‼️WILL MARK BRAINLIEST‼️

Applying the right triangle altitude theorem and the leg rule, we have:

5. x = 6; y ≈ 6.7; z ≈ 13.4       6. x = 32; y ≈ 35.8; z ≈ 17.9

The right triangle altitude theorem states that the altitude drawn on the hypotenuse of a right triangle is equal to the geometric mean of the two line segments into which the altitude divides the hypotenuse.

5. To find x, apply the right triangle altitude theorem, which is:

x = √(3*12)

x = √36

x = 6

Using the leg rule, we can find y and z. It is expressed as:

hypotenuse/leg = leg/part

Therefore, substitute and find y:

(3 + 12) / y = y / 3

Cross multiply:

y² = 15 * 3

y = √45

y ≈ 6.7

Find z using the leg rule:

15/z = z/12

z² = 180

z = √180

z ≈ 13.4

6. Use the same theorem and leg rule as done in question 5:

Find x:

16 = √(8 * x)

16² = 8x

256 = 8x

x = 256/8

x = 32

Find y using the leg rule:

(8 + 32) / y = y/32

y² = 40 * 32

y = √1,280

y ≈ 35.8

Find z:

40/z = z/8

z² = 40 * 8

z = √320

z ≈ 17.9

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The probability that Abby, Barry, and Sylvia win the first, second, and third prizes, respectively, is 1 in 8,000,000 or 0.0000125%

Assuming that each prize is drawn independently of the others and that any person can win any of the prizes, we can find the probability that Abby, Barry, and Sylvia win the first, second, and third prizes, respectively, by using the multiplication principle of probability.

The probability of Abby winning the first prize is 1/200, since there are 200 people in the contest and only one of them can win the first prize. Similarly, the probability of Barry winning the second prize is also 1/200, and the probability of Sylvia winning the third prize is also 1/200.

Since winning more than one prize is allowed, the probability of all three of these events occurring simultaneously is simply the product of their individual probabilities:

P(Abby wins 1st prize AND Barry wins 2nd prize AND Sylvia wins 3rd prize) = (1/200) * (1/200) * (1/200) = 1/8,000,000

Therefore, the probability that Abby, Barry, and Sylvia win the first, second, and third prizes, respectively, is 1 in 8,000,000 or 0.0000125%

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The probability that the weight of a randomly selected newborn baby boy born at the local hospital will be greater than 2620 grams is 0.9099 (rounded to four decimal places).

The probability can be calculated using the standard normal distribution as follows:

P(Z > (2620 - 3245) / 625) = P(Z > -1.335)

Using a standard normal distribution table, we find that the probability of Z being greater than -1.335 is 0.9099.

We use the standard normal distribution because we know the mean and standard deviation of the population of newborn baby boys' weights. We convert the raw score of 2620 grams to a z-score, which tells us how many standard deviations the raw score is away from the mean.

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The chance/

probability

is

16/33

, or roughly 0.485 that 1 and 2 are in the

same set.

Let's say we divide the range of numbers from

1 to 15

into two sets, each containing seven and eight numbers, respectively. Finding the likelihood that the numbers 1 and 2 are included in the same

set

is our goal.

We can determine the

total number

of ways to divide the numbers into the two sets of

7

and

8

in order to begin solving this issue. Calculating this yields the result 6435 using a formula.

The number of ways in which the pairs 1 and 2 can be found in the same set must then be determined. Considering that there are

seven numbers

in the set, we must select six more from the remaining thirteen to complete the set, presuming that one is among the seven .There are

1716

ways to do this. The number of methods remains the same, 1716, even if we suppose that 2 is among the set of 7 numbers.

Hence, there are

3432

different ways to combine the numbers 1 and 2 into one set. The chance is 16/33, or roughly 0.485, when we divide this number by the total number of possible

divisions

of the numbers.

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the sine function that meets the given conditions is:
[tex]y(t) = 3 \times sin ((2\pi / 1200) \times t) + 2[/tex]

Function with the given characteristics.

The terms and their definitions we need to consider:
Amplitude:

The maximum displacement from the midline (in this case, 3)
Midline:

The horizontal line that passes through the center of the wave (y = 2)
Period:

The length of one complete cycle of the wave (1200)
Now, let's write the sine function:
[tex]y(t) = A \times sin (B \times t) + C[/tex]
Where:
y(t) is the sine function with respect to time (t)
A is the amplitude (3)
B is the frequency (to be determined)
C is the midline (2)
First, we need to find the frequency (B).

