Write an equation of the function g(x)g(x) that is the graph of f(x)=|x|f(x)=|x| , but shifted left 4 units and shifted up 8 units.

Let f(x)=9x+7f(x)=9x+7.

Use algebra to find the inverse function f−1(x)f-1(x). Fill in the box with correct expression.

f−1(x)=

(c) Given the function
f(x)={3x+5x<03x+10x≥0f(x)={3x+5x<03x+10x≥0

Calculate the following values:
f(−1)=f(-1)=
f(0)=f(0)=
f(2)=

After solving this equation we get, x = 1.714 or x = 12/7

To solve the given equation.

2/3x + 1 = 5 - 2x

Add 2x on both side.

2/3x + 1 + 2x= 5 - 2x + 2x

7x/3 + 1 = 5

subtract 1 on both side.

7x/3 = 5 - 1

7x/3 = 4

x = 12/7 or 1.714

Therefore, after solving this equation we get, x = 1.714.

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Mary and Benjamin can completely frost a maximum of 215 cupcakes together in one hour.

To find out how many cupcakes Mary and Benjamin can frost together in one hour, we need to determine their individual cupcake-frosting rates and then add them up.

Mary takes 45 minutes to frost 100 cupcakes, so her rate is 100 cupcakes / 45 minutes = 20/9 cupcakes per minute.

Benjamin takes 80 minutes to frost 110 cupcakes, so his rate is 110 cupcakes / 80 minutes = 11/8 cupcakes per minute.

Working together, their combined rate is (20/9 + 11/8) cupcakes per minute.

To convert this rate to cupcakes per hour, we multiply by 60 since there are 60 minutes in an hour:

(20/9 + 11/8) * 60 = (160/72 + 99/72) * 60 = (259/72) * 60 = 3.5972 * 60 = 215.83 cupcakes per hour.

Therefore, Mary and Benjamin can completely frost approximately 215.83 cupcakes together in one hour. Since we can't have a fraction of a cupcake, we round down to the nearest whole number.

Therefore, Mary and Benjamin can completely frost a maximum of 215 cupcakes together in one hour.

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The domain of the composition function f o g(x) can be expressed as (-∞, -4] ∪ (-4, 2) ∪ (2, +∞).

To find the domain of f o g(x), we need to consider two things: the domain of f(x) and the domain of g(x), and find their intersection.

The function f(x) = 1/(x-2) has a restricted domain because the denominator cannot be equal to zero. Thus, x-2 ≠ 0, which means x ≠ 2. So the domain of f(x) is all real numbers except x = 2.

The function g(x) = √(x+4) involves taking the square root of a real number. For the square root to be defined, the expression inside the radical (x+4) must be non-negative. Therefore, x+4 ≥ 0, which implies x ≥ -4. Hence, the domain of g(x) is all real numbers greater than or equal to -4.

To find the domain of f o g(x), we need to find the intersection of the domains of f(x) and g(x). Since f(x) cannot have x = 2 and g(x) must have x ≥ -4, the domain of f o g(x) is the set of real numbers greater than or equal to -4, excluding x = 2. In interval notation, the domain can be expressed as (-∞, -4] ∪ (-4, 2) ∪ (2, +∞).

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The length of the rectangle is 15/16 cm, and the width is 9/16 cm, maintaining the ratio of 5:3 and resulting in a perimeter of 3 cm.

Let's denote the length of the rectangle as L and the width as W. According to the problem, the ratio between the length and width is given as 5:3.

We can express this relationship using the equation:

L/W = 5/3

To find the length and width, we also need to consider the perimeter of the rectangle. The formula for the perimeter of a rectangle is given by:

Perimeter = 2(L + W)

In this case, the perimeter is given as 3 cm. Substituting the values into the equation, we have:

2(L + W) = 3

Now, we have a system of two equations:

Equation 1: L/W = 5/3

Equation 2: 2(L + W) = 3

We can solve this system of equations to find the values of L and W.

