Use calculus to find the absolute maximum and minimum values of the function.
f(x) = 2x − 4 cos(x), −2 ≤ x ≤ 0
(a) Use a graph to find the absolute maximum and minimum values of the function to two decimal places.
maximum minimum (b) Use calculus to find the exact maximum and minimum values.
maximum minimum

Answer:

y = 2

Step-by-step explanation:

If you look at the highest point on the original and lowest on the reflection, there is a difference of 6. That is, 5 to -1 is a distance of 6. So, the parallelogram must have been reflected across the midpoint. 6/2 = 3, 5-3 =2

The measure of the side AC is 23. 69

Using the Pythagorean theorem, we have that the square of the longest side of a triangle is equal to the sum of the squares of the other two sides of the triangle.

The other two sides of the triangle are the opposite and the adjacent sides.

This is represented as;

x² = y² + z²

Now, substitute the values

25² = 8² + z²

Find the square values

625 = 64 +z²

collect the like terms

z² = 625 - 64

z² = 561

Now,find the square root of both sides

z =23. 69

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Answer:

To calculate the mean length of the insects found, we can use the following formula:

Mean length = (Sum of lengths) / (Number of lengths)

We can calculate the sum of lengths by adding up the lengths of all the insects and dividing by the number of insects:

Sum of lengths = Length (0) + Length (10) + Length (20)

Sum of lengths = 0 mm + 10 mm + 20 mm

Sum of lengths = 30 mm

Number of lengths = 3

Mean length = Sum of lengths / Number of lengths

Mean length = 30 mm / 3

Mean length = 10 mm

Therefore, the mean length of the insects found is 10 mm.

To use systematic random sampling to select dealers from the list of 35 members, we start with the 5th dealer and then select every 6th dealer thereafter. This means we would select dealers numbered 04, 10, 16, 22, 28, and 34. These 6 dealers would be included in the sample for estimating the mean revenue from dealer service departments.
To perform a systematic random sampling of the 35 Metro Toledo Automobile Dealers Association members, you will start with the 5th dealer and select every sixth dealer after that. The dealers included in the sample are as follows: 5, 11, 17, 23, 29, and 35.

random sample  is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. A simple random sample is an unbiased sampling technique. Simple random sampling is a basic type of sampling.

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The apothem of the polygon, give the area of the dodecagon, would be 3. 69 inches .

The apothem of a polygon can be defined as the radius of its inscribed circle, or as the perpendicular distance between any one of its sides and the center of the shape.

The apothem of a 12 - gon, given the area to be 140 square inches is therefore:

140 = 2 × x ² x tan ( 180 / 2 )

x = 3. 69 inches

In conclusion, the apothem would be 3. 69 inches .

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The probability of a z-score less than -1.48 or greater than 1.48 is approximately 0.1394.

The sample proportion, p-hat, is a random variable that follows a normal distribution with a mean of the true population proportion, p, and a standard deviation of:

[tex]\sqrt{[(p(1-p))/n]}[/tex]

where n is the sample size.

In this case, p = 0.07, n = 220, and the maximum difference allowed between the sample proportion and the population proportion is 0.04. We can write this as:

|p-hat - p| > 0.04

Simplifying this inequality, we get:

p-hat < p - 0.04 or p-hat > p + 0.04

To calculate the probability of either of these events occurring, we need to find the z-scores corresponding to the two cutoff points. Using the formula for the standard error of the sample proportion, we get:

SE = sqrt[(p(1-p))/n] = sqrt[(0.07 * 0.93)/220] = 0.027

The z-score corresponding to a sample proportion that is 0.04 below the population proportion is:

z1 = (p - 0.04 - p) / SE = -1.48

The z-score corresponding to a sample proportion that is 0.04 above the population proportion is:

z2 = (p + 0.04 - p) / SE = 1.48

The probability of either of these events occurring can be found using the standard normal distribution table or calculator. The probability of a z-score less than -1.48 or greater than 1.48 is approximately 0.1394. Therefore, the probability that the sample proportion will differ from the population proportion by more than 0.04 is approximately 0.1394, rounded to four decimal places.

