What the pOH of a solution containing [OH-] = 4.91 x
10-11 mol L-1 ?
Report your answer to at least 1 decimal place

The solution is, distance from the point to the line is 22√10 feet.

A straight line is an endless one-dimensional figure that has no width. It is a combination of endless points joined both sides of a point and has no curve.

here, we have,

The distance from point (x, y) to the line ax+by+c=0 is given by ...

 d = |ax +by +c|/√(a²+b²)

In general for, the equation of the line is ...

y = x/3 - 4

i.e. x -3y -12 = 0

so the distance formula is ...

 d = |x -y -12|/√(1² +(-3)²) = |x -y - 12|/√10

For the given point, the distance is ...

 d = |-6 -4 -12|/√10

   = 22/√10

 d = 22/√10 . . . .

distance from the point to the line = 22/√10

now, distance = 22/√10 * 10

                       =22√10 feet.

Hence, The solution is, distance from the point to the line is 22√10 feet.

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Answer:

4x+3y

Step-by-step explanation:

Combining like terms, we get:

5x - x + 4y - y = 4x + 3y

Therefore, 5x+4y-x-y simplifies to 4x+3y.

The variance of the error term is equal to the expected value of the squared error term, which is equal to the variance of Xi.

The asymptotic distribution of the coefficient β from bivariate OLS is given by √N(β – β) + d(0,σ2), where σ2 is the variance of the error term and is given by Var((Xi - E[X])) / Var(xi)2. If the error term is homoskedastic, then the variance of the error term does not depend on the value of Xi and is constant for all values of Xi. This implies that the variance of Y given X and r does not depend on r, as shown below:

Var(Y|X, r) = Var(βX + ε|X, r) = Var(ε|X, r) = σ2

Since the variance of the error term is constant and does not depend on the value of X or r, the variance of Y given X and r is also constant and does not depend on r.

If Yi is earnings and Xi is an indicator for whether someone has gone to college, homoskedasticity would imply that the variance of earnings for college and non-college workers is the same. This is unlikely to hold in practice, as there are likely to be other factors that affect earnings, such as occupation, experience, and location, that may differ between college and non-college workers and lead to different variances in earnings.

If the error term is homoskedastic and E[εi|Xi] = 0, then the variance of the error term is equal to the variance of Xi, as shown below:

Var(εi) = E[εi2] - (E[εi])2 = E[εi2] = E[(Xi - E[Xi])2] = Var(Xi)

This is because the error term is uncorrelated with Xi and has a mean of zero

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Yes, the fraction 4/9 will make each equation true.

Fraction is an element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

To see why, we can simplify the fraction 4/9 as follows:

4/9 = (4 x 7)/(9 x 7) = 28/63

Now, we can substitute 4/9 with 28/63 in each equation to see that they are all true:

63 x 28/63 = 28

18 x 28/63 = 8

96 x 28/63 = 42

36 x 28/63 = 16

We can also write it as:

63 × (4/9) = 28

18 × (4/9) = 8

96 × (4/9) = 42

36 × (4/9) = 16

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You can make 16 headbands using the blue fabric and 5 headbands using the yellow fabric, for a total of 21 headbands.

To calculate this, we need to convert 2/3 yard to feet since the fabric is given in feet. 2/3 yard is equal to 2 feet.

Next, we need to determine how much fabric is needed to make one headband, which is 2 feet.

To find out how many headbands we can make using the blue fabric, we divide the total length of blue fabric (12 feet) by the length needed for one headband (2 feet). 12 feet / 2 feet = 6 headbands per yard of blue fabric. Therefore, with 12 feet of blue fabric, we can make 6 x 12 = 72 headbands.

Similarly, to find out how many headbands we can make using the yellow fabric, we divide the total length of yellow fabric (4 feet) by the length needed for one headband (2 feet). 4 feet / 2 feet = 2 headbands per yard of yellow fabric. Therefore, with 4 feet of yellow fabric, we can make 2 x 4 = 8 headbands.