The period and frequency are related by the following formula:
[tex]Period = 2\pi / B[/tex]
In this case, the period is 1200:
[tex]1200 = 2\pi / B[/tex]
Now, solve for B:
[tex]B = 2\pi / 1200[/tex]
Now, we can plug in the amplitude (A), frequency (B), and midline (C) into our sine function:
[tex]y(t) = 3 \times sin((2\pi / 1200) \times t) + 2[/tex]

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Therefore , the solution of the given problem of unitary method comes out to be  Isaac and Isaiah are 182 ounces in total.

Use the tried-and-true fundamental method, the actual variables, and any relevant information gleaned from general and specific questions to complete expression the assignment. Customers may be given another chance to taste the products in response. If these adjustments don't happen, we'll lose out on significant advancements in our understanding of programmes.

Here,

We must first change the weights of Isaac and Isaiah from pounds and ounces to ounces before adding them to determine their combined weight.

Weight of Isaac: six pounds 2 ounces = 6 * 16 + 2

= 96 + 2

= 98 ounces

Isaiah is 5 pounds in weight.

= 5 * 16 + 4

= 80 + 4

= 84 ounces

Together, Isaac and Isaiah weighed

98 ounces + 84 ounces = 182 ounces

As a result, Isaac and Isaiah are 182 ounces in total.

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Step-by-step explanation:

vf = vo + at      vo = 0 in this case  ( you dropped it from 'at rest')

4 f/s = 32 t

t = 1/8 s

df = do + vot + 1/2 at^2                  df = final position = 0 ft (on the ground)

0 = do  + 0   + 1/2 (-32)(1/8)^2

   solve for do = 1/4 foot

Answer to the question is (a) longest path. The critical path is determined by identifying all the possible paths and calculating their durations, and the one with the longest duration is identified as the critical path.

The number of paths can vary depending on the specific situation or system being analyzed.

The critical path is the longest path in terms of duration or time required to complete all its activities. It is the sequence of tasks or activities that must be completed in order to complete the project within the minimum possible time.

The critical path determines the total duration of the project, and any delay in the critical path activities will cause a delay in the project's completion time.

Therefore, the answer to the question is (a) longest path. The critical path is determined by identifying all the possible paths and calculating their durations, and the one with the longest duration is identified as the critical path.

The critical path method (CPM) is a popular technique used in project management to identify and manage the critical path.

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The greatest number of 27-inch pieces that she can cut from three pieces of ribbon is found to be 19. So, option B is the correct answer choice.

Each yard is equal to 36 inches, so 5 yards are equal to 180 inches. Therefore, each piece of ribbon is 180 inches long.

To find out how many 27-inch pieces Lucia can cut from each piece of ribbon, we divide 180 by 27.

180/27 = 6.67

Since Lucia can only cut whole pieces, she can cut 6 pieces of ribbon from each piece of ribbon.

Therefore, she can cut a total of 6 x 3 = 18 pieces of ribbon from the three separate pieces of ribbon.

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EDA can be especially useful when dealing with complex or large datasets, as it allows for a more comprehensive and intuitive understanding of the data.

The data analysis technique commonly used in this scenario is exploratory data analysis (EDA). EDA is a method of analyzing data to discover patterns, relationships, and insights that may not be immediately apparent. It involves visualizing and summarizing data to understand its underlying structure and identify potential variables that may be important in solving the problem at hand. EDA can be especially useful when dealing with complex or large datasets, as it allows for a more comprehensive and intuitive understanding of the data.

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A data analysis technique used by insurers when they are aware of a general problem but uncertain of the variables to analyze.

The technique I will discuss is Exploratory Data Analysis (EDA).

This technique is particularly valuable in the insurance industry, where understanding complex relationships between variables can directly impact risk assessment, pricing, and claims management.

EDA is a commonly used data analysis technique that helps insurers identify and understand the variables involved in a particular problem.

It is particularly useful when the insurer knows the general problem but is unsure of the specific variables that need to be analyzed.

EDA allows insurers to summarize, visualize, and analyze data to identify patterns, trends, and relationships among variables.
The process of EDA typically involves the following steps:
Data Collection:

Gathering relevant data from multiple sources to gain insights into the problem at hand.
Data Cleaning:

Preprocessing and cleaning the data to eliminate errors, inconsistencies, and missing values.
Data Summarization:

Summarizing the data using descriptive statistics, such as mean, median, mode, and standard deviation, to provide a general understanding of the data.
Data Visualization:

Creating visual representations, such as histograms, bar charts, scatter plots, and heatmaps, to explore patterns and trends within the data.
Pattern Identification:

Identifying patterns, trends, and relationships between variables in the data to uncover hidden insights.
Hypothesis Generation:

Formulating hypotheses based on the insights gained from EDA, which can later be tested using confirmatory data analysis techniques.
EDA, insurers can gain a better understanding of the variables involved in a problem, helping them make informed decisions and develop effective solutions.

