If X has an exponential distribution with parameter , derive a general expression for the (100p)th percentile of the distribution. Then specialize to obtain the median.

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 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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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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Answer:

D

Step-by-step explanation:

The circle has center (-3, 2) and radius 5. The line is a vertical line passing through the point (-8, 0).

We can see from the graph that the line does not intersect the circle at any point. Therefore, the correct answer is:

D. The line does not pass through the circle.

The poster dimensions are 18 inches by 39 inches, or (18, 39).

We know that Megan cuts out a rectangle with dimensions,
The remaining paper is used to wrap an 11-inch cube with no paper wasted, and there are 3 square inches left over.

Multiplying by 6 to eliminate fractions, we get:
6ab - ab - 15a - 2b + 180 = 7980
Rearranging and factoring, we get:
(5a - b)(a - 6) = 7640
Since b>a, we know that 5a-b must be positive, and thus a-6 must also be positive.

Therefore, we can set up a system of equations:
5a - b = x
a - 6 = [tex]\frac{7640}{x}[/tex]
Solving for a and b, we get:

b = 5a - x
We know that the difference between the length and width of the poster is less than 50, so we can set up an inequality:
The remaining wrapping paper has an area equal to the original poster area minus the cutout area.

We know that the remaining paper is used to wrap an 11-inch cube, which has a surface area of 6(22)=726 square inches, and there are 3 square inches leftover.
Now, we are given that b>a and b-a<50. We can start by testing different integer values for a and b within the given range to see which pair of values satisfies the equation.
After testing values, we find that (a, b) = (18, 39) is the solution that satisfies the equation and the given constraints.

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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

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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The text is asking for the slope-intercept form of a linear equation that passes through two given points, (-5,5) and (4,-1). The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. To find the slope, we use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the given points. Once we find the slope, we can use it to write the equation in point-slope form: y - y1 = m(x - x1). Finally, we can rearrange the point-slope form to get the slope-intercept form.

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This result holds for any two harmonic functions f and g, and any closed surface S.

The flux of the vector field F through any closed surface S is zero, we can apply the divergence theorem:

∫∫_S F · n dS = ∫∫∫_V ∇ · F dV

So, n is the outward unit normal vector to the surface S, and V is the volume enclosed by S.

Let's compute the divergence of F:

∇ · F = ∇ · (f ∇g) - ∇ · (g ∇f)

= ∇f · ∇g + f ∇²g - ∇g · ∇f - g ∇²f

= f ∇²g - g ∇²f

So, we used the identities ∇ · (fG) = f ∇ · G + ∇f · G and ∇ · (∇f) = ∇²f.

Since both f and g are harmonic functions, we have ∇²f = ∇²g = 0, so ∇ · F = 0. Therefore, the flux of F through any closed surface S is zero:

∫∫_S F · n dS = ∫∫∫_V ∇ · F dV = ∫∫∫_V 0 dV = 0

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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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Part 1: Yes, all squares have 2 pairs of parallel sides.

Part 2: No, not all rectangles have 4 congruent sides.

Part 1: This is because all squares are rectangles, and all rectangles have 2 pairs of parallel sides. Additionally, since all four sides of a square are congruent, this means that the two pairs of sides are also congruent, making them parallel.

Part 2: While all squares are rectangles with 4 congruent sides, rectangles can have two pairs of parallel sides that are not congruent. For example, a rectangle with a length of 5 units and a width of 3 units has two pairs of parallel sides, but they are not congruent.

One pair of sides is longer than the other pair. Therefore, the fact that all squares are rectangles with 4 congruent sides does not mean that all rectangles have 4 congruent sides.

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The area of the region that is bounded by the curve r=5sin(θ) and lies in the sector 0≤θ≤π is 25/2 square units.

To find the area of the region that is bounded by the curve r=5sin(θ) and lies in the sector 0≤θ≤π, we can use the formula for the area of a polar region:

A = 1/2 ∫(b,a) r(θ)² dθ

where a and b are the values of θ that define the region.

