please help, i dont undertsand these!

If you roll a four-sided pyramid-shaped die three times, there are 64 possible outcomes that could occur.

If you roll a four-sided pyramid-shaped die three times, there are 64 possible outcomes that could occur.

A four-sided die in the shape of a pyramid is also called a tetrahedral die, and it has four triangular faces, each with a different number on it.

When rolling this die once, there are four possible outcomes since there are four faces and each face has a different number.

When rolling the die three times, we can use the multiplication principle to find the total number of possible outcomes. According to this principle, the total number of outcomes is equal to the product of the number of outcomes for each roll.

Therefore, the total number of possible outcomes when rolling a tetrahedral die three times is:

4 x 4 x 4 = 64

A pyramid-shaped four-sided die can be rolled three times for a total of 64 potential results.

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Answer:

3

Step-by-step explanation:

In the random sample of 17 students in this group, the probability of the average height being 73-75 inches is about 0.014, or 1.4%.  

Let the heights be normally distributed with a mean of 70 inches and a standard deviation of 3 inches.

If we sample 17 students from this group, the distribution of the sample mean will also be approximately normal, with a mean of 70 inches and a standard deviation of 3/sqrt(17) inches (according to the central limit theorem).

This sample average interval can be normalized by the following formula:

z = (x - μ) / (σ / sqrt(n))

where x =sample mean,

μ = population mean,

σ= population standard deviation,

and n = sample size.

For the lower bound of 73 inches, the standardized values ​​are:

z1 = (73 - 70) / (3 / sqrt(17)) ≈ 2.25

For the upper limit of 75 inches, the standardized values ​​are:

z2 = (75 - 70) / (3 / sqrt(17)) ≈ 3.87

Find the probability that the sample mean is between these two values. This probability can be found by finding the area under the standard normal curve between z1 and z2 using a standard normal distribution table or calculator.

P(2.25 < z < 3.87) ≈ 0.014

So, in her random sample of 17 students in this group, the probability of the average height being 73-75 inches is about 0.014, or 1.4%.  

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See the graphed function image below.

The total surface area of the tower, including the bottom base, is [tex]3600+400\sqrt{2}[/tex] square meters.

The surface area of the tower can be found by calculating the surface area of the prism and the surface area of the pyramid separately, and then adding them together.

The surface area of the square base of the prism is [tex]20^2 = 400[/tex] square meters.

The lateral surface area of the prism is the product of the perimeter of the base and the height, which is [tex]4*(20)*40 = 3200[/tex] square meters.

The surface area of the pyramid is given by the formula:

base area + 1/2(perimeter of base)(slant height)

The base area of the pyramid is [tex]20^2 = 400[/tex] square meters. The perimeter of the base is 4(20) = 80 meters. Therefore, the surface area of the pyramid is:

[tex]400 + [(\frac{1}{2})*(80)(10\sqrt{2})] = 400 + 400\sqrt{2}[/tex] square meters.

The total surface area of the tower is the sum of the surface area of the prism and the surface area of the pyramid, which is:

[tex]3200 + 400 + 400\sqrt{2} = 3600 + 400\sqrt{2}[/tex] square meters.

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We know that the line passes through the origin, which means that the coordinates of the point on the line are (0,0). We also know that the line has a slope of 2. Therefore, we can use the point-slope form of the equation of a line to find its equation:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is a point on the line. Plugging in the values we know, we get:

y - 0 = 2(x - 0)

which simplifies to:

y = 2x

Thus, the equation of the line containing the origin and with a slope of 2 is y = 2x.

By answering the presented question, we may conclude that Hence, the probability answer is that they are dependant since P(M) P(H) does not hold.

Probability theory is an area of mathematics that determines the likelihood that an event will occur or that a proposition is true. A probability is a number between 0 and 1, where 1 represents certainty and 0 represents how likely an event is to occur. A probability is a numerical expression for the chance or likelihood that a given event will occur. Probabilities may either be stated as percentages ranging from 0% to 100% or as numbers ranging from 0 to 1. the proportion of occurrences among all equally likely alternatives that lead to a certain event relative to all conceivable outcomes.