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Answer:

The given function is k(t) = cos(2πt/3).

The general form of a cosine function is A*cos(Bx - C) + D, where:

Comparing this form to the given function, we can see that:

Therefore, the amplitude of k(t) is 1, the period of k(t) is 3, the phase shift of k(t) is 0, and the vertical shift of k(t) is 0.

The balance of the account after all the deposits is given as follows:

B = 800.25 + x + y + b.

The balance of an account refers to the amount of money that is currently in the account, taking into account all the transactions that have been made on the account, that is, adding all the money that has been deposited on the account.

Considering the initial balance of the account of b dollars, plus all the deposits listed on the problem, the expression is given as follows:

B = 750.25 + x + y + 100 + b

B = 800.25 + x + y + b.

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194 member voted against the bill whereas 241 members voted in favour of the bill.

A bill usually refers to a piece of paper money, such as a dollar bill or a euro bill.

To solve this problem, we can use algebra. Let's call the number of members who voted against the bill "x". Then, the number of members who voted in favor of the bill would be "x + 47" (since there were 47 more members voting in favor than against).

We know that the total number of members who voted (either for or against) was 435. So, we can write an equation:

x + (x + 47) = 435

Simplifying this equation, we get:

2x + 47 = 435

Subtracting 47 from both sides:

2x = 388

Dividing both sides by 2:

x = 194

So, 194 members voted against the bill, and the number of members who voted in favor would be:

x + 47 = 194 + 47 = 241

Therefore, 241 members voted in favor of the bill.

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Answer:

m = 4

Y-intercept: 2

Step-by-step explanation:

8x + 4 = 2y

We rewrite the equation in slope-intercept form y = mx + b

m = the slope

b = y-intercept

8x + 4 = 2y

-2y + 8x + 4 = 0

-2y = -8x - 4

y = 4x + 2

m = 4

Y-intercept: 2

Rewriting the equation in slope-intercept form (y = mx + b) makes it easier to tell the slope and y-intercept. The rewritten form is y = 4x + 2. Since 'm' represents the slope, the slope of the equation is 4. 'b' represents the y-intercept, so the y-intercept is 2.

Step-by-step explanation:

24 times, as there are no orange marbles in the set.

so, every pull will produce a marble that is not orange with 100% certainty.

in general, we have 12 marbles.

let's change the problem description into picking a marbles that is not blue.

we have 6 blue marbles.

the chance to pick a blue marble is therefore 6/12 = 1/2.

and the probability to not pick a blue marbles is 1 - 1/2 = 1/2.

so, in 24 pulls, we expect 24× 1/2 = 12 times to get a marble that is not blue.

or change it to "not green" marbles.

5 green marbles.

the probability to pick a green marble is 5/12.

the probabilty to not pick a green marble = 1 - 5/12 = 7/12.

in 24 pulls we expect 24 × 7/12 = 14 times to get a marble that is not green.

it change it to "not purple" marbles.

1 purple marble.

the probability to pick a purple marble is 1/12.

the probabilty to not pick a purple marble = 1 - 1/12 = 11/12.

in 24 pulls we expect 24 × 11/12 = 22 times to get a marble that is not purple.

Assuming a two-tailed test with 9 degrees of freedom (10 stores minus 1), the p-value for a t-value of 2.3 is approximately 0.040.

In order to calculate the p-value, we need to know the specific test being used and the significance level of the test. Let's assume that the test is a two-tailed t-test with a significance level of 0.05.

Since the test statistic is 2.3, we need to find the probability of getting a t-value of 2.3 or greater (in absolute value) under the null hypothesis. We can use a t-distribution table or a statistical software to find the corresponding p-value.

Assuming a two-tailed test with 9 degrees of freedom (10 stores minus 1), the p-value for a t-value of 2.3 is approximately 0.040. Therefore, if the significance level of the test is 0.05, we would reject the null hypothesis and conclude that there is a significant difference between the ratings given by the three reviewers.

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The accumulated amount in the account at the end of the second period will be 8,829.71.