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Answer: y=mx+b

Step-by-step explanation: Y= 15.75x + 1200

Answer: y=mx+b

Step-by-step explanation: Y= 15.75x + 1200

To find the inverse Laplace transform of f(s), we need to first find the partial fraction decomposition of the function.
f(s) = 6s / (s² - 8s + 17) - 1 / (s² - 8s + 17)

To solve for the roots of the denominator, we can use the quadratic formula:
s = (8 ± √(8² - 4(1)(17))) / 2
s = 4 ± j
So the partial fraction decomposition of f(s) is:
f(s) = [A / (s - 4 - j)] + [B / (s - 4 + j)]
To solve for A and B, we can multiply both sides of the equation by the denominator and substitute the roots of the denominator:
6s = A(s - 4 + j) + B(s - 4 - j)
At s = 4 + j:
6(4 + j) = A(j)
At s = 4 - j:
6(4 - j) = B(-j)
Solving for A and B, we get:
A = 3 - j
B = 3 + j
So the partial fraction decomposition of f(s) is:
f(s) = [(3 - j) / (s - 4 - j)] + [(3 + j) / (s - 4 + j)]
Now we can take the inverse Laplace transform of each term using the table of Laplace transforms:
[tex]l^-1{[(3 - j) / (s - 4 - j)]} = e^(4t)cos(t) - e^(4t)sin(t)[/tex]
[tex]l^-1{[(3 + j) / (s - 4 + j)]} = e^(4t)cos(t) + e^(4t)sin(t)[/tex]

So the inverse Laplace transform of f(s) is:
[tex]f(t) = e^(4t)cos(t) - e^(4t)sin(t) - 1 / √13 * e^(4t)sin(t + arctan(3))[/tex]

Therefore, the answer to the question is:
The inverse Laplace transform of f(t) =[tex]l^-1{f(s)} is f(t) = e^(4t)cos(t) -e^(4t)sin(t) - 1 / √13 * e^(4t)sin(t + arctan(3)).[/tex]

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Small integral root of the characteristic equation by inspection; then factor by division5y^(3) - 4y" - 11y' - 2y =0. The solutions to the original equation are: y = -1/5 and y = -1.

To find a small integral root of the characteristic equation, we need to first write the equation in its characteristic form:

r^(3) - (4/5)r^(2) - (11/5)r - (2/5) = 0

Now, we can try plugging in small integers for r until we find a root that satisfies the equation. Trying r = 1, we get:

1^(3) - (4/5)(1)^(2) - (11/5)(1) - (2/5) = 0

This simplifies to:

-2/5 = 0

Since this is not true, we move on to the next integer. Trying r = -1, we get:

(-1)^(3) - (4/5)(-1)^(2) - (11/5)(-1) - (2/5) = 0

This simplifies to:

-2/5 = 0

Again, this is not true. We move on to r = 2:

2^(3) - (4/5)(2)^(2) - (11/5)(2) - (2/5) = 0

This simplifies to:

0 = 0

Since this is true, we have found a small integral root of the characteristic equation: r = 2.

Now, to factor the equation by division, we divide the original equation by (y-2), which gives us:

5y^(2) + 6y + 1 = 0

This can be factored into:

(5y + 1)(y + 1) = 0

Therefore, the solutions to the original equation are:

y = -1/5 and y = -1.

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Answer:In mathematics, the process of determining which metrics to use is slightly different but still follows some basic steps. Here is a summary:

Define the problem: Identify the specific problem or question that needs to be answered mathematically. For example, if the problem is to optimize a manufacturing process, the relevant metrics could be production output and defect rates.

Identify the variables: Determine the variables that are relevant to the problem or question. These variables could include quantities such as time, distance, temperature, or pressure.