To see if the two occurrences are independent, we may look at whether the likelihood of one event is changed by the occurrence of the other.

Let M represent a passing mathematics class and H represent a passing history class. The probability may therefore be stated as follows:

P(M) = 0.85 P(H) = 0.70 P(M H) = 0.50 (where signifies the "and" or intersection operation)

We may use the following formula to test for independence:

If M and H are independent, then P(M H) = P(M) P(H).

When we substitute the provided probabilities, we get:

0.50 = 0.85 × 0.70

0.50 ≠ 0.595

As a result, the equation is invalid, and the events are interdependent.

Hence, the answer is that they are dependant since P(M) P(H) does not hold.

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A sample of 719 times bigger would be needed to estimate the mean per capita income at the 95% level of confidence with an error of at most $0.68.

It is given to us that an economist wants to estimate the mean per capita income (in thousands of dollars) for a major city in California and that he believes that the mean income is $21.2, and the standard deviation is known to be $9.3,

Therefore, here we can make out that:

Number of samples required (n) is given by:

n = (Z²α/2) × б²/e²

It is given to us that, α = 0.05

Therefore, from data provided to us we can get to know that the critical values of Z = ± 1.96

б = 9.3

e = 0.68

when we substitute these values in the above equation, we get:

n = 1.96² × 9.3²/0.68² = 719

Therefore, a sample of 719 times bigger would be needed to estimate the mean per capita income at the 95% level of confidence with an error of at most $0.68.

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We should choose a sample size of at least 153 to be 95% confident that the true average waiting time is within 7 days of the sample mean.

In statistics, a confidence interval is a range of values that is used to estimate an unknown population parameter with a certain degree of confidence or probability. It is a statistical measure that indicates the level of uncertainty or variation associated with a particular estimate or sample.

A confidence interval is typically expressed as a range of values, along with a level of confidence, such as 95% or 99%. This means that if the same study were to be repeated many times, the calculated confidence interval would contain the true population parameter for the given confidence level in a specified proportion of the repetitions. For example, a 95% confidence interval means that if the same study were repeated many times, the interval would contain the true population parameter 95% of the time.

We can use the formula for the margin of error of a confidence interval to solve this problem:

Margin of error = z*(sigma/√(n))

where:

z = the z-score associated with the desired level of confidence (in this case, 1.96 for 95% confidence)

sigma = the population standard deviation (15 days)

n = the sample size we want to find

We want the margin of error to be no more than 7 days, so we can set up the following inequality:

7 <= 1.96*(15/√(n))

Solving for n, we get:

n >= (1.96*15/7)²

n >= 12.36²

n >= 152.7

Therefore, we should choose a sample size of at least 153 to be 95% confident that the true average waiting time is within 7 days of the sample mean.

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If the weight of coal in 43 cars selected at random had an average (x) of less than 61.5 tons, it might make you suspect that the loader had slipped out of adjustment.

Here's why:
Compare the average weight with the expected weight
First, you need to know the expected average weight of coal in the cars when the loader is functioning correctly. If the expected average is significantly higher than 61.5 tons, it could indicate an issue with the loader.
Assess the sample size
A sample size of 43 cars is relatively large, which gives you more confidence in the accuracy of the calculated average weight. If the sample size were smaller, it would be more likely that the result is due to random chance.
Evaluate the deviation from the expected weight
If the average weight is much lower than the expected weight, it would be more likely that the loader is out of adjustment. A small deviation could still be due to random chance or other factors, but a large deviation would suggest a potential issue with the loader.
Consider other factors
Before concluding that the loader is out of adjustment, it's essential to consider other factors that may have caused the lower average weight, such as changes in the type of coal, variations in car sizes, or even human error in weighing the coal.
Investigate further
If the deviation from the expected weight is significant and you cannot identify any other factors that might have caused the lower average weight, it would be reasonable to suspect that the loader may have slipped out of adjustment. In this case, you should conduct a thorough investigation, including inspecting the loader and checking its settings to determine if an adjustment is needed.

