If P --> Q is an existing functional dependency, which of the following is NOT an augmented functional dependency? OPA --> Q O P. X. Y --> Q OP.X --> O Q -->P OP.A-->P

Answer:

X=8

Bcuz 8x=64

64÷8=8

So that the answer is 8

Answer:

[tex]\bf x=8[/tex]

Step-by-step explanation:

[tex]\bf 8x=64[/tex]

Divide both sides by 8:

[tex]\bf \cfrac{8x}{8}=\cfrac{64}{8}[/tex]

Simplify:-

[tex]\bf x=8[/tex]

______________________

Hope this helps!

Answer:

4.75 liters of punch

Step-by-step explanation:

2.8 + 0.75 = 3.55

3.55 + 1.2 = 4.75 liters of punch

Answer:

  (a) AC = 4√2 cm

  (b) AM = 2√2 cm

  (c) EM = √41 cm

  (d) EF = 3√5 cm

Step-by-step explanation:

You want to solve for various lengths in the right square pyramid shown with base edge 4 cm and lateral edge 7 cm.

Each right triangle can be solved for unknown lengths using the Pythagorean theorem: the square of the hypotenuse is the sum of the squares of the other two sides.

Right triangles of interest here are ...

  ADC . . . . for finding AC and AM (isosceles right triangle)

  CME . . . . for finding EM

  FME . . . . for finding EF

AC is the hypotenuse of ∆ADC, so ...

  AC² = AD² +DC²

  AC = √(4² +4²)

  AC = 4√2 . . . . cm

M is the midpoint of AC, so ...

  AM = AC/2 = (4√2)/2

  AM = 2√2 . . . . cm

FM is half the length of one side of the base, so is 2 cm. CM = AM = 2√2.

  CE² = CM² +EM²

  EM = √(CE² -CM²) = √(7² -(2√2)²)

  EM = √41 . . . . cm

EF is the hypotenuse of ∆EMF.

  EF² = EM² +FM²

  EF = √(EM² +FM²) = √(41 +2²) = √45

  EF = 3√5 . . . . cm

The 1.5 liters of fruit juice is cheaper per liter compared to the 1 liter of fruit juice.

For the 1 liter of fruit juice, the cost per liter is R25/1 liter = R25/liter.

For the 1.5 liters of fruit juice, the cost per liter is R34/1.5 liters = R22.67/liter.

A supermarket is a large retail store that offers a wide range of food and household items to consumers. These stores are typically designed to be a one-stop shop for customers, allowing them to purchase everything they need in one convenient location. One of the primary advantages of shopping at a supermarket is the ability to choose from a wide variety of products at competitive prices.

Supermarkets usually have multiple departments, including a fresh produce section, a meat and seafood section, a bakery, and a deli. In addition to food items, supermarkets also offer a variety of household goods, such as cleaning supplies, personal care products, and pet food. Many supermarkets also offer loyalty programs, coupons, and other incentives to help customers save money.

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Answer:

x = +8

x = -8

That's the thing that I only know. Im a seventh grader(turned 13) so I wouldn't know this fully. Ask someone else to get it fully.

Step-by-step explanation:

5x + 6x - 10 + 12 = 90

11x = -88

x = -8

or

x = 8

The greatest possible number of different types of this caterpillar that could exist is 120.

If we assume that the order of the parts does not matter and that all caterpillars with the same color arrangement are considered identical, we can use combinations to find the number of different types of caterpillars that could exist.

First, we need to choose 2 out of the 5 parts to be green, which can be done in 5 choose 2 ways:

5 choose 2 = (5!)/(2!(5-2)!) = 10

For each green-yellow arrangement there are;

3! ways to permute the yellow parts and

2! ways to permute the green parts.

Therefore, the total number of different types of caterpillars is:

10 × 3! × 2! = 10 × 6 × 2 = 120

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The work needed to bring two point charges from infinity to their current positions is equal to the Coulomb potential energy of the system, given by: U = kQ1Q2/r, where k is the Coulomb constant, Q1 and Q2 are the two point charges, and r is the distance between them.

The electric potential due to the two charges at the vertex p is given by the sum of the potentials of each charge individually. V = kQ1/r1 + kQ2/r2, where k is the Coulomb constant, Q1 and Q2 are the two point charges, and r1 and r2 are the distances between the charges and the vertex p.The direction of the electric field due to the two charges at the vertex p is given by the vector sum of the electric field of each charge individually. The direction of the electric field due to each charge can be calculated by taking the vector difference between the position of the charge and the position of the vertex p.