Select the appropriate metrics: Choose the metrics that will be used to measure the variables. In mathematics, metrics can be measures of central tendency, variability, correlation, or other mathematical concepts.

Test and refine the metrics: Test the chosen metrics on real-world data to ensure that they are reliable and accurate. Refine the metrics as needed to improve their performance.

Use the metrics to make decisions: Once the metrics have been validated, use them to make data-driven decisions. This could include optimizing processes, predicting outcomes, or identifying patterns in data.

In terms of who would be involved in the process, it would depend on the nature of the problem or question being addressed. Typically, mathematicians, statisticians, data analysts, and subject matter experts would be involved in the process of selecting and refining the appropriate metrics.

Step-by-step explanation:

Area =  width X height = 7 cm X  7cm = 49 cm^2

the solutions to the quadratic equation 3x² - 12x + 7 = 0 are approximately x = 2 + (1/3)√(15) and x = 2 - (1/3)√(15).

what is approximately ?

"Approximately" is used to indicate that a value or quantity is not exact but is close enough to be used or considered as a reasonable estimate. It means that the value given may be slightly higher or lower than the actual value, but it is close enough to be useful for practical purposes.

In the given question,

The correct setup for the quadratic formula for the equation 3x² - 12x + 7 = 0 is:

x = (-b ± √(b² - 4ac)) / (2a)

Where a = 3, b = -12, and c = 7. Substituting these values into the formula, we get:

x = (-(-12) ± √((-12)² - 4(3)(7))) / (2(3))

Simplifying:

x = (12 ± √(144 - 84)) / 6

x = (12 ± √(60)) / 6

x = (12 ± 2√(15)) / 6

x = 2 ± (1/3)√(15)

Therefore, the solutions to the equation 3x² - 12x + 7 = 0 are approximately x = 2 + (1/3)√(15) and x = 2 - (1/3)√(15).

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Note that the equation in terms of y, volume of air in a person's lungs in mL and t, in seconds, to represent the given context is y = 800 cos (π/2 t) + 1600.

The function depicted in the graph exhibits a periodic nature, characterized by its oscillation between two extreme points.

The time period of the function is derived as T = 4 seconds, given that each complete oscillation occurs between one crest and the subsequent crest or from one through and the subsequent through.

The midline of this function is y = 1600 mL, which denotes its average value.

Furthermore, the amplitude of this function equals half the distance between both extreme values, equivalent to A = 800mL. Given that the function has a period of 4 seconds and that its first crest is located at (2.5, 2600), it follows that we may represent this function with an equation expressed as:

y = A Sin (2π/T ( t-  c)) + 1600


Where:

A = 800
T = 4
2.5, 2600 is a crest


Thus,

1000 = -A + 1600
A = 600

Solving for A, we get

c = 2.5 - T/4 = 0.5


replacing those values, we have:

y = 800 sin (π/2 (t - 0.5)) + 1600

This can be simplified further to read

y = 800 cos (π/2 t) + 1600.

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The probability of selecting a multiple of 10 is 2/11 or 0.182, which can also be expressed as 18.2%.

Given that numbers 20 through 30 were written on individual cards and placed in a bag

If you take one card from the bag, we have to find the probability that it will be a multiple of 10

There are two multiples of 10 between 20 and 30, which are 20 and 30.

The total number of cards in the bag is 11 (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30).

Therefore, the probability of selecting a multiple of 10 is 2/11 or 0.182, which can also be expressed as 18.2%.

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Answer:

x>4

Step-by-step explanation:

It seems there is only one possible solution to this very problem.

The experimental probability that the next car in the parking lot will be silver is given as follows:

p = 7/10.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Out of 40 cars in the parking lot, 28 are silver, hence the probability is given as follows:

p = 28/40

p = 7/10.

The complete problem is:

Marco counted 40 cars in the parking lot. 28 were silver. What is the experimental probability that the next car in the lot will be silver?