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In a right triangle, the sum of the measures of the two acute angles is always 90 degrees. So if ∠J and ∠K are acute angles in a right triangle, then we have:

m∠J + m∠K = 90°

Since m∠J is less than 60°, it follows that m∠K is greater than 30°, because their sum is 90°.

Now, let's consider the cosine of ∠J and ∠K. By definition, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. So we have:

cos(J) = adjacent side to ∠J / hypotenuse

cos(K) = adjacent side to ∠K / hypotenuse

Since ∠J and ∠K are acute angles in a right triangle, they each have a unique adjacent side. Since the triangle is a right triangle, the hypotenuse is always the same, regardless of which angle is being considered.

Therefore, in general, it is not true that cos(J) = cos(K) for all right triangles with acute angles ∠J and ∠K, where m∠J is less than 60°. It depends on the specific values of the adjacent sides.

Answer:

x = 3

Step-by-step explanation:

5(x + 2)- 12 = 3x + 4

5x + 10 - 12 = 3x + 4

5x - 3x = 4 - 10 + 12

2x = 6

x = 3

Answer:

x=3

Step-by-step explanation:

5(x+2)-12= 3x +4

5x + 10 -12= 3x + 4

10-12= -2x + 4

6-12= -2x

-6= -2x

-6/ -2= 3

The statement that best describes the 50% rule is: C. Individuals should spend up to only 50% of their medium-term savings then build savings back up.

The 50% rule is a personal finance guideline that suggests individuals should spend no more than 50% of their after-tax income on necessities, such as housing, food, transportation, and utilities.

The 50% rule is a personal finance guideline that suggests individuals should spend no more than 50% of their after-tax income on necessities, such as housing, food, transportation, and utilities. The other 50% should be split between savings and discretionary spending, such as entertainment and travel.

Therefore, the statement that best describes the 50% rule is:

C. Individuals should spend up to only 50% of their medium-term savings then build savings back up.

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Hue is arranging chair, She can form 6 rows of a given length with 3 chairs left over, or 8 rows of that same length if she gets 11 more chairs. There are 7 chairs in that row length.

A row is anything that is arranged in a series, such as a sequence of figures on a worksheet or stitches on a knitting needle. A row is anything that is arranged in a straight line, such as a line of penguins at the zoo, tulips in a yard, or tuba players moving in your town's Fourth of July procession.

Suppose the number of chairs in a row is 'x' and total number of chairs are 'y'.

Given:

Hue can form 6 rows of a given length with 3 chairs left over.

From that we get

y=6x+3 ………… (1)

Or she can form 8 rows of that same length if she gets 11 more chairs.

We get

y=8x-11 …………(2)

We can put the value of y from the equation 1 in equation 2

6x+3=8x-11

Or, 2x=14

Or, x=7

X is the number of chairs present in a row length. So that is 7.

Hence there are 7 chairs in that row length

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Gabrielle and Leslie are approximately 3.61 meters apart.

What is Pythagoras theorem?

The hypotenuse's square is equal to the sum of the squares of the other two sides if a triangle has a straight angle (90 degrees), according to the Pythagoras theorem. Keep in mind that BC2 = AB2 + AC2 in the triangle ABC signifies this. Base AB, height AC, and hypotenuse BC are all used in this equation. The longest side of a right-angled triangle is its hypotenuse, it should be emphasized.

We know that Mrs. Kelley is standing 6 meters in front of Gabrielle, and Leslie is sitting to Gabrielle's right. We are also given that Mrs. Kelley and Leslie are 7 meters apart.