In this case, the region is defined by 0 ≤ θ ≤ π and r(θ) = 5[tex]sin(\theta)^{(1/2)}[/tex], so we have:

A = 1/2 ∫(π,0) [5[tex]sin(\theta)^{(1/2)}[/tex]]² dθ

Simplifying the integrand:

A = 1/2 ∫(π,0) 25sin(θ) dθ

Using the identity ∫sin(θ)dθ = -cos(θ), we get:

A = 1/2 [-25cos(π) + 25cos(0)] = 25/2

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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.

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 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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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 optimal solution is to cut the wire into two pieces, one forming a square with side length x = 2/π feet, and the other forming a circle with radius r = (1 - 4/π)/π feet.

A two-dimensional figure's area is the amount of space it takes up. In other terms, it is the amount that counts the number of unit squares that span a closed figure's surface.

Let's first start by noting the formulas for the area of a circle and the area of a square in terms of their radius/length:

Area of circle = πr²

Area of square = x²

We also know that the total length of the wire is 2 feet, so the sum of the circumference of the circle and the perimeter of the square must equal 2:

Circumference of circle = 2πr

Perimeter of square = 4x

2πr + 4x = 2

Simplifying this equation, we get:

πr + 2x = 1

We want to maximize the sum of the areas of the circle and square, which is given by:

πr² + x²

We can use the equation we just derived to eliminate r from this expression:

π(1 - 2x)²/4 + x²

Expanding and simplifying this expression, we get:

(π/4)x² - πx + π/4

To find the value of x that maximizes this expression, we need to take the derivative with respect to x and set it equal to zero:

d/dx [(π/4)x² - πx + π/4] = (π/2)x - π = 0

Solving for x, we get:

x = 2/π

Now we can use the equation we derived earlier to find the corresponding value of r:

πr + 2x = 1

πr + 4/π = 1

πr = 1 - 4/π

r = (1 - 4/π)/π

So, the optimal solution is to cut the wire into two pieces, one forming a square with side length x = 2/π feet, and the other forming a circle with radius r = (1 - 4/π)/π feet.

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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

There are no values of c in the interval where f'(c) = 0. Therefore, the answer is C = (no values)

To use Rolle's theorem, we need to check that the function is continuous on the closed interval [-4,2] and differentiable on the open interval (-4,2). Both conditions are satisfied by f(x) = (2x³ + 21x - 2)/2, so we can proceed with finding the values of c where f'(c) = 0.

First, we find the derivative of f(x):
f'(x) = 6x² + 21/2

Next, we set f'(x) = 0 and solve for x:
6x² + 21/2 = 0
6x² = -21/2
x² = -7/4
x = ±√(-7/4) = ±(i√7)/2

Since these values are not in the given interval (-4,2], we conclude that the correct option is C.

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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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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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Step-by-step explanation:

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

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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The system of inequalities is solved and the solutions are ( -5, - 2 )

Given data ,

Let the first inequality be A , 3x+2y≥-19

Let the second inequality be B , x+3y<-11

On simplifying , we get

The solution to the inequality is the point of intersection of the graph

Now , on plotting the graph , we get

The point of intersection is P ( -5 , -2 ) and the solution is point P

And , the points which are not a solution is Q ( 1 , -5 ) , ( 0 , 0 )

Hence , the system of inequalities are solved

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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

Answer:

(-4, -3)

Step-by-step explanation:

When a point is reflected in the x-axis, the x-coordinate does not change, and the y-coordinate becomes negative:

Therefore, if point P(3, 4) is reflected in the x-axis, then:

When a point is rotated 90° clockwise about the origin, it produces a point that has the coordinates (y, -x):

Therefore, if point P'(3, -4) is rotated 90° clockwise about the origin, then:

The endpoint of the line segment is (43, 58).

To find the endpoint of the line segment, we need to first find the coordinates of the point that divides the line segment in a 1:5 ratio.

Let's call the endpoint we're looking for (x, y).

We know that the point (13, 14) divides the line segment into two parts with a ratio of 1:5. This means that the distance from (8, 19) to (13, 14) is one-sixth of the distance from (8, 19) to (x, y).