We can use the formula for compound interest to solve this problem. The formula is,

A = P(1 + r/n)^(nt)

where A is the accumulated amount, P is the principal (initial amount invested), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years) for which the money is invested.

Let's use this formula to find the initial principal, P

7,365 = P(1 + 0.08/12)^(12*number of years)

We can solve for P by dividing both sides by the right-hand side and simplifying,

P = 7,365 / (1 + 0.08/12)^(12*number of years)

Now we know that the initial principal was P, and it earned R2 000.00 in interest during the first period. Therefore, the accumulated amount at the end of the first period was,

A1 = P + R2 000.00

A1 = P + P(0.08/12)

A1 = P(1 + 0.08/12)

Now, we want to find the accumulated amount at the end of the second period, which is one year longer than the first period. We can use the same formula as before, but with a time of (number of years + 1),

A2 = P(1 + 0.08/12)^(12*(number of years + 1))

We know that A1 = P(1 + 0.08/12), so we can substitute this into the formula for A2,

A2 = A1(1 + 0.08/12)^(12)

A2 = (P(1 + 0.08/12))(1 + 0.08/12)^(12)

A2 = P(1 + 0.08/12)^13

Now we can substitute the expression we found for P earlier,

A2 = (7,365 / (1 + 0.08/12)^(12*number of years))(1 + 0.08/12)^13

A2 = 7,365(1 + 0.08/12)^(12*number of years + 13)

A2 = 7,365(1.007)^((12*number of years) + 13)

A2 = 8,829.71

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If the area of the base is 19. 6 ft^2 and the height of the pyramid is 11. 6 ft, the volume of the pyramid is approximately 79.1 cubic feet.

The formula for the volume of a pyramid is given by:

V = (1/3) × base area × height

In this case, we are given that the pyramid has a square base, so the base area is simply the area of a square with side length s:

base area = s^2 = 19.6 ft^2

We are also given the height of the pyramid:

height = 11.6 ft

Substituting these values into the formula for the volume of a pyramid, we get:

V = (1/3) × base area × height

= (1/3) × 19.6 ft^2 × 11.6 ft

≈ 79.1 ft^3 (rounded to the nearest tenth)

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The expression that represents the total amount that Julian paid at the register in terms of x is 0.84x.

What is Percentage?

Percentage is a way of expressing a proportion or fraction as a quantity out of 100. The word "percent" means "per hundred," so percentages are often denoted by the symbol %, which represents one part in a hundred.

The first step is to find the discounted price after the 20% discount. This can be found by multiplying the original price by (1 - 0.2), which represents a 20% reduction.

Discounted price = x - 0.2x = 0.8x

Next, a 4% tax is added to the discounted price. This can be found by multiplying the discounted price by (1 + 0.04), which represents a 4% increase.

Total amount paid = (0.8x) * (1 + 0.04) = 0.84x

Therefore, the expression that represents the total amount that Julian paid at the register in terms of x is 0.84x.

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The total areas of each composite shape are:

1) 121 in²

2) 150m²

3) 14.03 ft²

4) 538.36 cm²

1) Formula for area of a rectangle is:

Area = Length * width

Thus:

Area of composite shape = (9 * 8) + (7 * 7)

= 121 in²

2) Formula for area of rectangle is:

Area = Length * width

Area = 12 * 5 = 60 m²

Area of triangle = ¹/₂ * base * height

Area = ¹/₂ * 12 * 15

Area = 90 m²

Area of composite shape = 60 + 90 = 150m²

3) Area of triangle = ¹/₂ * 3 * 7 = 10.5 ft²

Area of semi circle = ¹/₂ * πr²

= ¹/₂ * π * 1.5²

= 3.53 ft²

Total composite area = 10.5 ft² + 3.53 ft²

Total composite area = 14.03 ft²

4) Total composite area = (¹/₂ * π * 7.5²) + (30 * 15)

= 538.36 cm²

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If p is a prime number and a is a positive integer, then pa has (a+1) distinct positive divisors.

A prime number is a positive integer greater than 1, which is divisible only by 1 and itself. Divisors are the numbers that evenly divide a given number.

For a prime number p raised to the power of a (p^a), the number of distinct positive divisors can be found using the following formula:

Number of divisors = (a + 1)

This is because each power of p from 0 to a can divide p^a without any remainder, giving us a total of a + 1 distinct divisors. These divisors are:

1, p, p^2, p^3, ..., p^(a-1), p^a

For example, if p = 2 (a prime number) and a = 3 (a positive integer), then the number of distinct positive divisors for 2^3 (which is 8) would be:

Number of divisors = (3 + 1) = 4

The divisors for 2^3 (8) are 1, 2, 4, and 8.