From the diagram, we can see that Mrs. Kelley and Gabrielle form a right triangle with Leslie as the vertex. Using the Pythagorean theorem, we can find the distance between Gabrielle and Leslie:

[tex]d^2 = (7)^2 - (6)^2d^2 = 49 - 36d^2 = 13[/tex]

d ≈ 3.61

Therefore, Gabrielle and Leslie are approximately 3.61 meters apart.

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If the point Q(10,0) lies on the circle, then the other points that lie on the circle C are (a) (5 , 5√3) and (b) (6,4).

We know that, if "circle C" is centered at the origin, its equation is of the form ⇒ x² + y² = r², where r is = radius of circle,

We know that point Q(10,0) lies on circle C, so we substitute its coordinates into the equation of the circle to find the value of r:

⇒ 10² + 0² = r²,

⇒ r² = 100,

⇒ r = 10,

So, the equation of circle C is : x² + y² = 100,

Now, we check each of the given points to see which ones lie on the circle:

Option (a) : (5 , 5√3)

⇒ 5² + (5√3)² = 100,

⇒ 25 + 75 = 100,

The point (5 , 5√3)lie on circle.

Option (b) : (6,4)

⇒ 6² + 4² = 100,

⇒ 36 + 16 = 100,

The point (6,4) lies on circle.

Option (c) : (4,5√3)

⇒ 4² + (5√3)² = 100,

⇒ 16 + 75 = 91 ≠ 100,

This point (4,5√3) does not lie on circle.

Option (d) : (3 , √34)

⇒ 3² + (√34)² = 100,

⇒ 43 ≠ 100

This point (3 , √34) does not lie on circle.

Therefore, (a) (5 , 5√3) and (b) (6,4) are the point that "lies-exactly" on the circle.

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The given question is incomplete, the complete question is

Circle C is centered at the origin. If Q(10,0) lies on circle C, which of the following points also lies on circle C?

(a) (5 , 5√3)

(b) (6,4)

(c) (4,5√3)

(d) (3 , √34)

If  x is the unknown number and Nineteen more than a number. Thus, Translation is 19 units on the right to get x + 19.

Each point in a figure is moved the same distance in a single direction using a type of transformation known as translation.

In geometry, a function that transfers an object a specific distance is referred to as translation. Nothing else is done to change the object. It is not scaled, reflected, rotated, or refracted.

Give that:

x is the unknown number and Nineteen more than a number.

Then, there is a shift of 19 units on the right of x. add 19 to unknown number x to get the translation.

x + 19

Thus, Translation is 19 units on the right to get x + 19.

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Answer: 700π/3 OR 233.33π

Step-by-step explanation:

Volume of Cylinder = [tex]\pi r^2h[/tex]

Volume of Hemisphere = [tex]\frac{2}{3}\pi r^3[/tex]

[tex]V_{c}[/tex]  = 5² x 6 x π

[tex]V_{c}[/tex]  = 25 x 6 x π

[tex]V_{c}[/tex]  = 150π

[tex]V_{h}[/tex] = (2 x 5³ x π) / 3

[tex]V_{h}[/tex] = (2 x 125 x π) / 3

[tex]V_{h}[/tex] = 250π/3

[tex]V_{T}[/tex] = [tex]V_{c}[/tex] + [tex]V_{h}[/tex]

[tex]V_{T}[/tex] = 150π + 250π/3

[tex]V_{T}[/tex] = 700π/3 OR 233.33π

The rate of the function is 4 dollars per gallon of gas. 4 is the cost per gallon of gas, and 8 is the fixed cost of the car wash.

A gallon of gas = 20 pounds of CO₂!

20 pοunds οf carbοn diοxide are prοduced when 6.3 pοunds οf petrοl are burned.

The twο οxygen atοms accοunt fοr a large pοrtiοn οf the weight οf carbοn diοxide (CO₂) (the O₂). Atοms οf carbοn and hydrοgen are jοined tο fοrm petrοl mοlecules. The carbοn and hydrοgen in the gas mοlecules in petrοl burn apart. H₂O, οr water, is created when twο hydrοgen atοms unite with οne οxygen atοm.