The magnitude of the electric field due to the two charges at the vertex p is given by the sum of the magnitudes of the electric fields of each charge individually. Etotal = √(E12 + E22), where E1 and E2 are the electric fields due to the two charges.

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So the transformed rectangle has vertices A'(40, 160), B'(40, 130), C'(80, 130), and D'(80, 160).

In mathematics, a transformation rule (also known as a transformation function, transformation formula, or simply a transformation) is a mathematical rule or formula that describes how to map or transform a set of points in one coordinate system to another set of points in a different coordinate system. Transformations can be applied to various mathematical objects, such as points, lines, curves, shapes, or functions, and can be used to achieve various purposes, such as to change the size, shape, position, or orientation of an object, to create a mirror image or a rotation of an object, or to change the coordinate system of an object.

Here,

There appears to be an error in the coordinates of points B and D that you provided, as they are both (-90,25). I will assume that the correct coordinates of point D are (-50,55) to form a rectangle ABCD.

To apply the given transformation rules to each of the four vertices of the rectangle, we add 130 to the x-coordinate and 105 to the y-coordinate. Therefore, the coordinates of the transformed rectangle are:

A' = (-90 + 130, 55 + 105) = (40, 160)

B' = (-90 + 130, 25 + 105) = (40, 130)

C' = (-50 + 130, 25 + 105) = (80, 130)

D' = (-50 + 130, 55 + 105) = (80, 160)

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Complete question:

Please help! The transformation rules says: (x, y)→(x + 130, y + 105 )  

Given rectangle ABCD with A(-90,55) B(-90,25) C(-50,25) D(-50,25).

Find the coordinates of transformed rectangle.

The z-score for the 25th percentile of a standard normal distribution is approximately -0.625. Here option A is the correct answer.

To find the z-score for the 25th percentile of a standard normal distribution, we need to use a standard normal distribution table or calculator. The 25th percentile corresponds to a cumulative area under the standard normal curve of 0.25.

Using a standard normal distribution table or calculator, we can find that the z-score corresponding to a cumulative area of 0.25 is about -0.68. This means that approximately 25% of the area under the standard normal curve lies to the left of -0.625.

So, among the given options, the correct answer is Option A, -0.625, Option D, -0.50, which is also incorrect. Option E, 0.00, is definitely incorrect because the 25th percentile is to the left of the mean.

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The probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is 0.6827.

Here the given data are

mean [tex]\mu = $380[/tex],

standard deviation [tex](\sigma) = $50[/tex]

Let X be the random variable which denotes the amounts of money spent on clothing in a year by college students. The distribution of X is normal distribution.

We need to find the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year. If we have the standard normal distribution, we can easily calculate the probability from the normal distribution table. Otherwise, we have to use the standard normal distribution and convert the values to standard units. This process is called standardization. We will use the z-score formula for standardization.

Let’s standardize the given values.

Lower value [tex](X_1) = $300[/tex]

Upper value [tex](X_2) = $400[/tex]

Population mean [tex](\mu) = $380[/tex]

Population standard deviation [tex](\sigma) = $50z_1 = (X_1 - \mu) / \sigma z_1 = ($300 - $380) / $50z_1 = -1.6z_2 = (X_2 - \mu) / \sigma z_2 = ($400-$380)/$50z_2 = 0.4[/tex]

Now we need to find the area between these two z-scores using the standard normal distribution table.The probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is

[tex]P(-1.6 < Z < 0.4).P(-1.6 < Z < 0.4) = P(Z < 0.4) - P(Z < -1.6)\\P(Z < 0.4) = 0.6554\\P(Z < -1.6) = 0.0548\\P(-1.6 < Z < 0.4) = 0.6554 - 0.0548 = 0.6006[/tex]

Therefore, the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is 0.6006 (approx.) or 0.6827 (approx.).

Hence, the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is 0.6827.

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It is Arithmetic Sequence. An ordered group of numbers with a shared difference between each succeeding term is known as an arithmetic sequence.

For the given sequence

d= 49-56 = -7

d= 42-49 = -7

Thus, there is -7 as a common difference between the terms.

The distance between succeeding terms in an arithmetic series is always the same. It is often referred to as an arithmetic series or arithmetic progression. The following statement can be used to represent an arithmetic sequence: a, (a + d), (a + 2 d), (a + 3 d),..., where a is the first term and d is the constant difference between values.