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The critical points for the function f(x, y) are (0, ∛6) and (0, -∛6). To get the critical points for the function f(x, y) = x^3 * y^3 - 9x^2 - 3y - 8, follow these steps:

Step:1. Compute the partial derivatives with respect to x and y:
  - f_x(x, y) = ∂f/∂x = 3x^2 * y^3 - 18x
  - f_y(x, y) = ∂f/∂y = x^3 * 3y^2 - 3
Step:2. Set both partial derivatives equal to 0 to find critical points:
  - 3x^2 * y^3 - 18x = 0
  - x^3 * 3y^2 - 3 = 0
Step:3. Solve the system of equations:
  For the first equation, factor out 3x:
  - 3x(y^3 - 6) = 0
  So, either x = 0 or y^3 - 6 = 0, which gives y = ±∛6.
For the second equation, factor out 3:
  - 3(x^3y^2 - 1) = 0
  So, x^3y^2 - 1 = 0.
Step:4. Combine the information from the two equations:
  - If x = 0, the second equation becomes -1 = 0, which is not possible.
  - If y = ±∛6, the second equation becomes x^3(6 - 1) = 0, which gives x = 0.
So, the critical points for the function f(x, y) are (0, ∛6) and (0, -∛6).

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The critical points for the function f(x, y) are (0, ∛6) and (0, -∛6). To get the critical points for the function f(x, y) = x^3 * y^3 - 9x^2 - 3y - 8, follow these steps:

Step:1. Compute the partial derivatives with respect to x and y:
  - f_x(x, y) = ∂f/∂x = 3x^2 * y^3 - 18x
  - f_y(x, y) = ∂f/∂y = x^3 * 3y^2 - 3
Step:2. Set both partial derivatives equal to 0 to find critical points:
  - 3x^2 * y^3 - 18x = 0
  - x^3 * 3y^2 - 3 = 0
Step:3. Solve the system of equations:
  For the first equation, factor out 3x:
  - 3x(y^3 - 6) = 0
  So, either x = 0 or y^3 - 6 = 0, which gives y = ±∛6.
For the second equation, factor out 3:
  - 3(x^3y^2 - 1) = 0
  So, x^3y^2 - 1 = 0.
Step:4. Combine the information from the two equations:
  - If x = 0, the second equation becomes -1 = 0, which is not possible.
  - If y = ±∛6, the second equation becomes x^3(6 - 1) = 0, which gives x = 0.
So, the critical points for the function f(x, y) are (0, ∛6) and (0, -∛6).

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Using proportion, Tomas can distribute the juice to approximately 22 friends.

Given that,

Total amount of juice = 3 quarts

Amount of juice distributed to each friend = 1/8 liters

1 liter = 1.057 quarts

1/8 liters = 1.057/8 quarts

              = 0.132125 quart

Amount of juice with each friend = 0.132125 quart

Number of friends who got 0.132125 quart = 1

Number of friends who got 3 quarts = 3 / 0.132125 = 22.7 friends

                                                                                    ≈ 22 friends

Hence 3 quarts of juice can be distributed to 22 friends.

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The complete question in english is given below.

Tomas wants to distribute 3 quarts of orange juice to his friends in 1-eighth-liter glasses.

How many friends can he distribute the juice?

The iterated integral is [tex]\frac{2}{3}\left(1+e^5\right)^{\frac{3}{2}}-\frac{2}{3}[/tex].

What is Integrate?

In calculus, integration is the process of finding the integral of a function. The integral is the inverse of the derivative, and it represents the area under a curve between two points. Integration is a fundamental concept in calculus, and it has many applications in various fields such as physics, engineering, economics, and more.

The integral of a function f(x) over an interval [a, b] is denoted by ∫(a to b) f(x) dx, and it is defined as the limit of a sum of areas of rectangles as the width of the rectangles approaches zero. In other words, it is the sum of infinitely many small areas under the curve.