From Equation 1, we get:

L = (5/3)W

Substituting this value of L into Equation 2, we have:

2((5/3)W + W) = 3

(10/3)W + 2W = 3

(16/3)W = 3

W = (3 * 3) / 16

W = 9/16

Now, substituting this value of W back into Equation 1, we can find L:

L = (5/3)(9/16)

L = (45/48)

L = 15/16

Therefore, the length of the rectangle is 15/16 cm, and the width is 9/16 cm, maintaining the ratio of 5:3 and resulting in a perimeter of 3 cm.

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Answer:

Step-by-step explanation:

$4.7 billion ..

If demand deposits are moved into money market deposits, M1: ↓ (Decrease), M2: NC (No Change)

When demand deposits are moved into money market deposits, the impact on the monetary aggregates M1 and M2 can be described as follows:

M1 (Money Supply Aggregate 1):

- ↓ (Decrease)

M2 (Money Supply Aggregate 2):

- NC (No Change)

Moving demand deposits into money market deposits results in a decrease in M1. This is because demand deposits are included in M1 as they are considered readily accessible for making transactions. However, money market deposits are not included in M1 as they typically have more restrictions on withdrawals and are less liquid.

On the other hand, the shift from demand deposits to money market deposits does not affect M2. M2 includes a broader range of financial assets, such as savings deposits, time deposits, and money market mutual funds. Money market deposits are already part of M2, so the overall composition of M2 remains unchanged.

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Answer:

6n^2 - 5n + 3

Step-by-step explanation:

To find the nth term rule for the given quadratic sequence, we need to determine the pattern or relationship between the terms. Looking at the sequence:

4, 22, 52, 52, 94, 148,...

We can observe that the first term is 4, the second term is 22 (which is 18 more than the first term), and the third term is 52 (which is 30 more than the second term).

To find the nth term rule, we will first find the differences between consecutive terms:

1st difference: 18, 30, 0, 42, ...

We notice that the 2nd difference (the differences between the differences) is constant, which suggests that the sequence follows a quadratic pattern.

2nd difference: 12, -30, 42, ...

Now, to find the nth term rule, we can use the general form of a quadratic sequence:

an = dn^2 + en + f

By substituting the values of the terms into the equation, we can find the coefficients d, e, and f.

Let's use the first three terms to form three equations:

For the 1st term (n = 1):

4 = d(1)^2 + e(1) + f

4 = d + e + f ...(1)

For the 2nd term (n = 2):

22 = d(2)^2 + e(2) + f

22 = 4d + 2e + f ...(2)

For the 3rd term (n = 3):

52 = d(3)^2 + e(3) + f

52 = 9d + 3e + f ...(3)

Solving these three equations simultaneously will give us the values of d, e, and f.

Subtracting equation (1) from equation (2):

18 = 3d + e ...(4)

Subtracting equation (1) from equation (3):

48 = 8d + 2e ...(5)

Now, subtracting equation (4) from equation (5):

30 = 5d

d = 6

Substituting the value of d into equation (4):

18 = 3(6) + e

e = -5

Substituting the value of d into equation (1):

4 = 6 + (-5) + f

f = 3

Therefore, the nth term rule for this quadratic sequence is:

an = 6n^2 - 5n + 3

The quadratic function is A = x² and the maximum area is 3164.0625 feet²

Given data:

To find the quadratic function that can be used to find the maximum area of the tennis court, we need to express the area of the court as a function of one variable, which we'll call x.

Let's assume that the length of the tennis court is x feet. In that case, the width of the court will also be x feet to maximize the area (since a square shape yields the maximum area for a given perimeter).

The perimeter of the tennis court consists of two lengths and two widths, which adds up to 2x + 2x = 4x. We know that the total fencing available is 225 feet. Therefore, we can set up the equation:

4x = 225

Simplifying the equation, we find:

x = 225/4

x = 56.25

Now, we can express the area of the tennis court, A, as a quadratic function of x:

A = x * x

A = x²

Substituting the value of x we found:

A = (56.25)²

A = 3164.0625

Therefore, the quadratic function that represents the maximum area of the tennis court is A = x², and the maximum area is 3164.0625 square feet.