To determine the sum of an arithmetic sequence, it is generally simple to add or subtract all the terms in a short series together. An individual can quickly determine the sum of an arithmetic series for a particular number of terms by using the generic formula for the nth term of an arithmetic sequence.

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The total surface area of the remaining portion is approximately 23.14 in.

To find the total surface area of the remaining portion of a wooden sphere after two equal caps have been removed, use the formula SA = 4πr2. A sphere is symmetrical, and thus, the diameter of the wooden sphere is equal to the diameter of the remaining portion. The radius of the remaining portion is equal to half the diameter of the sphere minus the radius of the cap.

The diameter of the wooden sphere is 11.8 in. As such, the radius of the sphere is 5.9 in. If two equal caps are removed, the diameter of the remaining portion is equal to 11.8 in - 2x R_cap, where R_cap is the radius of the cap. Since the caps are equal, we can simplify the formula to

D = 11.8 - 2R_cap. R_cap is equal to the radius of a circle with area equal to the surface area of one cap. As such, we can use the formula SA = 2πrh + πr2 to find the surface area of the cap. We know the diameter of the sphere is 11.8 in. Thus, the radius of the sphere is 5.9 in. We also know that the height of the cap is 5.9 in. Since the caps are equal, we can use the formula to find the surface area of one cap and multiply by 2 to get the total surface area of both caps.

SA_cap = 2π(5.9 in)(5.9 in) + π(5.9 in)

2SA_cap = 2π(34.84 in2) + π(34.84 in2)

SA_cap = 2π(34.84 in2) + 109.45 in2SA_cap ≈ 219.74 in

Since the surface area of the cap is equal to 219.74 in, we can use the formula to find the radius of the cap.

219.74 in = 2πrh + πr22(219.74 in2)

= 2π(5.9 in)h + π(5.9 in)22(219.74 in2)

= 37.699 in2 + 109.45 in23r2

= 72.533 in2r ≈ 4.545 in

Using the formula D = 11.8 - 2R_cap, we can find the diameter of the remaining portion of the wooden sphere.

D = 11.8 - 2(4.545 in)D ≈ 2.71 in

The radius of the remaining portion of the wooden sphere is equal to 5.9 in - 4.545 in. Thus, the radius of the remaining portion of the sphere is 1.355 in. Finally, we can find the total surface area of the remaining portion of the sphere.

SA = 4πr2SA = 4π(1.355 in)2SA ≈ 23.14 in

Therefore, the total surface area of the remaining portion is approximately 23.14 in.

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Answer:

To rewrite the quadratic equation in the form y = a(x - p)²+q, we need to complete the square.

y = 2x^2 + 16x + 26

y = 2(x^2 + 8x) + 26

y = 2(x^2 + 8x + 16 - 16) + 26 // Adding and subtracting (8/2)^2 = 16 inside the parentheses

y = 2((x + 4)^2 - 16) + 26

y = 2(x + 4)^2 - 32 + 26

y = 2(x + 4)^2 - 6

Therefore, the quadratic equation y = 2x ^ 2 + 16x + 26 rewritten in the form y = a(x - p)²+q is y = 2 * (x + 4) ^ 2 - 6, so the answer is D

Answer:

y= 2(x+4)^2 -6

Step-by-step explanation:

y= 2x^2 + 16x + 26

It is in the form y= ax^2 + bx + c

To rewrite in the form y=a(x-p)^2 + q

We need to fin p and q. We already have a in the original equation.

In y= 2x^2 + 16x + 26,    a=2.

The formula say that:  p=-b/2a

p= -16/(2*2)

p=-16/4

p=-4

In the formula, we replace a and  y= 2(x-(-4))^2 +q

Obtaining, y= 2 (x+4)^2 + q

Now, to find q we need to obtain a point from the original equation. Commonly the y-intercept. In the form y= ax^2 + bx + c ; C is the y-intercept.

y-intercept: (0,c)

Therefore, in y= 2x^2 + 16x + 26

y-intercept: (0,26)

In the equation we already have:

y= 2(x+4)^2 +q

26= 2(0+4)^2 + q

26=2(4)^2 +q

26= 2(16) + q

26= 32 + q

-6 = q

Joining all the results, we obtain:

y= 2(x+4)^2 -6

Given:

The inequality is

[tex]-3(2x - 5) < 5(2 - x)[/tex]

To find:

The correct representations of the given inequality.