Integrate with respect to w first, treating v as a constant:

[tex]$$\int_0^{e^v} \sqrt{1+e^v} d w=\left[w \sqrt{1+e^v}\right]_0^{e^v}=e^v \sqrt{1+e^v}[/tex]

[tex]$$2. Integrate the result from step 1 with respect to $\mathrm{v}$ :$$[/tex]

[tex]$$\int_0^5 e^v \sqrt{1+e^v} d v=\left[\frac{2}{3}\left(1+e^v\right)^{\frac{3}{2}}\right]_0^5=\frac{2}{3}\left(1+e^5\right)^{\frac{3}{2}}-\frac{2}{3} .$$[/tex]

Therefore, the value of the iterated integral is

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The total distance traveled over the interval 0 < t < 8 is: 32/3 + 16/3 + 176/3 = 224/3.

To find the displacement over the given interval, we need to integrate the velocity function:

∫v(t)dt = ∫(t^3 - 10t^2 + 24t)dt = (1/4)t^4 - (10/3)t^3 + 12t^2

Now we can evaluate the displacement over the different intervals:

0 < t < 4:

(1/4)(4)^4 - (10/3)(4)^3 + 12(4)^2 = 32/3

4 < t < 6:

(1/4)(6)^4 - (10/3)(6)^3 + 12(6)^2 - [(1/4)(4)^4 - (10/3)(4)^3 + 12(4)^2]

= -16/3

6 < t < 8:

(1/4)(8)^4 - (10/3)(8)^3 + 12(8)^2 - [(1/4)(6)^4 - (10/3)(6)^3 + 12(6)^2]

= 176/3

Therefore, the displacement over the entire interval 0 < t < 8 is:

32/3 - 16/3 + 176/3 = 64/3

To find the distance traveled over the given interval, we need to break down the motion into the different intervals of direction:

0 < t < 4: The particle is moving in the positive direction, so the distance traveled is the same as the displacement, which is 32/3.

4 < t < 6: The particle is moving in the negative direction, so the distance traveled is the absolute value of the displacement, which is 16/3.

6 < t < 8: The particle is moving in the positive direction, so the distance traveled is the same as the displacement, which is 176/3.

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The total distance traveled over the interval 0 < t < 8 is: 32/3 + 16/3 + 176/3 = 224/3.

To find the displacement over the given interval, we need to integrate the velocity function:

∫v(t)dt = ∫(t^3 - 10t^2 + 24t)dt = (1/4)t^4 - (10/3)t^3 + 12t^2

Now we can evaluate the displacement over the different intervals:

0 < t < 4:

(1/4)(4)^4 - (10/3)(4)^3 + 12(4)^2 = 32/3

4 < t < 6:

(1/4)(6)^4 - (10/3)(6)^3 + 12(6)^2 - [(1/4)(4)^4 - (10/3)(4)^3 + 12(4)^2]

= -16/3

6 < t < 8:

(1/4)(8)^4 - (10/3)(8)^3 + 12(8)^2 - [(1/4)(6)^4 - (10/3)(6)^3 + 12(6)^2]

= 176/3

Therefore, the displacement over the entire interval 0 < t < 8 is:

32/3 - 16/3 + 176/3 = 64/3

To find the distance traveled over the given interval, we need to break down the motion into the different intervals of direction:

0 < t < 4: The particle is moving in the positive direction, so the distance traveled is the same as the displacement, which is 32/3.

4 < t < 6: The particle is moving in the negative direction, so the distance traveled is the absolute value of the displacement, which is 16/3.

6 < t < 8: The particle is moving in the positive direction, so the distance traveled is the same as the displacement, which is 176/3.

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The number of data pairs (n), the mean of x (X, the mean of y (y'), the standard deviation of x (s), and the standard deviation of y (sy).