To find the lengths of the sides of the resulting fence, we know that the length and width of the court are both x. Substituting the value of x:

Length = 56.25 feet

Width = 56.25 feet

Hence, the lengths of the sides of the resulting fence are both 56.25 feet.

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The given LP problem can be solved using graphical and Big M methods. The graphical solution involves plotting the constraints and determining the feasible region and optimal solution. The Big M method involves introducing artificial variables and using a two-phase approach to find the optimal solution. In this case, the LP problem has a unique optimal solution.

(a) To solve the problem graphically, we first plot the constraints on a graph. The first constraint, 3x1 + x2 ≥ 4, can be represented by a line with points (0, 4) and (4/3, 0). The second constraint, 2x1 + x2 ≤ 4, is represented by a line with points (0, 4) and (2, 0). The third constraint, x1 + x2 = 3, is a straight line passing through (3, 0) and (0, 3). The feasible region is the intersection of the shaded regions determined by these three constraints. We can then find the optimal solution by evaluating the objective function at the corners of the feasible region. The maximum value occurs at the corner (4/3, 0), where z = 4/3.

(b) To solve the LP problem using the Big M method, we introduce artificial variables to convert the inequalities into equalities. We then use a two-phase approach to find the optimal solution. In the first phase, we minimize the sum of the artificial variables by adding them to the objective function with a large coefficient (M). The constraints are solved to obtain an initial feasible solution. In the second phase, we remove the artificial variables and solve the modified objective function. In this case, since the initial feasible solution has no artificial variables in the optimal solution, the second phase is not necessary. The optimal solution obtained from the first phase is x1 = 1, x2 = 2, with z = 0.

(c) In the graphical solution, the basic solutions occur at the corners of the feasible region. The basic solution (4/3, 0) is feasible, and the objective function value at this point is z = 4/3. In the Big M method, the basic solution obtained in the first phase is x1 = 1, x2 = 2, which is also feasible. The objective function value at this basic solution is z = 0.

(d) The LP problem does not have alternative optimal solutions because there is only one feasible region and the objective function has a unique maximum value. Therefore, the problem does not suffer from infeasibility or unboundedness.

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Answer:

-2

Step-by-step explanation:

This is the steepness of the line.

Take 2 points (0,5) and (1,3).  If you start at (0,5) you would move down 2 and right one to get to point (1,3)

-2/1 = -2

Helping in the name of Jesus.

Answer: Gradient = -2

Step-by-step explanation:

Gradient = [tex]\frac{y2-y1}{x2-x1}[/tex]

We take to point on the line (1, 3), (2, 1)

=> gradient = [tex]\frac{1-3}{2-1}[/tex] = -2

The region enclosed by y = 5x and y = 3x² is integrated with respect to x. The area of the region is 5/6 square units.

To find the area of the region enclosed by the given curves, we need to determine the limits of integration. First, we set the two curves equal to each other: 5x = 3x². Rearranging, we get 3x² - 5x = 0, which factors to x(3x – 5) = 0. This equation yields two solutions: x = 0 and x = 5/3.

To find the area, we integrate the difference between the upper curve (y = 5x) and the lower curve (y = 3x²) with respect to x over the interval [0, 5/3]. Thus, the area can be found by integrating the expression 5x – 3x² with respect to x over the given interval. Evaluating the integral gives us an area of 5/6 square units.

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The inverse function of f(x) = (5x²) / 9 is given by:

f^(-1)(x) = ±√[(9x) / 5].

To find the inverse of the function f(x) = (5x²) / 9, we'll follow these steps:

Step 1: Replace f(x) with y: y = (5x²) / 9.

Step 2: Swap the x and y variables: x = (5y²) / 9.

Step 3: Solve the equation for y. Let's start by multiplying both sides of the equation by 9 to eliminate the fraction: 9x = 5y².

Step 4: Divide both sides of the equation by 5: (9x) / 5 = y².

Step 5: Take the square root of both sides to solve for y: y = ±√[(9x) / 5].