Solution:

We have,

[tex]-3(2x - 5) < 5(2 - x)[/tex]

Using distributive property, we get

[tex]-3(2x)-3(-5) < 5(2)+5(-x)[/tex]

[tex]-6x+15 < 10-5x[/tex]

Therefore, the correct option is C.

Isolate variable terms.

[tex]15-10 < 6x-5x[/tex]

[tex]5 < x[/tex]

It means, the value of x is greater than 5.

Since 5 is not included in the solution set, therefore, there is an open circle at 5.

So, the graphical represents of the solution is a A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.

Therefore, the correct option is D.

In monopolistic competition, the end result of entry and exit is that firms end up with a price that lies on the downward-sloping portion of the average cost curve.

Monopolistic competition is a market condition in which many small firms compete with each other by selling slightly varied, but essentially comparable goods or services at somewhat different prices. These companies enjoy some market power, but they are not monopolies because their products or services are close substitutes for each other.

The equilibrium price in a monopolistically competitive market is a long-run, but not a short-run, outcome of entry and exit. Because the market is monopolistic, entry and exit do not have an immediate impact on the price; it simply alters the number of producers operating in the market. Over time, the entry and exit of producers in the industry will increase or decrease the number of substitutes available, driving demand curves and resulting in the price of the commodity settling on the down-sloping portion of the average cost curve in the long run.

Therefore, it can be concluded that the end result of entry and exit in monopolistic competition is that companies end up with a price that lies on the downward-sloping portion of the average cost curve.

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The estimated probability of a success is 40% since the numbers 1 and 2 represent a success out of the integers 1 to 5. Therefore, the success outcomes (1 and 2) make up 2 out of 5 possible outcomes, or 40%.

To find the estimated probability of a success, we need to determine the proportion of successes in the generated list.

Out of the numbers 1 to 5, two numbers represent success (1 and 2). Therefore, the probability of success for each individual number is 2/5 or 0.4.

Since we are considering a randomly generated list of integers, we can assume that each number is equally likely to be generated. So, the estimated probability of a success can be calculated by finding the proportion of 1's and 2's in the list.

Let's assume that the list has n elements. If we generate the list multiple times, we can expect that the proportion of successes will approach the true probability of success, which is 0.4.

For example, if we generate a list of 10 integers and get the following numbers: 2, 5, 1, 3, 1, 4, 2, 5, 3, 1, then we have 4 successes out of 10 numbers. So, the proportion of successes in this list is 4/10 or 0.4, which matches the true probability of success.

Therefore, the answer is A. 40%.

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The amount of yoghurt left is 12. 18 ounces

The algebraic expressions are defined as those expressions that are made up of variables, terms, constants, factors and coefficients.

The expressions are also composed of some arithmetic or mathematical operations, such as;

From the information given,

Let the total number of ounces of yoghurt be x

Let the number of ounces of yoghurt sold be y

We have that;

The number left= x - y

= 98.38 - 86.2

= 12. 18 ounces of yoghurt

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Answer:

Step-by-step explanation:

[tex]f(-3)=-(-)3-1=2\\\\f(-2)=-(-2)-1=1\\\\f(0)=-0-1=-1\\\\f(2)=-2-1=-3\\\\f(4)=-4-1=-5\\\\f(x)=5\rightarrow 5=-x-1\rightarrow6=-x \rightarrow x=-6[/tex]

Answer:

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Let the cost price of the camera be x.

When the selling price is 60000 there is a loss of 20%. Let's show this as equation and find the cost price:

The cost price is Rs 75000, and we need a profit of 12%, it gives us the selling price of:

In the following question, among the various parts to solve- a) the probability that the individual waits more than 7 minutes is 0.3. b)the probability that the individual waits between 2 and 7 minutes is 0.5.

a) The probability that an individual will wait more than 7 minutes can be found as follows:

Given that the waiting time of an individual is a continuous uniform distribution and that a bus arrives at the bus stop every 10 minutes.Since the waiting time is a continuous uniform distribution, the probability density function (PDF) can be given as:fX(x) = 1/(b-a)where a = 0 and b = 10.