To calculate the correlation coefficient r, you will need the following information: the number of data pairs (n), the mean of x (X), the mean of y (y'), the standard deviation of x (sx), and the standard deviation of y (sy). Use the given formula: r = [1/(n-1)] Σ(xi - X)(yi - y') / (s * sy)

Plug in the values for each variable, calculate the sum, and divide by (n-1). Then, divide the result by the product of the standard deviations (s × sy). Round your final answer to two decimal places. Keep in mind that the correlation coefficient r can range from -1 to 1.

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The solution is: the length VW is 5.

Here, we have,

We are given the length of a line segment VX = 13

We have a point W in the line

The line is divided into two

VX = VW + WX

VX = 13

WX = 8

Hence,

13 = VW + 8

VW = 13 - 8

VW = 5

Therefore, the length VW = 5

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complete question:

Point W is on line segment V X. Given W X = 8 and VX = 13, determine the length VW

Step-by-step explanation:

If the population of ants in a colony doubles every week and there are initially 135 ants, we can use the following formula to model the growth after n weeks using N as the population of ants:

N = 135 * 2^n

Where n is the number of weeks after the initial population count.

For example, after 1 week, the population would be:

N = 135 * 2^1 = 270

After 2 weeks, the population would be:

N = 135 * 2^2 = 540

And so on. This formula assumes that the ants have no natural predators, diseases, or other factors that would limit their growth, and that the rate of doubling is constant over time.

Answer:

540

Step-by-step explanation:

sorry im in a rush bye gtg :D pls give brainliest

Based on the survival experience data provided, the probability that a 75-year-old male will survive to age 80 is 0.769. It is important to note that this probability is based on the survival experience of a total 1,000 males who retire at age 65. However, this data provides a general estimate of the probability of survival based on the sample population studied.

We can calculate the probability that a 75-year-old male will survive to age 80 by using the following formula:

P(Age 75 survives to age 80) = (Number of males surviving at Age 80) / (Number of males surviving at Age 75)

Using the data provided in the table, we can determine that the number of males surviving at Age 80 is 596 and the number of males surviving at Age 75 is 775.

Therefore, the probability that a 75-year-old male will survive to age 80 is:

P(Age 75 survives to age 80) = 596 / 775 = 0.769

Thus, the correct answer is option D. It is important to note that this probability is based on the survival experience of a total 1,000 males who retire at age 65.

This means that there are other factors that may influence an individual's probability of survival, such as their lifestyle habits, medical history, and genetics. However, this data provides a general estimate of the probability of survival based on the sample population studied.

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Complete Question:

The following table shows the survival experience of a total 1,000 males who retire at age 65. Based on this data, the probability that a 75-year old male will survive to age 80 is:

Age - Number of male surviving

65 - 1000

70 - 907

75 - 775

80 - 596

85 - 383

A. 0.596

B. 1-0.596 = 0.404

C. 1-0.775 = 0.225

D. 0.769

The answer of the given question based on probability is ,  option (d) 12.50%, which is just slightly lower than the calculated value.

Probability is measure of likelihood or chance of event occurring. It is number between 0 and 1, where 0 represents impossible event and 1 represents certain event. In other words, the probability of an event happening is the ratio of the number of favorable outcomes to the total number of possible outcomes. Probability theory is  branch of mathematics that deals with study of random phenomena and their analysis, like  flipping of coin or the rolling of dice.

To calculate the expected return for the asset, we need to multiply each return by its corresponding probability and then sum up the results.

Expected return = (0.05 x 0.1) + (0.1 x 0.1) + (0.1 x 0.505) + (0.25 x 0.25) = 0.005 + 0.01 + 0.0505 + 0.0625 = 0.128

Therefore, the expected return for the asset is 12.8%.

The closest option to this answer is (d) 12.50%, which is just slightly lower than the calculated value.

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the experimental probability of rolling a 5 is 18/200 or 9/100, which can be simplified to 0.09 or 9%.

what is experimental probability  ?