Therefore, the inverse function of f(x) = (5x²) / 9 is given by:

f^(-1)(x) = ±√[(9x) / 5].

However, it's important to note that the inverse of f(x) is not a function because it fails the horizontal line test.

The square root (√) introduces both positive and negative values, so for any given x, there are two possible y-values. Therefore, the inverse is not a function.

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Answer:

.

Step-by-step explanation:

.............................

The value that is not the same as the other three values will be; (D) sin-260°.

To evaluate the given values:

(A) sin100°

(B) sin 80°

(C) sin-80° = - sin80°

(D) sin-260° = -sin(260° + 360°) = -sin(620°)

Since 360° = 1 full rotation, we can subtract 360° from 620° to get a reference angle between 0° and 360°:

Therefore the trigonometric function are;

620° - 360° = 260°

So,-sin(620°) = -sin260°

Hence, we have:

(A) sin100° ≈ 0.9848

(B) sin 80° ≈ 0.9848

(C) sin-80° = - sin80° ≈ -0.9848

(D) sin-260° = -sin(260° + 360°) = -sin(620°)

Therefore, option (D) sin-260° is not the same as the other three values.

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a. The mean is 11.00 (correct)

b. The mean is 14.70 (incorrect)

c. The median (the 50th percentile) is 7.00 (correct)

d. The median (the 50th percentile) is 10.75 (incorrect)

5, 6, 7, 15, 22

The mean is calculated by adding up all the numbers and dividing by the total count:

Mean = (5 + 6 + 7 + 15 + 22) / 5 = 11

So option a. The mean is 11.00 is correct.

The median (the 50th percentile) is the middle number when the dataset is arranged in ascending order. Since we have an odd number of values, the median is the middle value itself.

So option d. The median (the 50th percentile) is 10.75 is incorrect.

The median of the dataset is 7 since it is the middle number when the dataset is arranged in ascending order.

So option c. The median (the 50th percentile) is 7.00 is correct.

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The measure of an exterior angle of a polygon is always greater than 0 degrees.

By definition, an exterior angle is formed by extending one side of the polygon and the adjacent side. The sum of all exterior angles in any polygon is always 360 degrees.

Therefore, it is not possible for the measure of an exterior angle to be 0 degrees. If the measure of an exterior angle is given as 0 degrees, it would be an invalid or impossible situation.

The measure of the corresponding interior angle would be 180 degrees (180 - 0 = 180), which is the maximum possible measure for an interior angle in a polygon.

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a.  This type of instruction allows for different actions to be taken based on a specific condition.

b. Setting the value of "count" to 0.

c.  It allows for a block of instructions to be repeated multiple times based on a specific condition or until the condition is no longer true.

d. It performs a mathematical calculation using variables and assigns the result to another variable..'

a) Condition/Conditional instruction: "If the mixture is dry, add 1 cup of water." This instruction involves a condition that checks whether the mixture is dry. If the condition evaluates to true, the action of adding 1 cup of water is performed. This type of instruction allows for different actions to be taken based on a specific condition.

b) Assignment/Instruction: "Set the count to 0." This instruction involves assigning a specific value (0) to the variable "count." It is an assignment that assigns a particular value to a variable, in this case, setting the value of "count" to 0.

c) Looping/Iterative instruction: "If y >= x, repeat steps 3-6." This instruction establishes a loop that repeats steps 3-6 as long as the condition (y >= x) evaluates to true. It allows for a block of instructions to be repeated multiple times based on a specific condition or until the condition is no longer true.

d) Arithmetic/Calculation instruction: "Add a to b and place in c." This instruction involves performing an arithmetic operation (addition) by adding the values of variables 'a' and 'b,' and storing the result in variable 'c.' It performs a mathematical calculation using variables and assigns the result to another variable..'

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Conjecture: The pattern in the sequence is that each term is the reciprocal of a power of 2, where the power increases by 1 for each subsequent term.