Therefore, we can make a total of 72 + 8 = 80 headbands.

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Complete Question:

You need 2/3 yard of fabric to create a headband: You have 12 feet of blue fabric and 4 feet of yellow fabric; How many headbands can you make with all of the fabric?

The answers are:

35. f(x)⋅g(x) = 9x2 - 0.25

36. (f(x))2 = 9x2 + 3x + 0.25

37. (g(x))2 = 9x2 - 3x + 0.25

To MULTIPLYING FUNCTIONS, we simply multiply the corresponding terms of each function together. Let's use the given functions f(x)=3x+0.5 and g(x)=3x−0.5 and perform the indicated operations.

35. f(x)⋅g(x) = (3x+0.5)(3x−0.5) = 9x2 - 0.25

36. (f(x))2 = (3x+0.5)2 = 9x2 + 3x + 0.25

37. (g(x))2 = (3x-0.5)2 = 9x2 - 3x + 0.25

So, the answers are:

35. f(x)⋅g(x) = 9x2 - 0.25

36. (f(x))2 = 9x2 + 3x + 0.25

37. (g(x))2 = 9x2 - 3x + 0.25

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(a) Revenue will be zero when p = $0 and $2000.

(b) Revenue will exceed $1,400,000 when p > 500 or p > 700

(a) To find the prices at which revenue is zero, we need to set R(p) equal to 0 and solve for p:0 = -4p^2 + 8000p0 = 4p(p - 2000)So either 4p = 0 or p - 2000 = 0.

Solving for p gives us:

p = 0 or p = 2000

Therefore, the prices at which revenue is zero are $0 and $2000.

(b) To find the range of prices for which revenue exceeds $1,400,000, we need to set R(p) greater than 1,400,000 and solve for p:

1,400,000 < -4p^2 + 8000p

Rearranging the equation gives us:

0 < 4p^2 - 8000p + 1,400,000

Factoring the left side of the equation gives us:

0 < (p - 500)(4p - 2800)

So either p - 500 > 0 or 4p - 2800 > 0.

Solving for p gives us:

p > 500 or p > 700

Since we want the range of prices for which revenue exceeds $1,400,000, we need to take the larger value of p.

Therefore, the range of prices for which revenue exceeds $1,400,000 is p > 700, or prices greater than $700.

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The solution for c in the equation is 35/3

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

\(c = 5\frac{5}{6} \times 2 \)

Express the equation properly

So, we have the following representation

c = 5 5/6 * 2

Convert the fraction to improper​ fraction

So, we have the following representation

c = 35/6 * 2

Evaluate the products

c = 35/3

Hence, the solution is 35/3

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The Simplex Algorithm to solve the given linear programming problem gives the solution [tex]x_{1}[/tex] = 16, , [tex]x_{2}[/tex] = 0,  [tex]x_{3}[/tex]  = 11; while the minimum value of objective function is "13.2".

The solution to the given linear programming problem can be obtained using the Simplex algorithm. Using the initial basic feasible solution ([tex]x_{1}[/tex] = 0,[tex]x_{2}[/tex] = 0, [tex]x_{3}[/tex] = 0), the following sequence of steps can be used to solve the problem:

The solution of the problem is: [tex]x_{1}[/tex] = 16, [tex]x_{2}[/tex] = 0, [tex]x_{3}[/tex] = 11.

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The expanded and exponential forms of the given algebraic expressions are given below:

1. Expanded form:  -3 * y * y * y * y * - 4 * x

Exponential form: 12xy⁴

2. 5x²y⁴ * 5xy⁵

Expanded form: 5 * x * x * y * y * y * y * 5 * x * y * y * y * y * y

Exponential form: 5¹⁺¹ * x² * y⁴⁺⁵ = 5²x²y⁹

3. Expanded form: 2 * t * u * u * u * u * 3 * t * t * t * 4 * t * u * u

Exponential form: 24t³u⁶

The expanded and exponential forms of the given algebraic expressions are given below:

1. - 3y⁴ * - 4x

Expanded form:  -3 * y * y * y * y * - 4 * x

Exponential form: -3 * y * y * y * y * - 4 * x = 12xy⁴

2. 5x²y⁴ * 5xy⁵

Expanded form: 5 * x * x * y * y * y * y * 5 * x * y * y * y * y * y

Exponential form: 5¹⁺¹ * x² * y⁴⁺⁵ = 5²x²y⁹

3. 2tu⁴ * 3t³ * 4tu²

Expanded form: 2 * t * u * u * u * u * 3 * t * t * t * 4 * t * u * u

Exponential form:  2 * 3 * 4 * t¹⁺¹⁺¹ * u⁴⁺² = 24t³u⁶

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Answer:

1.5

Step-by-step explanation:

See the screenshots for step by step explanation

 Use symoballab it is helpfull

None of the given options represents an amount of money equivalent to fife tenths of a dollar.

What is Money?

In mathematics, money is a unit of measure used to represent the value of goods or services. It is typically denoted in a currency unit, such as dollars, euros, or yen, and is often used to express prices, wages, or financial transactions. Money is a form of quantitative data and can be manipulated using various mathematical operations, such as addition, subtraction, multiplication, and division.

Fife tenths of a dollar is equivalent to $0.50, since there are 10 tenths in a dollar.

The amount of money that the digit 5 represents $0.50.

Looking at the given options, the only option that contains a total value of $0.50 is option B:

1 dollar bill = $1.00

quarter = $0.25

dime = $0.10

nickel = $0.05

5 Pennies = $0.05

Adding these values together, we get a total of $1.45, which is more than $0.50. Therefore, option B is incorrect.

Option D contains a total value of:

10 dollar bill = $10.00

quarter = $0.25

5 nickels = $0.25

2 Pennies = $0.02

Adding these values together, we get a total of $10.52, which is more than $0.50. Therefore, option D is incorrect.

Option A contains a total value of:

5 dollar bill = $5.00

quarter = $0.25

dime = $0.10

2 nickels = $0.10

4 Pennies = $0.04

Adding these values together, we get a total of $5.49, which is more than $0.50. Therefore, option A is incorrect.

The only remaining option is option C, which contains a total value of:

Two 20 dollar bills = $40.00

10 dollar bill = $10.00

quarter = $0.25

dime = $0.10

2 Pennies = $0.02

Adding these values together, we get a total of $50.37, which is more than $0.50. Therefore, option C is also incorrect.

Thus, none of the given options represents an amount of money equivalent to fife tenths of a dollar.

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Answer is SAS, we can solve this question by SAS theorem of triangle, for that we have to know more about triangle.

A triangle in geometry is a three-sided polygon or object with three edges and three vertices.

If two triangles satisfy one of the following conditions, they are congruent:

a. Each of the three sets of corresponding sides is equal. (SSS)

b. The comparable angles between two pairs of corresponding sides are equal.(SAS)

c. The corresponding sides between two pairs of corresponding angles are equal. (ASA)

d. One pair of corresponding sides (not between the angles) and two pairs of corresponding angles are equal. (AAS)

e. In two right triangles, the Hypotenuses pair and another pair of comparable sides are equal. (HL)

So, △MNP≅△OPN and NO II MP and NO ≅ MP

then, ∠ ONP = ∠ MPN and Side NP is common.

Therefore we can solve it by SAS Theorem.

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Simplifying the equation of v 4. 4 ( v - 16. 8 ) - -2. 3 = 3. 62, we find that v = 21.45.

To solve for v, we will use the following steps:

Simplify the left-hand side of the equation by distributing the 4.4:

4.4v - 73.92 + 2.3 = 3.62

Simplify further by combining like terms:

4.4v - 71.62 = 0

Add 71.62 to both sides of the equation:

4.4v = 71.62

Solve for v by dividing both sides by 4.4:

v = 71.62 / 4.4

v = 16.27727...

Round the answer to two decimal places:

v = 21.45

Therefore, the solution to the equation 4.4(v - 16.8) - (-2.3) = 3.62 is v = 21.45.