Each petrοl carbοn atοm jοins twο οxygen atοms that are already present in the atmοsphere. It prοduces CO₂.

To find the rate of the function, we need to calculate the slope of the line that represents the total cost of Owen's stop at the gas station as a function of the number of gallons of gas. We can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. We can use the coordinates (0, 8) and (7, 36) to calculate the slope:

slope = (36 - 8) / (7 - 0) = 28 / 7 = 4

Therefore, the rate of the function is 4 dollars per gallon of gas.

To find the cost per gallon of gas, we can use the y-intercept of the line, which represents the fixed cost of the car wash. From the graph, we can see that the y-intercept is 8 dollars. Therefore, the gas costs (y - 8) dollars per gallon.

We can write this equation as:

y = 4z + 8

where y is the total cost of Owen's stop at the gas station, z is the number of gallons of gas, 4 is the cost per gallon of gas, and 8 is the fixed cost of the car wash.

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The complete question is:

Answer:

Length is 7 units

Step by Step explanation:

Assuming you mean the Area is 28 square units

Then

L = w + 3

A = w(w + 3)

28 = w2 + 3w

We can subtract 28 from both sides of the equation to obtain the Quadratic

w2 + 3w - 28 = 0

Factoring to give

(w + 7)(w - 4) = 0

The width cannot be -7

w = 4

L = w + 3

L = 4 + 3 = 7

The width of the rectangle is 4 units and the length of the rectangle is 7 unites.

Answer: First section: (x³+x²-x-1); Second section: (2x³-6x+4); Total volume: (3x³+x²-7x+3)

Step-by-step explanation:

The picture I attached was the lines I drew to make finding the volume easier.

To find the second upper side I subtracted x+1 from 2x+3, which came to x+2.

This is why I scrapped out the 2x+3 in the bottom to make it less confusing.

Now you should be able to separate the two prisms.

For the first section, multiply (x+1)(x+1)(x-1). I'll do the right two first because it's easier.

(x+1)(x²-1x+1x-1)  -1x and 1x cancel out. So now it's (x+1)(x²-1).

The volume of the first section is (x³+x²-x-1).

For the second section, multiply (x+2)(2x-2)(x-1).

The first two turn out as (2x²-2x+4x-4), which is (2x²+2x-4). Then, multiply all that by (x-1): (2x²+2x-4)(x-1).

You should get (2x³-2x²+2x²-2x-4x+4), which is then (2x³-6x+4).

So the volume of the second section is (2x³-6x+4).

Then it asks you to find the total volume, so just add the two volumes: (x³+x²-x-1) + (2x³-6x+4). Then add like-terms.

(3x³+x²-7x+3) should be the total volume.

For the explaining part (being pretty vague), just say how the volumes for both sections were added by like-terms to sum up to the total volume of the plant stand.

Hope that had helped (or at least worked) :p

a

Answer:

10% of 30

=10/100×30

=3

The zeros of the quadratic function  [tex]f(x) = 2x^2 - 5x + 2 are x = 1/2 and x = 2.[/tex]

A quadratic function is a mathematical function of the form:

[tex]f(x) = ax^2 + bx + c[/tex]

where a, b, and c are constants, and x is the variable. The highest power of the variable x in a quadratic function is 2, which means it is a second-degree polynomial.

In a quadratic function, the graph is a parabola, which can open up or down depending on the sign of the leading coefficient a. If a > 0, the parabola opens up, and if a < 0, the parabola opens down. The vertex of the parabola represents the minimum or maximum value of the function, depending on the direction of the parabola.

To find the zeros of the quadratic function:

[tex]f(x) = 2x^2 - 5x + 2[/tex]

We need to solve the equation:

[tex]2x^2 - 5x + 2 = 0[/tex]

We can factor this quadratic equation as follows:

[tex]2x^2 - 4x - x + 2 = 02x(x - 2) - 1(x - 2) = 0(2x - 1)(x - 2) = 0[/tex]

So the zeros of the quadratic function are:

x = 1/2 or x = 2.