In the given sequence 1, 1/2, 1/4, 1/8, we can observe that each term is the reciprocal of a power of 2. The first term, 1, is equivalent to 2^0, and the second term, 1/2, is equivalent to 2^(-1), and so on. Therefore, the pattern suggests that each term is the reciprocal of 2 raised to the power of n, where n represents the position of the term in the sequence.

Using this conjecture, we can find the next item in the sequence. Since the last term in the given sequence is 1/8, which is equivalent to 2^(-3), the next term should be the reciprocal of 2 raised to the power of (-3 + 1), which simplifies to 2^(-2). Therefore, the next item in the sequence is 1/4, or 2^(-2).

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The scale factor and the perimeters are = 1/2 and 34 and 17 units.

Given that are two polygons are similar MNPQ ~ XYZW, we need to find the scale factor and the perimeter of each polygon.

Scale factor = ratio of the lengths of the sides of the similar polygons.

Scale factor = MQ / WX = 8/4 = 2

Now,

The perimeter of the polygon XYZW = 10 + 9 + 8 + 7 = 34 units.

We know that the ratio of the perimeters of the similar polygons is equal to the scale factor,

So,

The perimeter of the polygon MNPQ = 34 / 2 = 17 units.

Hence the scale factor and the perimeters are = 1/2 and 34 and 17 units.

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The sketch of the function from 0 to 2π resembles one complete cycle of the sine function, with a period of 4π and an amplitude of 1.

To find the period and amplitude of the function y = sin(2θ/4), we can manipulate the given function to match the standard form of the sine function: y = A sin(Bθ + C), where A is the amplitude, B is the frequency (or the reciprocal of the period), and C is the phase shift.

Comparing the given function to the standard form, we have:

A = 1 (amplitude)

B = 2/4 = 1/2 (frequency)

The amplitude of the function is 1, which represents the maximum vertical distance the graph reaches from the midline.

To find the period, we use the formula T = 2π/B, where B is the frequency. In this case, the frequency is 1/2, so the period is:

T = 2π/(1/2) = 4π

Therefore, the period of the function is 4π, representing the length of one complete cycle of the sine function.

To sketch the function from 0 to 2π, we can plot points for several values of θ within that interval and connect them to form a smooth curve. Since the function is y = sin(2θ/4), we can simplify it to y = sin(θ/2) to make it easier to work with.

Using this simplified form, we can calculate the y-values for various values of θ:

θ = 0: y = sin(0/2) = sin(0) = 0

θ = π/2: y = sin((π/2)/2) = sin(π/4) = √2/2

θ = π: y = sin((π)/2) = sin(π/2) = 1

θ = 3π/2: y = sin((3π/2)/2) = sin(3π/4) = √2/2

θ = 2π: y = sin((2π)/2) = sin(π) = 0

Plotting these points on the graph and connecting them with a smooth curve, we get a graph that starts at 0, reaches a maximum value of 1, returns to 0, reaches a minimum value of -1, and returns to 0 again. The graph repeats this pattern every 4π, which matches the period we calculated.

The amplitude of the function, which is 1, represents the distance from the midline to the maximum or minimum value of the graph.

Therefore, the sketch of the function from 0 to 2π resembles one complete cycle of the sine function, with a period of 4π and an amplitude of 1.

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(a) The inverse of A is:

A⁻¹ = [-2 1 0

1 0 1

1/2 0 1]

(b) The solution to the system of linear equations is x = [-3, 2, 3/2].

To find the inverse of A, we can use Gaussian Elimination. We can start by adding the first row to the second row. This gives us:

```

[1 0 1 4

1 1 2 -3

1 1 3 1]

```

Next, we can subtract the second row from the third row. This gives us:

```

[1 0 1 4

1 1 2 -3

0 0 1 2]

```

Finally, we can divide the third row by 2. This gives us the row reduced form of the matrix:

```

[1 0 0 -2

0 1 0 1

0 0 1 1/2]

```

As we can see, the third column of the row reduced matrix is the inverse of the first column of the original matrix. Therefore, the inverse of A is:

```

A⁻¹ = [-2 1 0

1 0 1

1/2 0 1]