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Answer:

If 2 is a rational root of the equation x^3 - 5x^2 - 4x + 20 = 0, then (x - 2) is a factor of the polynomial. This is because of the rational root theorem, which states that any rational root of a polynomial with integer coefficients must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 20 and the leading coefficient is 1, so the possible rational roots are ±1, ±2, ±4, ±5, ±10, and ±20.

Since 2 is a root, we can use long division or synthetic division to divide x^3 - 5x^2 - 4x + 20 by (x - 2). We get:

code

2 | 1 -5 -4 20 | 2 -6 -20 |------------ | 1 -3 -10 0

Therefore, we have:

x^3 - 5x^2 - 4x + 20 = (x - 2)(x^2 - 3x - 10)

Now, we need to solve the quadratic equation x^2 - 3x - 10 = 0. We can use the quadratic formula:

x = (3 ± sqrt(3^2 - 4(1)(-10))) / 2 x = (3 ± sqrt(49)) / 2 x = (3 ± 7) / 2

So the solutions to the equation x^3 - 5x^2 - 4x + 20 = 0 are:

x = 2, x = (3 + 7)/2 = 5, x = (3 - 7)/2 = -2

Therefore, the solution set is {2, 5, -2}.

Step-by-step explanation:

Step 1: Use the Rational Root Theorem to find possible rational roots The Rational Root Theorem states that any rational root of a polynomial with integer coefficients must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 20 and the leading coefficient is 1, so the possible rational roots are ±1, ±2, ±4, ±5, ±10, and ±20. Since we are given that 2 is a root, we can conclude that (x - 2) is a factor of the polynomial.

Step 2: Use long division or synthetic division to divide the polynomial by (x - 2) We can use long division or synthetic division to divide the polynomial x^3 - 5x^2 - 4x + 20 by (x - 2). Both methods will give the same result, but I'll show the synthetic division method here:

code

2 | 1 -5 -4 20 | 2 -6 -20 |------------ | 1 -3 -10 0

The first row of the division represents the coefficients of the polynomial x^3 - 5x^2 - 4x + 20, starting with the highest degree term. We divide the first coefficient by the divisor, which gives us 1/1 = 1. Then we multiply the divisor (2) by the quotient (1), which gives us 2. We write 2 below the next coefficient (-5), and subtract to get (-5) - 2 = -7. We bring down the next coefficient (-4) to get -7 - (-4) = -3. We repeat the process with -3 as the new dividend, and so on, until we get a remainder of 0 in the last row.

The second row shows the partial quotients (in this case, just one quotient of 1), and the third row shows the coefficients of the quotient polynomial x^2 - 3x - 10. The last row shows the remainder, which is 0 in this case.

Step 3: Factor the quotient polynomial We now have x^3 - 5x^2 - 4x + 20 = (x - 2)(x^2 - 3x - 10), since (x - 2) is a factor of the polynomial. We can factor the quadratic polynomial x^2 - 3x - 10 by finding two numbers that multiply to -10 and add to -3. These numbers are -5 and 2, so we can write:

x^2 - 3x - 10 = (x - 5)(x + 2)

Step 4: Find the solutions to the equation We now have:

x^3 - 5x^2 - 4x + 20 = (x - 2)(x^2 - 3x - 10) = (x - 2)(x - 5)(x + 2)

The solutions to the equation are the values of x that make the polynomial equal to 0. These values are the roots of the equation. We have:

x - 2 = 0, so x = 2 is a root x - 5 = 0, so x = 5 is a root x + 2 = 0, so x = -2 is a root

Therefore, the solution set is {2, 5, -2}.

The solution set of the equation x^(3)-5x^(2)-4x+20=0 is found to be {2, 5, -2}.

One rational root of the given equation is 2. We can use synthetic division to find the other roots.

Set up the synthetic division table by placing the root (2) in the leftmost column and the coefficients of the equation (1, -5, -4, 20) in the top row.