Therefore, the zeros of the quadratic function is:[tex]f(x) = 2x^2 - 5x + 2 are x = 1/2 and x = 2.[/tex]

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The correct answer to the given question about equation of Candice taking a hike is option D)  0.38x + 13 = 87

The cost for Candice to enter the park is a fixed fee of $13.00, and the additional cost for the tour group to hike is $0.38 per mile. If Candice has $87.00 to spend, we can use the following equation to determine how many miles she can hike:

0.38x + 13 = 87

Here, x represents the number of miles Candice can hike.

The left-hand side of the equation represents the total cost, which includes the fixed entrance fee of $13.00 plus the variable cost of $0.38 per mile multiplied by the number of miles hiked, represented by 0.38x.

The right-hand side of the equation represents the total amount Candice can spend, which is $87.00.

By solving for x, we can find the number of miles Candice can hike within her budget.

Therefore, the correct answer is D. 0.38x + 13 = 87

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For the given sample space we have the probabilities as follows: 1. P(sum=11) = 6/64 = 3/32 2. P (sum < 6) = 10/64 3. P ( sum is greater than or equal to 7) = 49/64 4.  P(sum is a multiple of 3 or a multiple of 4) = 33/64.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

For the sample space of rolling two 8 sided die we have:

1. P (sum = 11):

The samples that give the sum equal to 11 are:

(3, 8), (4, 7), (5, 6), (6, 5), (7, 4), and (8, 3)

Thus,

P(sum=11) = 6/64 = 3/32

2. P (sum < 6):

(1, 1), (1, 2), (1, 3), (1, 4)

(2, 1), (2, 2), (2, 3)

(3, 1), (3, 2)

(4, 1)

Thus, P (sum < 6) = 10/64

3. P ( sum is greater than or equal to 7):

(1, 6), (1, 7), (1, 8) = 3

(2, 5), (2, 6) (2, 7), (2, 8) = 4

(3, 4), (3, 5) (3, 6) (3, 7) (3, 8) = 5  

(4, 3) (4, 4) (4, 5) (4, 6), (4, 7) (4, 8) = 6

(5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (5, 7) (5, 8) = 7

(6, 1) upto (6, 8) = 8

(7, 1) upto (7, 8) = 8

(8, 1) upto (8, 8) = 8

Total = 3 + 4 + 5 + 6 + 7 + 8 + 8 + 8 = 49

Thus, P ( sum is greater than or equal to 7) = 49/64

4. P (sum is a multiple of 3 or a multiple of 4):

Multiple of 3: 3, 6, 9, 12, ........

(1, 2) (1, 5) (1, 8) (2, 1) (2, 4) (2, 7) (3, 1) (3, 3) (3, 6) (4, 2) (4, 5) (4, 8) (5, 1) (5, 4) (5, 7), (6, 3), (6, 6) (7, 2) (7, 5) (7, 8) (8, 1) (8, 4), (8, 7) = 23

Multiple of 4: 4, 8, 12, 16, .........

(1, 3) (1, 7) (2, 2) (2, 6) (3, 1) (3, 5) (4, 4) (4, 8), (5, 3), (5, 7), (6, 2), (6, 6) (7, 1) (7, 5) (8, 4) (8, 8) = 16

Common terms = 6

Thus, P(sum is a multiple of 3 or a multiple of 4) = 23 + 16 - 6 / 64 = 33/64

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The volume of a hemisphere is 66.9cm³

Volume of cube is 768cm³

Composite volume is 834.9cm³

The volume of a cube is the amount of space enclosed within the boundaries of a cube. A cube is a three-dimensional object with six congruent square faces, all of which meet at right angles.  Therefore, the formula for calculating the volume of a cube is:

Volume = side length x side length x side length = side³

Given:

Sides of cube

a=8cm

b=8cm

c=12cm

Radius of hemisphere=4cm

The volume of a hemisphere = (1/3)πr³cubic centimeters.