Experimental probability is the measure of the likelihood of an event based on actual experimental results or observations, rather than theoretical probabilities. It is calculated by dividing the number of times an event occurs by the total number of trials or observations.

In the given question,

The experimental probability of rolling a 5 can be found by taking the number of times a 5 was rolled and dividing it by the total number of rolls.

From the given data, we can see that a 5 was rolled 18 times out of a total of 200 rolls (sum of frequencies).

Therefore, the experimental probability of rolling a 5 is 18/200 or 9/100, which can be simplified to 0.09 or 9%.

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where the above conditions are given

2.2.1 To calculate the electrical rating of the small front plate of the stove in kilowatt (kW), we need to divide the wattage by 1000:

1900 W / 1000 = 1.9 kW

From the log table, we can see that the geyser uses the most electricity, with a total of 336 kWh used for the week.

To reduce consumption in this area, Mr Thorn could consider using the geyser less frequently or reducing the temperature setting.

the cost of electricity of the refrigerator with freezer is;

total kWh x cost per kWh:

13.9 kWh x R0.945/kWh = R13.13

the total kWh Mr Thorn used when boiling his kettle for the week, we need to convert the time used in minutes to hours:

40 minutes = 40/60 hours = 0.67 hours

Calculating  the total kWh:

1.4 kW x 0.67 hours = 0.938 kWh

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By algebraic manipulation ,  the number of items that can be produced at a $267 production cost per item is approximately **8** or **414.

The term "algebraic manipulation" refers to the changing of algebraic expressions, frequently into a simpler or more manageable form. To get an algebraic expression in the required form, variables must be rearranged and replaced. The expression's value stays the same during this rearrangement. Algebraic manipulation is a skill that is developed via practise and problem-solving.

We can manipulate the variables in algebra to isolate n and utilise it to solve the equation 135n+56000/n = 267.

To do this, one method is to multiply both sides by n, yielding the result 135n² + 56000 = 267n.

The following quadratic equation is obtained by rearranging the terms:

135n² - 267n + 56000 = 0.

We can use factoring or the quadratic formula to resolve this.

n = (267 √(2672 - 4*135*56000)) / (2*135) 8.04 or n 413.7 are the answers.

Accordingly, depending on how many products you wish to create, the number of items that can be produced at a $267 production cost per item is approximately **8** or **414.

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If x is contractible and y is path-connected, then [x, y ] has a single element.

Suppose that x is contractible and y is path-connected. We want to show that [x, y] has a single element.

Since x is contractible, there exists a constant map f: X → {x₀}, where x₀ is some fixed point in x. In other words, f takes every point in x to x₀.

Now, let g: [0,1] → y be any path in y. Since y is path-connected, we can always find such a path.

We can define a map h: [0,1] → [x,y] as follows:

h(t) = (f(1-t), g(t))

Note that h is well-defined, continuous, and takes h(0) = (x₀, g(0)) and h(1) = (x₀, g(1)).

Now suppose that there exist two elements a = (a₁, a₂) and b = (b₁, b₂) in [x, y]. We want to show that a = b.

Since a and b are in [x,y], we have a₁, b₁ ∈ x and a₂, b₂ ∈ y.

Since x is contractible, we have the constant map f: x → {x₀}. So, a₁ and b₁ both map to x₀.

Since y is path-connected, we have the path g: [0,1] → y from a₂ to b₂. Therefore, we can define the map h: [0,1] → [x, y] as:

h(t) = (x₀, g(t))

Note that h is well-defined, continuous, and takes h(0) = a and h(1) = b.

Since [0,1] is connected, the image of h, which is a subset of [x, y], must be connected as well. But the only connected subset of [x, y] with more than one point is the entire interval [x, y] itself. Since h takes distinct endpoints a and b to the same connected subset of [x, y], it must be the case that a = b. Therefore, we have shown that any two elements of [x,y] are equal, which means that [x,y] has a single element.

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