2|1-5-420|2-6-201-3-100

Multiply the root (2) by each of the numbers in the bottom row and place the result in the row below. Then add the numbers in the top and bottom rows to get the numbers in the final row.

The final row represents the coefficients of the reduced equation: x^(2)-3x-10=0. We can use the quadratic formula to find the remaining roots:

x = (-(-3) ± √((-3)^(2) - 4(1)(-10))) / (2(1))

x = (3 ± √(9 + 40)) / 2

x = (3 ± √49) / 2

x = (3 ± 7) / 2

x = 5 or x = -2

So the solution set of the equation is {2, 5, -2}.

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The approximate solution using the three-point formula with step size h = 1/2 is y(t) = 0 for all t.

The boundary value problem given is y′′ − y = 0 with boundary conditions y(0) = 0 and y(2) = e2 − e−2.

(a) To find the exact solution y(t), we can use the characteristic equation r^2 - 1 = 0. This gives us r = 1 and r = -1. The general solution is therefore y(t) = c1e^t + c2e^-t.

Using the boundary conditions, we can find the constants c1 and c2.

For y(0) = 0, we have 0 = c1 + c2, which gives us c2 = -c1.

For y(2) = e2 − e−2, we have e2 − e−2 = c1e^2 + c2e^-2. Substituting c2 = -c1, we get e2 − e−2 = c1e^2 - c1e^-2.

Solving for c1, we get c1 = (e2 − e−2)/(e^2 - e^-2) = 1/2. Therefore, c2 = -1/2.

The exact solution is y(t) = (1/2)e^t - (1/2)e^-t.

(b) To use the three-point formula with step size h = 1/2, we can set up a table with tn and yn values.

tn | yn
---|---
0  | 0
1/2| y1
1  | y2
3/2| y3
2  | e2 - e-2

The three-point formula is yn+1 = yn-1 + 2h(y′n). We can use this formula to find the values of y1, y2, and y3.

For y1, we have y1 = 0 + 2(1/2)(y′0) = y′0. Since y′0 = y′(0) = (1/2)e^0 - (1/2)e^0 = 0, we have y1 = 0.

For y2, we have y2 = y0 + 2(1/2)(y′1) = 0 + 2(1/2)(0) = 0.

For y3, we have y3 = y1 + 2(1/2)(y′2) = 0 + 2(1/2)(0) = 0.

Therefore, the approximate solution using the three-point formula with step size h = 1/2 is y(t) = 0 for all t.

It is important to note that the three-point formula is not accurate for this particular boundary value problem due to the size of the step and the nature of the differential equation. A smaller step size or a different numerical method may yield a more accurate approximation.

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he correct answer is d. 4/ x^2 -2x+1 + 1.

To find f(g(x)), we need to substitute g(x) into f(x).

f(x) = x^2

g(x) = 2/x-1

So, f(g(x)) = (2/x-1)^2

= 4/x^2 - 4/x + 1

Therefore, the correct answer is d. 4/ x^2 -2x+1 + 1.

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Answer:

8290

Step-by-step explanation:

Explanation is on the pic

We can conclude that the square pyramid has a volume of 111.5 in³.

A square pyramid in geometry is a pyramid with a square base.

It is a right square pyramid with C4v symmetry if the apex is perpendicular to and above the square's center.

It is an equilateral square pyramid, the Johnson solid J1 if all edge lengths are equal.

A polyhedron with seven faces is called a heptahedron.

One "normal" heptahedron exists, and it is formed up of a surface with one side made up of four triangles and three quadrilaterals.

So, 6.6 inches wide is the square pyramid's base.

The square pyramid's height is 7.7 inches or h.

V = a²(h/3) is the formula for the square pyramid's volume.

By changing the values given:
V = 6.6²(7.7/3)

V = 43.56(2.56)

V = 111.51 ≈ 111.5

Therefore, we can say that the square pyramid has a volume of 111.5 in³.

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The correct answer is option A. 495, -305.