=1/3×π×4³

=66.9cm³

Volume of cube=a×b×c

=8×8×12

=768cm³

Composite volume=The volume of a hemisphere + Volume of cube

=66.9+768

=834.9cm³

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The time clocks necessarily follow two formats which are 12-hour format and 24-hour format.

12-hour:

half past three in the afternoon

9 minutes to midnight

quarter after 00:00 H (this should be "quarter after 12:00 am")

quarter to 19:00 H

half past 13:00 H

5 to 23:00 H

24-hour:

03:02 before 08:40

15:30

23:51

00:15

18:45

13:30

22:55

The most typical time format in the United States, Canada, and some other English-speaking nations is the 12-hour clock. It employs the digits 1 through 12 to represent the hours, and "am" or "pm" to denote morning or afternoon/evening, respectively.

The majority of non-American nations, including most of Europe and the rest of the globe, use the 24-hour clock as their primary time system. The hours are denoted by the digits 0-23, with no distinction made between morning and afternoon/evening.

Decimal time: In some nations, including France, the day is split into ten hours, each hour into one hundred minutes, and each minute into one hundred seconds.

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(a) A/x + Bx + [tex]C/(1 + x) + Dx^2 + Ex/(1 - x) + Fx^2[/tex]

(b)A/x + [tex]B/x^2 + C/x^3[/tex] + D/(x - 2) + [tex]E/(x^2[/tex]+ 2x + 4) + F/(x + 2) + [tex]G/(x^2[/tex] - 2x + 4)

(a) To perform partial fraction decomposition on the function:

[tex](x^4 (x^3 + x)) / (x^2 - x^4)[/tex]

we first factor the denominator:

[tex]x^2 - x^4 = x^2(1 - x^2) = x^2(1 + x)(1 - x)[/tex]

The partial fraction decomposition of the function is:

[tex](x^4 (x^3 + x)) / (x^2 - x^4) = A/x + Bx + C/(1 + x) + Dx^2 + Ex/(1 - x) + Fx^2[/tex]

where A, B, C, D, E, and F are constants to be determined.

(b) To perform partial fraction decomposition on the function:

[tex]4 / (x^6 - 8x^3)[/tex]

we first factor the denominator:

[tex]x^6 - 8x^3 = x^3(x^3 - 8) = x^3(x - 2)(x^2 + 2x + 4)(x + 2)(x^2 - 2x + 4)[/tex]

The partial fraction decomposition of the function is:

[tex]4 / (x^6 - 8x^3)[/tex] = [tex]A/x + B/x^2 + C/x^3[/tex] + D/(x - 2) + [tex]E/(x^2 + 2x + 4) + F/(x + 2)[/tex] + [tex]G/(x^2 - 2x + 4)[/tex]

where A, B, C, D, E, F, and G are constants to be determined.

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The following is a conditional statement: If an is true, then b is true. A conditional statement is a statement that can be written in the form “If P then Q,” where P and Q are sentences.

If p, then q, or "p implies q," is the conditional of q by p, and it is symbolised by p q. If both p and q are true, then it is false; otherwise, it is true. Contrapositive: "If q then p" is the opposite of a conditional statement of the form "If p then q."

Has a truth value of true if both a and b are true.

In addition to b's truth value, an is untrue.

if an is true and b is false, and has a truth value of false.

Then, with relation to the following claims:

1. p -> q

This sentence is true because p and q are both true.

2. p -> q

This is a false assertion because p is true and q is false..

3. p->q

Regardless of the truth value of q, this assertion is true because p is false.

4. ∼p -> q

P is the inverse of p. P is false if and only if p is true. And regardless of the truth value of q, this assertion is accurate.

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The complete question is

What is the truth value for the following conditional statement?

1.)

p: true

q: true

p → q

2.)

p: true

q: false

p → q

3.)

p: false

q: false

p → q

4.)

p: true

q: true

∼p → q