Using the distance formula, we can calculate the distance between the two points:

distance between (8, 19) and (13, 14) = [tex]\sqrt{(13-8)^{2}+(14-19)^{2} }[/tex] = [tex]\sqrt{74}[/tex]

We also know that the distance from (8, 19) to (x, y) is six times the distance from (8, 19) to (13, 14). So:

distance between (8, 19) and (x, y) = 6 * distance between (8, 19) and (13, 14)

= 6 * [tex]\sqrt{74}[/tex]

Now we can use the midpoint formula to find the coordinates of the point that divides the line segment in a 1:5 ratio:

midpoint = ((1/6)*x + (5/6)*13, (1/6)*y + (5/6)*14)

= ((x+65)/6, (y+70)/6)

We know that the midpoint of the line segment is (13, 14), so:

(x+65)/6 = 13 and (y+70)/6 = 14

Solving for x and y, we get:

x = 43 and y = 58

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Answer: The missing sides measure 3.1 and 5.2cm.

Step-by-step explanation: If the longest side of the triangle measured 6.2cm, and was twice as long as the shortest side, we should divide 6.2 by 2. Our shortest side is 3.1cm and our longest 6.2cm. The perimeter is 14.5, so we add up 6.2 and 3.1 to get 9.3, and then subtract that from 14.5. 14.5 - 9.3 = 5.2, so all the sides are 3.1, 5.2, and 6.2cm. We can be sure that this is the answer when we add 3.1 + 5.2 + 6.2 it equals 14.5. The sides of a scalene triangle all have different sides, so this is correct. If you give your answer choices I can tell you which is correct also.

ProjW(v1) = [(v1 · [-y+z, y, z]) / ([1, 1, -1] · [1, 1, -1])] [-y+z, y, z]

= [(1/3)(-y+z)] [-y+z, y, z]

= [-y^2/3+y*z/3, y

To find the orthogonal complement of W, we need to find all vectors in R^3 that are orthogonal (perpendicular) to every vector in W.

Let v = [x, y, z] be a vector in R^3. To be orthogonal to W, v must be orthogonal to every vector in W, so we need to find a condition that determines which vectors in R^3 satisfy this requirement.

A vector in W is of the form [x, y, z] = [-y+z, y, z], since x+y-z=0 implies x=-y+z. The dot product of v with [x, y, z] is:

v · [-y+z, y, z] = (-x(y-z) + y^2 + z^2)

For v to be orthogonal to every vector in W, this dot product must be zero for every choice of x, y, and z. In particular, it must be zero for x = y = z = 0. Therefore, we have:

v · [-y+z, y, z] = (-x(y-z) + y^2 + z^2) = 0

This simplifies to:

y^2 - x(y-z) - z^2 = 0

This is a quadratic equation in y, with coefficients that depend on x and z. For this equation to have a solution for every choice of x and z, the discriminant must be non-negative:

x^2 + 4z^2 ≥ 0

This condition is satisfied for all x and z, so the orthogonal complement of W is the set of all vectors v = [x, y, z] in R^3 that satisfy the equation:

y^2 - x(y-z) - z^2 = 0

To find a basis for W⊥, we can use the Gram-Schmidt process to orthogonalize the standard basis vectors e1 = [1, 0, 0], e2 = [0, 1, 0], and e3 = [0, 0, 1] with respect to W. Any resulting vectors that are linearly independent will form a basis for W⊥.

Starting with e1, we need to find the projection of e1 onto W, which is:

projW(e1) = [(e1 · [-y+z, y, z]) / ([1, 1, -1] · [1, 1, -1])] [-y+z, y, z]

= [(1(1)+0(-1)+0(1)) / (1+1+1)] [-y+z, y, z]

= (1/3)[-y+z, y, z]

= [-y/3+z/3, y/3, z/3]

Then, we subtract this projection from e1 to get a vector that is orthogonal to W:

v1 = e1 - projW(e1) = [1, 0, 0] - [-y/3+z/3, y/3, z/3] = [1+y/3-z/3, -y/3, -z/3]

Next, we apply the same process to e2 and e3, using the previous vectors as the new basis for the subspace:

projW(v1) = [(v1 · [-y+z, y, z]) / ([1, 1, -1] · [1, 1, -1])] [-y+z, y, z]

= [(1/3)(-y+z)] [-y+z, y, z]

= [-y^2/3+y*z/3, y

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