Hence the PDF of the waiting time can be given as:fX(x) = 1/10The probability that an individual waits more than 7 minutes can be obtained using the complementary probability. This is given by:P(X > 7) = 1 - P(X ≤ 7)The probability that X ≤ 7 can be obtained using the cumulative distribution function (CDF), which is given as:FX(x) = P(X ≤ x) = ∫fX(t) dtwhere x ∈ [a,b].In this case, the CDF of the waiting time is given as:FX(x) = ∫0x fX(t) dt= ∫07 1/10 dt + ∫710 1/10 dt= [t/10]7 + [t/10]10= 7/10Using this, the probability that an individual waits more than 7 minutes is:P(X > 7) = 1 - P(X ≤ 7)= 1 - 7/10= 3/10= 0.3So, the probability that the individual waits more than 7 minutes is 0.3.

b) The probability that the individual waits between 2 and 7 minutes can be calculated as follows:P(2 < X < 7) = P(X < 7) - P(X < 2)Since the waiting time is a continuous uniform distribution, the PDF can be given as:fX(x) = 1/10Using the CDF of X, we can obtain:P(X < 7) = FX(7) = (7 - 0)/10 = 0.7P(X < 2) = FX(2) = (2 - 0)/10 = 0.2Therefore, P(2 < X < 7) = 0.7 - 0.2 = 0.5So, the probability that the individual waits between 2 and 7 minutes is 0.5.

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Answer: Xavier will receive $3.21 in change.

Step-by-step explanation:

To find the change Xavier receives, we need to subtract the cost of the dog collar from the amount he paid with his $10 bill:

Change = $10 - $6.79 = $3.21

Therefore, Xavier will receive $3.21 in change.

The 95% confidence interval for the population mean is equal to (30.4,39.6).

A confidence interval is a range of values that surrounds the point estimate, such as a sample mean, and provides a sense of the precision of the estimate. The confidence interval (CI) contains the estimated parameter at a given level of confidence, usually 95% or 99%.

The 95% confidence interval for the population mean is given by:

x(bar) ± Zα/2 * σ/√n

Where,

x(bar) = 35σ = 100n = 25Zα/2 = Z0.025 = 1.96

Substituting the values, we get:

CI = 35 ± 1.96 * 100/√25

CI  = (30.4,39.6)

Hence, the 95% confidence interval for the population mean is (30.4,39.6).

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Considering the volume of the rectangular prism, the height of the sand inside the box is of 2 feet.

The volume of a rectangular prism, with dimensions length, width and height, is given by the multiplication of these dimensions, according to the equation presented as follows:

Volume = length x width x height.

The parameters for this problem are given as follows:

Hence the height of the sand inside the box is given as follows:

2 x 1.5 x h = 6

3h = 6

h = 2 ft.

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The formula for the real portion of a complex function can be used to write re(e(1/z)) in terms of x and y: f(z) + f(z*) = re(f(z)) / 2

where z* denotes z's complex conjugate.

By applying this formula to the provided function, we obtain:

re(e(1/z)) = (e(1/z) + e(1/z*)) / 2

re(e(1/z)) = (e(x-iy) + e(x+iy)) / 2

re(e(1/z)) = (ex (cos y + I sin y) + ex (cos y - I sin y)) /2

As a result, re(e(1/z)) can be represented as ex cos y in terms of x and y.

Because it fulfills Laplace's equation, which asserts that the total of a function's second-order partial derivatives is equal to zero, this function is harmonic in every domain excluding the origin.

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Answer:

3. Yes because it is the same thing. (short answer just elaborate more)

4. No, a trapezoid cannot be a parallelogram. Trapezoid has only one pair of parallel sides while in a parallelogram there are two pairs of parallel sides.

5. see last sentence in 4

6. group 1: the diamond and the square tilted, and the square.

   group 2: the second one, the cup like shape, the cup shape but taller(the last one)

7. hard to see

8. hard to see

Step-by-step explanation:

Cοnsequently, 960 tiles οf a 50 by 25 cm size will be needed tο cοver the 15m x 8m flοοr.

Cοmparing a quantitative measurement with a recοgnized standard amοunt οf sοme kind is the act οf measurement. Fοr instance, in the measurement 10 kg, kg is indeed the basic measure used tο describe mass οf a physical quantity, and 10 is the size οf the physical quantity.