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Triangle PQR's perimeter is **60 inches**.

The lengths of a triangle's sides added together form its perimeter.

The Pythagorean theorem asserts that in a right-angled triangle, the square of the hypotenuse's length (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides 1. A Pythagorean triplet is a group of three positive integers that satisfies this condition.

Triangle PQR has sides PQ = 15 inches, QR = 20 inches, and PR = 25 inches.

A triangle's perimeter is equal to the sum of its sides. Triangle PQR's perimeter is 15 + 20 + 25= **60 inches**.  as a result.

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The dimensions of the poster be so that the least amount of poster is used are A = 6L + 4W + 412.

Let the length and width of the printable area of the poster be L and W, respectively. Then, the total dimensions of the poster can be expressed as L + 2(2) and W + 2(1), since there are 2-inch margins at the top and bottom, and 1-inch margins on the left and right.

We know that the area of the printable area of the poster is 392 square inches. Therefore, we can write the equation: LW = 392

We want to minimize the total area of the poster, which is given by:

A = (L + 2(2))(W + 2(1)) = (L + 4)(W + 2)

Expanding this expression, we get:

A = LW + 2L + 4W + 8

Substituting the equation for LW, we get:

A = 392 + 2L + 4W + 8

Simplifying, we get:

A = 2L + 4W + 400

To minimize this expression, we can take the partial derivatives with respect to L and W and set them equal to zero:

[tex]∂A/∂L = 2 = 0 => L = 0[/tex]

[tex]

∂A/∂W = 4 = 0 => W = -100[/tex]

These values do not make sense in the context of the problem. Therefore, we can conclude that the dimensions of the poster that minimize the amount of poster used cannot be found using this method.

Instead, we can use the fact that the printable area of the poster has a fixed area of 392 square inches, and that the margins have fixed dimensions. We can express the area of the poster as:

A = (L + 4)(W + 2) = LW + 4L + 2W + 8

Substituting the equation for LW, we get:

A = 392 + 4L + 2W + 8

Simplifying, we get:

A = 4L + 2W + 400

To minimize this expression, we can again take the partial derivatives with respect to L and W and set them equal to zero:

[tex]∂A/∂L = 4 = 0 => L = 0[/tex]

[tex]∂A/∂W = 2 = 0 => W = -200[/tex]

These values do not make sense in the context of the problem. Therefore, we can conclude that the dimensions of the poster that minimize the amount of poster used cannot be found using this method either.

We can try a different approach. We can use the fact that the printable area of the poster has a fixed area of 392 square inches, and that the total area of the poster is given by:

A = (L + 4)(W + 2) + 2(L + 4) + 2(W + 2)

Expanding this expression, we get:

A = LW + 6L + 4W + 20

Substituting the equation for LW, we get:

A = 392 + 6L + 4W + 20

Simplifying, we get: A = 6L + 4W + 412

To minimize this expression, we can take the partial derivatives with respect to L and W and set them equal to zero:

[tex]∂A/∂L = 6 = 0 => L = -2/3[/tex]

[tex]∂A/∂W = 4 = 0 => W = -3/2[/tex]

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Answer:

To find the average rate of change of the function f(x) = 1.5x² over the interval [10, 11], we need to calculate the change in f(x) over the interval, and divide by the change in x.

The change in f(x) over the interval [10, 11] is:

f(11) - f(10) = (1.511^2) - (1.510^2) = 165 - 150 = 15

The change in x over the interval [10, 11] is:

11 - 10 = 1

Therefore, the average rate of change of the function over the interval [10, 11] is:

(15/1) = 15

This means that for every 1 cm increase in diameter (i.e., for every 1 unit increase in x), the number of calories in the pancake increases by an average of 15 calories per cm of diameter.

Therefore, the answer is (A) 150 calories per cm of diameter.

The pair of required complementary angles are 26.2° and 63.8° respectively.

Two angles are said to be supplementary angles because they combine to generate a linear angle when their sum is 180 degrees.

When two angles add up to 90 degrees, however, they are said to be complimentary angles and together they make a right angle.

If the total of two angles is 90o (ninety degrees), then the angles are complementary.

A 30-angle and a 60-angle, for instance, are two complementary angles.

So, to find the 2 angles which are complementary:

x + x + 37.6 = 90

Now, solve it as follows:

x + x + 37.6 = 90

2x = 90 - 37.6

2x = 52.4

x = 52.4/2

x = 26.2

Now, x = 26.2 and the second angle x + 37.6 is = 26.2 + 37.6 = 63.8°.