```

We can now use this inverse to solve Ax=b. We can write this equation as:

```

[1 0 1 4

1 1 2 -3

1 -1 1 1] * x = [4 -3 1]

```

Multiplying both sides of this equation by A⁻¹, we get:

```

x = A⁻¹ * b = [-2 1 0

1 0 1

1/2 0 1] * [4 -3 1] = [-3 2 3/2]

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Answer:

Step-by-step explanation:

Orange : apple is 5:2, the person has 27 liters of orange and 9 liters of apple

5:2

10:4

15:6

20:8

22.5:9

22.5+9 = 31.5

C is the answer

The units-of-activity depreciation method calculates depreciation based on the usage or activity level of an asset. So, the units-of-activity depreciation for the year is $13,080.

To calculate the units-of-activity depreciation for the year, we start by subtracting the estimated residual value from the initial cost to determine the depreciable amount: $175,000 - $11,500 = $163,500.

Next, we calculate the depreciation rate per mile by dividing the depreciable amount by the estimated useful life in miles: $163,500 / 30,000 miles = $5.45 per mile.

Finally, we multiply the depreciation rate per mile by the actual miles driven during the year (2,400 miles) to determine the units-of-activity depreciation: $5.45 per mile * 2,400 miles = $13,080.

Therefore, the units-of-activity depreciation for the year is $13,080. This method is suitable for assets whose wear and tear or usage is directly related to the number of units or activities performed. In this case, the truck's depreciation expense is based on the actual miles driven during the year, reflecting its usage and decreasing value based on that usage.

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To conduct the survey and collect information on environmental issues from 50 individuals, you can follow these steps.

1. Prepare a questionnaire on environmental issues: Water depletion, soil degradation, biodiversity, wells, afforestation, chemical fertilizers.

2. Approach 50 individuals for opinions on environmental issues.

3. Record responses and compile data from interviews or surveys.

4. Create a double bar graph with issues on the horizontal axis.

5. Use side-by-side bars for "Agree" and "Disagree" responses.

6. Label the graph with title, axes, and legend.

7. Use colors or patterns to differentiate bars for clarity.

1. Prepare a questionnaire with the list of environmental issues mentioned  -  Depletion in water , Soil degradation , Biodiversity , Construction of Wells , Afforestation , and Chemical Fertilizers.

2. Approach 50 individuals , either through in-person interviews , online surveys , or any other suitable method , and ask them to mark their answers as "Agree" or "Disagree" for each environmental issue.

3. Record the responses for each individual and compile the data.

4. Once you have the data , create a double bar graph to represent the information. Use the horizontal axis to represent the environmental issues (e.g. , Depletion in water , Soil degradation , etc.) , and the vertical axis to represent the count or percentage of respondents.

5. Create two bars side by side for each environmental issue , one representing the count or percentage of individuals who agreed and the other representing those who disagreed.

6. Label the graph appropriately , including a title , axis labels , and a legend to differentiate between the "Agree" and "Disagree" bars.

7. Use different colors or patterns to make the bars visually distinguishable.

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The mean is 1.354 seconds and sample standard deviation is approximately 0.0164 seconds

Let us find the mean which is sum of all the observations by total number of observations.

Mean = (1.36 + 1.04 + 1.22 + 1.29 + 1.48 + 1.55 + 0.97 + 1.18 + 1.13 + 1.32) / 10

= 13.54 / 10

= 1.354

The mean of the data set is 1.354 seconds.

We will use the formula for the sample standard deviation, which involves finding the differences between each value and the mean, squaring those differences, summing them up, dividing by (n-1), and taking the square root.

Deviation = [(1.36 - 1.354)² + (1.04 - 1.354)² + (1.22 - 1.354)² + (1.29 - 1.354)² + (1.48 - 1.354)² + (1.55 - 1.354)² + (0.97 - 1.354)² + (1.18 - 1.354)² + (1.13 - 1.354)² + (1.32 - 1.354)²] / 9