To find a positive angle that is coterminal to the given angle, we can add 360 degrees to the given angle. This is because coterminal angles are angles that have the same initial side and terminal side, but differ in the number of rotations. Adding 360 degrees to the given angle gives us the same initial side and terminal side, but with one extra rotation.

95 + 360 = 455

To find a negative angle that is coterminal to the given angle, we can subtract 360 degrees from the given angle. This gives us the same initial side and terminal side, but with one less rotation.

95 - 360 = -265

Therefore, the positive angle that is coterminal to the given angle is 455, and the negative angle that is coterminal to the given angle is -265.

Answer: A. 495, -305.

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Answer:

We do not have enough information to find the exact value of f(2.5) without additional assumptions about the nature of the exponential function. However, we can make an estimate using the given data and the properties of exponential functions.

First, we can write the general form of an exponential function as:

f(x) = a * b^x

where a is the initial value or y-intercept, and b is the base or growth factor. We can use the two given data points to set up a system of equations and solve for a and b:

f(-1) = a * b^(-1) = 18

f(5) = a * b^5 = 75

Dividing the second equation by the first equation, we get:

f(5) / f(-1) = (a * b^5) / (a * b^(-1)) = b^6 = 75 / 18 = 25 / 6

Taking the sixth root of both sides, we get:

b = (25 / 6)^(1/6) ≈ 1.472

Substituting this value of b into the first equation, we get:

a = f(-1) / b^(-1) = 18 / 1.472 ≈ 12.223

Therefore, we have the exponential function:

f(x) ≈ 12.223 * 1.472^x

Using this function, we can estimate the value of f(2.5) as:

f(2.5) ≈ 12.223 * 1.472^(2.5) ≈ 34.311

Note that this is only an estimate, and the exact value of f(2.5) may be different depending on the specific nature of the exponential function.

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:

Answer

the answer is b

If the selection is made randomly, there are 30045015 ways to select a committee of size 10 from a group of 10 men and 20 women. The probability that the committee consists of 5 men and 5 women is 0.129. The probability that there is no woman in the committee is  0.000000033.

1. The total number of ways to select a committee of size 10 from a group of 10 men and 20 women is given by the combination formula:
C(n,k) = n! / (k! * (n-k)!)
Where n is the total number of people (10 men + 20 women = 30) and k is the size of the committee (10).
C(30,10) = 30! / (10! * (30-10)!)
C(30,10) = 30! / (10! * 20!)
C(30,10) = 30045015
Therefore, there are 30045015 ways to select a committee of size 10 from a group of 10 men and 20 women.

2. The probability that the committee consists of 5 men and 5 women is given by:
P(5 men and 5 women) = C(10,5) * C(20,5) / C(30,10)
P(5 men and 5 women) = (10! / (5! * 5!)) * (20! / (5! * 15!)) / (30! / (10! * 20!))
P(5 men and 5 women) = (252 * 15504) / 30045015
P(5 men and 5 women) = 0.129

3. The probability that there is no woman in the committee is given by:
P(no woman) = C(10,10) * C(20,0) / C(30,10)
P(no woman) = (10! / (10! * 0!)) * (20! / (0! * 20!)) / (30! / (10! * 20!))
P(no woman) = (1 * 1) / 30045015
P(no woman) = 0.000000033
Therefore, the probability that there is no woman in the committee is 0.000000033.

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The closest distance that a cinema can put seats is 10.667 m.

One of the six mathematical functions used to express the side ratios of right triangles, the trigonometric function includes the sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). The figure shows these six trigonometric functions in respect to a right triangle. Surveying, engineering, and navigation issues where one of a right triangle's acute angles and the length of a side are known but the lengths of the other sides need to be determined can be solved with ease using trigonometry.

As per the given data:

the angle of elevation from a customer's eyes

to the top of the screen ≤ 31°.

Top of the cinema screen = 7.6 m above floor.

Customers' eyes =  1.2 m above floor.

For the closest distance that a cinema can put the seats (B):

the angle of elevation from a customer's eyes

to the top of the screen (maximum) = 31°.