Calculate the flοοr's tοtal square fοοtage in meters, then divide it intο the area οf each tile tο determine the necessary number οf tiles. The measurements must first be changed tο a cοmparable unit. By dividing by 100, we may cοnvert centimeters tο meters:

15m = 1500cm

8m = 800cm

In square meters, the flοοr space is as fοllοws:

1500cm x 800cm = 1200000cm² = 120m²

The area οf the each tiles in square meters must nοw be determined. The size οf each tile, which is 50 by 25 centimetres, is as fοllοws:

50cm x 25cm = 1250cm² = 0.125m²

Lastly, by dividing the entire flοοr area even by area οf each tile, we can determine the necessary number οf tiles:

120m² / 0.125m² = 960 tiles

The flοοr will therefοre need 960 tiles that are 50 cm by 25 cm in size tο cοver its 15 m × 8 m surface.

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The height of the rock formation is approximately 17.45 feet (rounded to two decimal places).

Trigonometry is a branch of mathematics that deals with the study of the relationships between the sides and angles of triangles. It is used to solve problems involving triangles, such as finding missing sides or angles, determining distances or heights, and more. Trigonometry has applications in various fields, such as engineering, physics, architecture, and astronomy, among others.

In the given question,

We can use trigonometry to solve this problem. Let h be the height of the rock formation in feet, and let x be the horizontal distance from Ramona to the base of the rock formation in feet. Then, we have:

tan(10) = h/x

Rearranging this equation, we get:

h = x * tan(10)

We need to find the value of h, so we need to find the value of x. We can use the angle of elevation and the distance that Ramona walked up the hill to find x. We have a right triangle with height h, base x, and hypotenuse 100 ft. The angle opposite the height h is 10 degrees. So, we have:

tan(10) = h/x

sin(10) = h/100

Rearranging the second equation, we get:

h = 100 * sin(10)

Substituting this into the first equation, we get:

x * tan(10) = 100 * sin(10)

Dividing both sides by tan(10), we get:

x = 100 * sin(10) / tan(10)

Plugging this value of x into the equation for h, we get:

h = x * tan(10) = (100 * sin(10) / tan(10)) * tan(10) = 100 * sin(10)

Therefore, the height of the rock formation is approximately 17.45 feet (rounded to two decimal places).

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We have that, given the system with a component that can fail, we can find the following probabilities

a) To approximate the probability that the system will continue to function after 3625 days, we can use the exponential distribution and its associated properties. The probability that the system is still working after 3625 days is equal to the probability that none of the 120 components have failed, which is equal to:

Probability = e(-120*(3625-25)/25) = 0.091

b) To approximate the probability that the system will continue to function after 4250 days, we must take into account the time required to replace a defective component. Since the time required to replace a defective component is a random variable that is uniformly distributed over (0,0,5), the expected time to replacement is 0.25. Therefore, the probability that the system is still working after 4250 days is equal to:

Probability = e(-120*(4250-25)/25) * e(-120*(0.25)/25) = 0.018

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The oldest and youngest the

time periods are Silurian and Tertiary respectively. Order the time periods from oldest to youngest is equals to

Silurian--> Triassic --> Jurassic -->Tertiary.

We have a table which shows different geologic time periods.

Period Number of Years Ago

Jurassic 2.08 × 10⁸

Silurian 4.38× 10⁸

Tertiary 6.64× 10⁷

Triassic 2.45× 10⁸

We have to order the time periods from oldest to youngest. Time period is the length of time during which an activity occurs and geologic time period showing the geologic eons, eras, periods, epochs, and associated. Now, we check the time periods and put in order.

So, 2.08 × 10⁸ = 2,08000,000

4.38× 10⁸ = 4,38000,000

6.64× 10⁷ = 6,6400,000

2.45× 10⁸ = 2,45000,000

Here, longest or oldest time period

= 4,38000,000

Youngest time period = 6,6400,000

Therefore, the order of time periods is Silurian --> Triassic -->Jurassic -->Tertiary. Hence, required time periods are Silurian and Tertiary.

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Yes, W_2 = {p_2 = a_2t_2 + a_1t + a_0 ∈ ℙ_2 | a_0 = 2} is a subspace of the vector space ℙ_2.

Yes, W_1 = {[a b c; d 0 0] ∈ M_{2x3} | b = a + c} is a subspace of the vector space M_{2x3}.

Vector spaces are closed under vector addition and scalar multiplication, and in this case, ℙ_2 is the set of all 2nd degree polynomials with the usual polynomial addition and scalar multiplication with reals.

Vector spaces are closed under vector addition and scalar multiplication, and in this case, M_{2x3} is the set of all 2x3 matrices with the standard matrix addition and scalar multiplication with reals.

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