135 ( divide both sides by 15 )

x = 9

Then

∠ A = 9x - 5 = 9(9) - 5 = 81 - 5 = 76°

1. Por lo tanto, la suma de las dos expresiones es 9x² + 5x + 18.

2. Por lo tanto, el resultado de la expresión es -22.

3. Por lo tanto, el resultado de la expresión es 24.

4. Por lo tanto, el valor de x es 3.

5. Por lo tanto, el resultado de la expresión es 45.

1. Para sumar las dos expresiones, simplemente sumamos los términos semejantes:

(4x² + 8x +12) + (5x² − 3x + 6) = 9x² + 5x + 18

Por lo tanto, la suma de las dos expresiones es 9x² + 5x + 18.

2. Evaluando la expresión numérica:

5 - 21 + (-3) (2) = 5 - 21 + (-6)

Luego, sumamos los términos:

= -16 - 6 = -22

Por lo tanto, el resultado de la expresión es -22.

3. Evaluando la expresión numérica:

(-3) (-2) (-1) (4) = 24

Por lo tanto, el resultado de la expresión es 24.

4. Para encontrar el valor de x, debemos despejarlo de la ecuación:

18 + 3x = 27

Restando 18 de ambos lados, obtenemos:

3x = 9

Dividiendo ambos lados por 3, obtenemos:

x = 3

Por lo tanto, el valor de x es 3.

5. Evaluando la expresión numérica:

3² + 4 (3)² = 3² + 4 (9)

Primero elevamos al cuadrado 3, luego multiplicamos 4 por 9 y sumamos ambos términos:

= 9 + 36

Por lo tanto, el resultado de la expresión es 45.

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The equation for the best fit of the scatter plot is y = x - 3

What is the scatter plot?

In a scatter plot, dots are used to represent the values of two different numerical variables. The values of each data point are represented by the position of each dot on the horizontal and vertical axes. Use scatter plots to see how different variables are related to one another.

Given data,

Let the scatter plot be represented as A

Now, the value of A is plotted below

where the slope of the plot is given by

m = y / x

m = 1

And, the y-intercept of the plot is when x = 0

So , when x = 0 , y = -3

Hence, the scatter plot is y = x - 3

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The inequality |x + 7| ≥ 2 has the solution set x ≤ –9 or x ≥ –5, as required.

We can write an absolute value inequality with the given solution set as follows:

|x + 7| ≥ 2

Here's how we arrived at this inequality:

First, we find the midpoint of the given solution set:

midpoint = (-9 + (-5)) / 2 = -7

Next, we find the distance from the midpoint to each endpoint of the solution set:

distance to -9 = |-7 - (-9)| = 2

distance to -5 = |-7 - (-5)| = 2

The maximum distance is 2, so we use c = 2 in the inequality.

Since the midpoint is to the left of both endpoints, we use the form |x - b| ≥ c.

Plugging in b = -7 and c = 2, we get:

|x + 7| ≥ 2

This inequality has the solution set x ≤ –9 or x ≥ –5, as required.

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The ratios of each type are:

1/4 has big beak (BB)

2/4  has big short beak (Bb)

1/4 has short beak (bb)

A Punnett square is a diagram used to predict the possible outcomes of a genetic cross between two individuals.

               B          b

B            BB        Bb

b             Bb        bb

We have 1 BB, this implies that we have one bird with big beak.

We have 2 Bb, this implies that we have two bird with big short beak.

We have 1 bb, this implies that we have one bird with short beak.

The ratios of each type are:

1/4 (i.e. 25%) has big beak (BB)

2/4 (i.e. 50%) has big short beak (Bb)

1/4 (i.e. 25%) has  short beak (bb)

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Answer:

40

Step-by-step explanation:

Hope this helps! =D

Brainliest! =D

The graph of the piecewise function can be seen in the image attached below where (x+2)² terminates at x = -1,  |x - 1| between the interval of x = -1 to x = 1, and  [tex]-\sqrt[3]{x}[/tex] from +x-axis to infinity [tex]\infty[/tex]

A piecewise function is a type of function that has different expressions or formulas for different parts or intervals of its domain. This means that the formula that defines the function changes depending on which part of the domain we are looking at.

Now, to graph the piecewise function, we need to start by identifying the different intervals on which the function is defined and the corresponding expressions that define the function on each interval.

If we have a piecewise function f(x) defined as:

The graph of the piecewise function can be seen in the image attached below.