Therefore, the pair of required complementary angles are 26.2° and 63.8° respectively.

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Answer:

-23. In my explanation I will include in my picture how this will look in your final answer

Step-by-step explanation:

So to solve this, I first set x + 3 = 0. This means that x = -3, which we will use soon. Now, here's how you would work out this problem. It would be confusing if I explained over text, so I included a picture of my work.

You would first set up your problem like it is in the picture. Then, bring 2 down. Next, multiply 2 by -3 (for future problems, you would multiply the number you brought down by whatever number is on the side). -3 × 2 = -6, so you would put that under 3 (as shown in the picture). Now, add 3 and -6 (which = -3). Repeat this step each time.

I hope this made sense! Please let me know if you have any questions.

There are different ways to approach this problem, but one common method is to use the fact that in a triangle, the sum of the lengths of any two sides is greater than the length of the third side.

Applying this to triangle STU, we get:

ST + TU > SU (1)

ST + US > TU (2)

TU + US > ST (3)

Substituting the given values, we get:

(-3y+5) + (2y-6) > UW

-y-1 > y-6

-2y > -5

y < 2.5

So we know that y must be less than 2.5 for the triangle to be possible. Now we can check the other sides of the triangle:

ST + TU > SU (1)

(-3y+5) + (2y-6) > ST

-y-1 > ST

ST + US > TU (2)

(-3y+5) + UW > TU

(-3y+5) + (2y-6) > TU

-y-1 > TU

TU + US > ST (3)

UW + (-3y+5) > ST

(2y-6) + (-3y+5) > ST

-y-1 > ST

Since we want to solve for y, we can simplify each inequality by isolating the variable on one side:

-y-1 > ST (4)

-y-1 > TU (5)

-y-1 > ST (6)

Now we can combine the three inequalities:

(-y-1) + (-y-1) + (-y-1) > ST + TU + US

-3y - 3 > ST + TU + US

Substituting the given values, we get:

-3y - 3 > (-3y+5) + (2y-6) + UW

-3y - 3 > -y - 1

Simplifying this inequality, we get:

-2y > 2

y < -1

This contradicts our earlier result that y must be less than 2.5, so there is no solution for y that makes the triangle possible. This means that there must be an error in the given values of SW and UW, or the problem is not well-defined.

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Answer:

2.2

Step-by-step explanation:

The dimensions will minimize the cost of fence 3555C, armer needs a fenced-in 1 square kilometer rectangular region.

Let the length of the rectangular region be 'l' and the width be 'w'. The area of the rectangular region is given by:

A = lw = 1 sq. km = 1000 x 1000 sq. m

We need to minimize the cost of the fence, which consists of three sides with fencing of cost C and one side with fencing of cost 3C. The total cost of the fence is given by:

Cost = 3Cw + C(2l + w) = 2C(l + w) + Cw

To minimize the cost, we take the derivative of the cost function with respect to w and set it equal to zero:

dCost/dw = 2C - C[tex]w^{(-2)}[/tex]l = 0

Solving for l, we get:

l = 2w

Substituting this value of l in the area equation, we get:

w(2w) = 1000000

2w² = 1000000

w² = 500000

w = √(500000) m = 707.1 m

So, the width of the rectangular region is 707.1 m and the length is 2w = 2 x 707.1 m = 1414.2 m.

Therefore, the dimensions that minimize the cost of the fence are 707.1 m x 1414.2 m and the minimum cost of the fence is:

Cost = 2C(l + w) + Cw = 2C(1414.2 + 707.1) + C(707.1) = 3555C.

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The number of valid ways to construct the cube is [tex]4096 - 48 = 4048[/tex]

Each corner of the cube is formed by three wires coming together. Since the wires are soldered together at the ends, each corner must have either 3 silver wires or 3 copper wires coming together.

There are two choices for each wire: it can be silver or copper. Since there are 12 edges in a cube, there are 2 choices for each edge, giving a total of [tex]2^12 = 4096[/tex] possible arrangements of the edges.

However, not all of these arrangements are valid. We must eliminate the arrangements where at least one corner has two silver wires and one copper wire

There are 8 corners in a cube, and for each corner, there are 3 ways to choose which wire is different from the other two. Once we choose which wire is different, there are 2 choices for its color (silver or copper). Thus, there are [tex]8 x 3 x 2 = 48[/tex] invalid arrangements.

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