= [0.000036 + 0.000088 + 0.000157 + 0.000068 + 0.000167 + 0.000167 + 0.001409 + 0.000136 + 0.000074 + 0.000073] / 9

= 0.002415 / 9

= 0.0002683

Standard Deviation = √(0.0002683)

= 0.0164

Hence, the sample standard deviation of the data set is 0.0164 seconds.

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The relation between the sides of a triangle is that the sum of the length of the two sides of a triangle is always greater than the other side.

With the help of the triangle inequality theorem,  it can be proved.

We know that,

Triangle has 3 sides.

Let's suppose any triangle ΔABC, where the length of  AB = c, BC=a and AC=b.

We need to prove,

a<b+c or, |BC|<|AB|+|AC|.

We know, that perpendicular is the shortest distance between any vertex to the opposite side.

Draw ΔABC, and extend AC to an external point D such that AB=AD.

Now, |CD|=|AC|+|AD|.

⇒|CD|=|AC|+|AB| [∵As per the drawing AB=AD].

⇒∠DBA<∠DBC[From the picture as ∠DBC = ∠DBA+∠ABD] ...... (i)

⇒∠ADB<∠DBC[For ΔADB, AB=AD. hence, ∠DBA=∠ADB].......(ii)

Again we know that the length of the side of any greater angle is always greater.

⇒|BC|<|CD|

⇒|BC<|AC|+|AB| [From (i) and (ii)]

Hence, we can say the measure of the sum of two sides of a triangle is always greater than the measure of the third side.

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Tangent Half-Angle Identity relates the tangent of an half angle to suitable cosines and sines. Pythagorean identity is a trigonometric identity that relates the sine and cosine using Pythagorean theorem which states that s[tex]sin^{2} + cos^{2} = 1[/tex].

In this case, we have been given the identity as tan A/2=1-cos A/sin A, so after rationalizing the RHS with the numerator, we get:

[tex]\frac{1-cos A}{sin A} * \frac{1+cos A}{1+cos A}[/tex]

[tex]\frac{1 - cos^{2}A }{sin A(1+cos A)}[/tex]

[tex]\frac{sin^{2}A }{sinA(1+cosA)}[/tex]

[tex]\frac{sin A}{1 + cos A}[/tex]

Hence Proved

Therefore, with the help of Tangent Half-Angle Identity and a Pythagorean identity we proved that tan A/2 = sin A / 1 + cos A.

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Jarod's average speed in miles per hour for the last 2 hours of his trip was 70.

To find Jarod's average speed for the last 2 hours of his trip, we can use the formula for average speed, which is total distance divided by total time.

We know that Jarod drove at a speed of 58 miles per hour for the entire 5-hour trip, and for the first 3 hours, he drove at a speed of 50 miles per hour.

To find the total distance, we can multiply the average speed (58 mph) by the total time (5 hours). This gives us 58 mph * 5 hours = 290 miles.

Now, we need to find the total distance covered in the first 3 hours. We can multiply the speed (50 mph) by the time (3 hours). This gives us 50 mph * 3 hours = 150 miles.

To find the total distance covered in the last 2 hours, we can subtract the distance covered in the first 3 hours from the total distance. This gives us 290 miles - 150 miles = 140 miles.

Finally, to find the average speed for the last 2 hours, we divide the total distance (140 miles) by the total time (2 hours). This gives us 140 miles / 2 hours = 70 miles per hour.

Therefore, Jarod's average speed in miles per hour for the last 2 hours of his trip was 70.

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To calculate the number of households that have basic cable TV, we can calculate 85% of the total number of households.

First, let's convert 115.4 million households to a numerical value:

115.4 million = 115,400,000 households

Now, we can calculate the number of households with basic cable TV by multiplying the total number of households by the percentage of households with basic cable TV:

Number of households with basic cable TV = 85% of 115,400,000

Number of households with basic cable TV = 0.85 * 115,400,000

Calculating this gives us:

Number of households with basic cable TV = 97,990,000

Therefore, approximately 97,990,000 households have basic cable TV.

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