Distance between customer eyes and top of the screen (P) = 7.6 - 1.2 = 6.4 m

∴ With θ = 31° , tanθ = (P/B)

tan31° = (6.4/d)

d = (6.4/tan31°)

d = 10.667 m

Hence, the closest distance that a cinema can put seats is 10.667 m.

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It will take 23.50 years for the investment to double.

The investment has the formula y= 5000(1.03)ˣ . Therefore, let's find the time in years the investment will double in value when starting at 5000 units.

The double of 5000 units is 10000 units.

Therefore,

y = 5000(1.03)ˣ

where

Hence,

y = 5000(1.03)ˣ

10000 = 5000(1.03)ˣ

divide both sides by 5000

10000 / 5000 = (1.03)ˣ

2 = (1.03)ˣ

log both sides

x = In 2 / In 1.03

x = 23.4977

Therefore,

x = 23.50 years

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Answer:

Step-by-step explanation:

Let x be the first term.  Let y be the common difference between each number in the sequence. x and the next three terms would be:

x, x+y,  x+2y,  and x+3y

The sum of the 4 terms is 4x + 6y and is equal to 38

4x + 6y = 38

4x  = 38 - 6y

x  = (19/2) - (3/2)y   [x is isolated here, to the left, for use in a lovely substitution coming up]

or x = 9.5 - 1.5y  [simplified]

===

The sum of the first 7 terms would be the first 4 [from above:   4x + 6y] plus the next 3 terms;

4x + 6y

 x + 4y

 x + 5y

 x + 6y

7x + 21y

7x + 21y is equal to 98

7x + 21y = 98

====

We have two equations and two unknowns, so we should be able to find an answer by substitution:

---

From above:

x  = (19/2) - (3/2)y  

7x + 21y = 98

Now use the first definition of x in the second equation:

7x + 21y = 98

7( (19/2) - (3/2)y) + 21a = 98

66.5 - 10.5y + 21y = 98

10.5y = 31.5

y = 3

Now use this value of y in either equation to find x:

7x + 21*(3) = 98

7x + 63 = 98

7x = 35

x =  5

====

Check:

Do the first 4 terms sum to 38?

5 + 8 + 11 + 14 = 38  YES

Do the first 7 terms sum to 98?

38 + 17 + 20 + 23 =   YES

Answer:

The graph of y=1/2x-10 is a straight line with a slope of 1/2 and y-intercept of -10. It slopes upward from left to right and intersects the y-axis at -10.

On the other hand, the graph of y=1/x is a hyperbola that passes through the origin and has asymptotes at x=0 and y=0. It consists of two branches that curve towards the asymptotes in opposite directions.

Compared to the graph of y=1/x, the graph of y=1/2x-10 is a simpler and more predictable shape. It does not have asymptotes and has a constant slope. It also does not pass through the origin, unlike the graph of y=1/x.

To find the exact values of sin(theta/2) and tan(theta/2), we can use the double angle formulas for sine and tangent. The double angle formulas are:
sin(theta/2) = √[(1-cos(theta))/2]
tan(theta/2) = sin(theta)/(1+cos(theta))
We are given that tan(theta) = -1/3. Using the Pythagorean identity, we can find the value of cos(theta):
tan^2(theta) = 1/9
1 + tan^2(theta) = 1 + 1/9 = 10/9
sec^2(theta) = 10/9
sec(theta) = √(10/9) = √10/3
cos(theta) = 1/sec(theta) = 3/√10 = 3√10/10
Now we can plug in the value of cos(theta) into the double angle formulas to find sin(theta/2) and tan(theta/2):
sin(theta/2) = √[(1-(3√10/10))/2] = √[(10-3√10)/20] = √[(10-3√10)]/√20
tan(theta/2) = (-1/3)/(1+(3√10/10)) = (-10/3)/(10+3√10) = (-10)/(10+3√10)
Therefore, the exact values of sin(theta/2) and tan(theta/2) are:
sin(theta/2) = √[(10-3√10)]/√20
tan(theta/2) = (-10)/(10+3√10)

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