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Therefore, the expected value of the event P(2k) for 2 ≤ k ≤ 6, given the event E, is 96/1 or simply 96.

Probability theory is widely used in many fields, including mathematics, statistics, physics, engineering, economics, and social sciences. It provides a framework for analyzing and understanding uncertain events and making predictions based on available information. Probability theory is also used in decision-making, risk management, and game theory, among other applications.

Here,

We can start by calculating the probability of each event in the sample space:

P(1) = 1/32

P(4) = 1/8

P(8) = 1/4

P(16) = 1/2

P(32) = 1

P(64) = 1

Next, we can calculate the probability of the event E by summing the probabilities of the individual outcomes in E:

P(E) = P(1) + P(8) + P(32) + P(64)

= 1/32 + 1/4 + 1 + 1

= 33/32

Now, we can calculate the expected value of the event P(2k) for 2 ≤ k ≤ 6:

E[P(2k)] = Σ[2 ≤ k ≤ 6] P(2k) * ΘS(2k)

= P(4)*ΘS(4) + P(8)*ΘS(8) + P(16)*ΘS(16) + P(32)*ΘS(32) + P(64)*ΘS(64)

= (1/8)*4 + (1/4)*8 + (1/2)*16 + (1)*32 + (1)*64

= 4/8 + 8/4 + 16/2 + 32 + 64

= 1/2 + 2 + 8 + 32 + 64

= 106.5

Therefore, the expected value of the event P(2k) for 2 ≤ k ≤ 6, given the event E, is 106.5. However, since the event E has a probability greater than 1, this result does not make sense. So, we need to normalize the probabilities of the events by dividing them by P(E) to obtain the correct expected value:

P'(1) = P(1)/P(E) = (1/32)/(33/32) = 1/33

P'(8) = P(8)/P(E) = (1/4)/(33/32) = 8/33

P'(32) = P(32)/P(E) = 1/(33/32) = 32/33

P'(64) = P(64)/P(E) = 1/(33/32) = 32/33

Now, we can calculate the normalized expected value:

E[P(2k)] = Σ[2 ≤ k ≤ 6] P'(2k) * ΘS(2k)

= P'(8)*ΘS(8) + P'(32)*ΘS(32) + P'(64)*ΘS(64)

=(8/33)*8 + (32/33)*32 + (32/33)*64

= (64/33) + (1024/33) + (2048/33)

= 3136/33

= 96/1

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Therefore, sin θ = 4/5, csc θ = 5/4, and cot θ = -3/4 using trigonometric ratio.

We know that the point (-3, 4) is on the terminal side of the angle θ, and we can use the Pythagorean theorem to find the value of the hypotenuse:

r² = x² + y²

r² = (-3)² + 4²

r² = 9 + 16

r² = 25

r = 5

Now we can find the values of sin θ, csc θ, and cot θ:

sin θ = y/r = 4/5

csc θ = r/y = 5/4

cot θ = x/y = -3/4

Therefore, sin θ = 4/5, csc θ = 5/4, and cot θ = -3/4.

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Answer:

The equation x - y = 7 represents a linear equation with two variables, x and y. To find three ordered pairs (x, y) that satisfy this equation, we can arbitrarily choose values for one variable and solve for the other.

Ordered Pair 1:

Let's choose x = 0:

x - y = 7

0 - y = 7 (Substituting x = 0)

y = -7

So, the first ordered pair is (0, -7).

Ordered Pair 2:

Let's choose y = 0:

x - y = 7

x - 0 = 7 (Substituting y = 0)

x = 7

So, the second ordered pair is (7, 0).

Ordered Pair 3:

Let's choose x = 5:

x - y = 7

5 - y = 7 (Substituting x = 5)

y = -2

So, the third ordered pair is (5, -2).

Therefore, three ordered pairs (x, y) that satisfy the equation x - y = 7 are: (0, -7), (7, 0), and (5, -2).

The cube's volume can be expressed as V = s³ = x³ since the height of the cube is x and all sides have equal length. The surface area of the cube can be expressed as SA = 6s² = 6x².

b. The sphere's volume can be expressed as V = (4/3)πr³, where r is half the diameter or x/2. Therefore, V = (4/3)π(x/2)³ = (1/6)πx³.

The surface area of the sphere can be expressed as SA = 4πr² = 4π(x/2)² = πx².

c. The ratio of the cube's volume to its surface area can be expressed as V/SA = (x³)/(6x²) = x/6, with the restriction that x > 0.

d. The ratio of the sphere's volume to its surface area can be expressed as V/SA = [(1/6)πx³]/(πx²) = x/6, with the restriction that x > 0.

e. Since the ratios of the volumes to the surface areas of the containers are the same, the efficiency of the packaging will depend only on the amount of material required for each shape. The cube will require less material since it has a smaller surface area, making it the more efficient choice.

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The equation of exponential function of following table is represented by the table below is y = A(Bˣ).

An exponential function is a mathematical function denoted by f(x) = eˣ (where the argument x is written as an exponent). Unless otherwise stated, the term usually refers to positive-valued functions of real variables, but can be extended to complex numbers and generalized to other mathematical objects such as matrices and Lie algebras. Although the exponential function arose from the notion of exponentiation (iterative multiplication), its modern definition (which has several equivalent characterizations) allows it to be extended exactly to any real argument, including irrational numbers. Its ubiquitous appearance in pure and applied mathematics has led mathematician Walter Rudin to believe that the exponential function is "the most important function in mathematics."

To find the equation of exponential form:

For case 1,

Exponential form: y = A(Bˣ)

y = 0.1, x = 0

[tex]0.1 = A(B^{0})[/tex]

0.1 = A(1)

A = 0.1

For case 2,

Exponential form: y = A(Bˣ)

y = 0.05, x = 1

[tex]0.05 = 0.1(B^{1})[/tex]

B = 0.5

For case 3,

y = 0.025, x = 2

[tex]A = 0.1(B^{2})[/tex]

[tex]0.025 = 0.1(0.5^{2})[/tex]

0.025 = 0.025

For case 4,

y = 0.0125, x = 3

0.0125 = 0.1(0.5³)

0.0125 = 0.1(0.125)

0.0125 = 0.0125

We calculated the value of A and B, and proved it in case 3 and case 4.

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The complete question is as follows:

Find the equation of the exponential function represented by the table below:

Answer:

955 11 7 11

3101 thats my answer

Step-by-step explanation:

10 and 3 thank you

Step-by-step explanation:

you remember the priorities of mathematical operations :

1. brackets

2. exponents

3. multiplications and divisions

4. additions and subtractions

it is irrelevant how close or how far apart the operands are written.

I think, missing operators are multiplications.

so, by writing really everything

(7 × 6) ÷ 2 × [{45 + 3 × (7 × 2 - 15 + 6)} + 4]

we get in a first step :

(42) ÷ 2 × [{45 + 3 × (14 - 15 + 6)} + 4]

second step

21 × [{45 + 3 × (5)} + 4]

third step

21 × [{45 + 15} + 4]

fourth step

21 × [60 + 4]

fifth step

21 × 64

sixth step

1,344

Answer:

$525

Step-by-step explanation:

To calculate the sales commission for this employee, we need to first calculate the total sales made by the employee for the week. The employee made sales of 800 on Monday, 600 on Tuesday, 1300 on Wednesday, 800 on Thursday and 800 on Friday. The total sales made by the employee for the week is:

800 + 600 + 1300 + 800 + 800 = 4500

Since the employee does not earn sales commission for the first $1000 of sales but earns a 15% sales commission on all sales made thereafter, we need to calculate the commission earned on sales above $1000.

The total sales above $1000 is:

4500 - 1000 = 3500

The commission earned on sales above $1000 is:

3500 x 15% =$525

Therefore, the employee earned a sales commission of $525 this week.

Answer:

the on side to side is the x axes and the y one is up and down

Step-by-step explanation:

The volume of the triangular prism will be 400 cubic feet.

Given that:

Width, W = 15 feet

Height, h = 8 feet

Base length, b = 6²/₃ feet

Volume is a measurement of three-dimensional space that is utilized. The concept of length is linked to the notion of capacity.

The volume of the triangular prism is calculated as,

V = 1/2 · b · h · W

V = 0.5 · 6²/₃ · 8 · 15

V = 400 cubic feet

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Answer:

Angle 1 = 51.6°

Angle 2 = 38.4°

Step-by-step explanation:

All complimentary angles add up to 90°.

If two complimentary angles were congruent then their measures would both be 45°.

Since one angle's measure is 6.6° more than the other's measure, it's measure is 6.6° more than 45°

Vice versa for the second angle, its measure will be 6.6° less than 45°