If H is the circumcenter of triangle BCD find each measure

The answers for finite functions are a.256 b.24 one-to-one functions

(a) To count the number of functions from A to B, we need to find the number of possible outputs for each input. Since there are 4 elements in A and 4 elements in B, there are 4 choices for each element in A.

Thus, there are 4^4 = 256 functions from A to B that can be defined.

(b) To count the number of one-to-one functions from A to B, we need to ensure that each element in A is mapped to a unique element in B. The first element in A can be mapped to any of the 4 elements in B. However, once we have chosen an element in B to map the first element in A to, we only have 3 choices left for the second element in A (since we cannot map it to the same element as the first).

Similarly, once we have chosen an element in B to map the first two elements in A to, we only have 2 choices left for the third element in A. Finally, once we have chosen an element in B to map the first three elements in A to, there is only 1 choice left for the fourth element in A.

Thus, there are 4*3*2*1 = 24 one-to-one functions from A to B that can be defined.

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Answer:

5% is the answer I think

Step-by-step explanation:

I hope this helps

Answer:

5

Step-by-step explanation:

Y= 4/80

4/80 * 100/100 = 5/100

Y= 5

The length of one side of the square is 3, as solving for "x" in the equation A(x) = x² - 6x + 9 yields (x - 3)² = 0, which has a solution of x = 3.

The area of a square is typically calculated using the formula A = s², where "s" represents the length of one side of the square.

In this problem, we are given the area of the square as A(x) = x² - 6x + 9.

To find the length of one side of the square, we need to solve for "x" in the equation A(x) = x² - 6x + 9.

Setting A(x) equal to zero

x² - 6x + 9 = 0

Factoring the quadratic

(x - 3)² = 0

Expanding the squared term

x - 3 = 0

Solving for "x"

x = 3

Therefore, the length of one side of the square is 3.

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The coordinates of the reflected points are:

Point A': (4, -4)

Point B': (0, 0)

Point C': (-2, -5)

As we know that a point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.

As per the question, given that ΔABC has vertices at (-4, 4), (0,0), and (-5,-2).

To find the coordinates of the reflected points, we need to swap the x and y-coordinates of each point.

Point A (-4, 4) becomes A' (4, -4)

Point B (0, 0) remains the same B' (0, 0)

Point C (-5, -2) becomes C' (-2, -5)

Therefore, the coordinates of the reflected points are:

Point A': (4, -4)

Point B': (0, 0)

Point C': (-2, -5)

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Answer:

(-5, 11.63) and (-5, -7.63)

Step-by-step explanation:

To find the coordinates of point A, we need to use the distance formula to find the distance between points A and B, and then use that distance to determine the possible y-coordinates of point A.

The distance between points A and B is given by:

distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁, y₁) are the coordinates of point A and (x₂, y₂) are the coordinates of point B. We know that the x-coordinate of point A is -5, and the coordinates of point B are (3, 2). So we can plug these values into the distance formula:

distance = √[(3 - (-5))² + (2 - y)²]

Simplifying the expression inside the square root:

distance = √[64 + (2 - y)²]

Now we need to find the possible values of y that make the distance equal to 13. We can set up an equation:

√[64 + (2 - y)²] = 13

Squaring both sides:

64 + (2 - y)² = 169

Expanding the square:

64 + 4 - 4y + y² = 169

Rearranging the terms:

y² - 4y - 101 = 0

Using the quadratic formula:

y = (4 ± √(4² - 4(1)(-101))) / (2(1))

Simplifying:

y = (4 ± √409) / 2

So the possible y-coordinates of point A are:

y = (4 + √409) / 2 ≈ 11.63

y = (4 - √409) / 2 ≈ -7.63

Therefore, the possible coordinates of point A are (-5, 11.63) and (-5, -7.63).

The zeroes on the graph of the function are x = 1.5, x = -1, and x = 5.

We have,

Zeroes of a function refer to the values of the input variable (also known as the independent variable) that make the output of the function equal to zero.

In other words, they are the values of the input variable that result in a function output of zero.

Now,

From the graph,

The point at which the y-axis is zero are:

The blue line:

x = 1.5

The red line:

x = -1 and 5

Thus,

The zeroes on the graph of the function are x = 1.5, x = -1, and x = 5.

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The equation  -5-4x=5y graph is given in attachment whose slope is -4/5

The given equation is -5-4x=5y

We have to convert to slope intercept form

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

So isolate y in the equation

Divide both sides by 5

-1-4/5x=y

y=-4/5 x -1

Slope is -4/5

Hence, the equation  -5-4x=5y graph is given in attachment

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The associated matrix A of the linear transformation T is:
A =
[  0   -8   0   62 ]
[  2   -7   0   0  ]
[  0    5   0   0  ]
[  9   -3   1   0  ]

To find the associated matrix A, we need to apply T to the standard basis vectors e1 = (1,0,0,0), e2 = (0,1,0,0), e3 = (0,0,1,0), and e4 = (0,0,0,1), and write the resulting vectors in terms of the standard basis.

T(e1) = (62, 0, 0, 9)
T(e2) = (-8, 2, 5, -3)
T(e3) = (0, 0, 0, 0)
T(e4) = (0, 0, 0, 0)

Thus, the first column of A is T(e1) written in terms of the standard basis, which is (62, 0, 0, 9), the second column is T(e2) written in terms of the standard basis, which is (-8, 2, 5, -3), the third and fourth columns are T(e3) and T(e4) written in terms of the standard basis, which are (0, 0, 0, 0) and (0, 0, 0, 0) respectively.

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We cannot conclude that the standard deviation of capsule weights of the drug is different from 0.8 g.

Procedure: One variance χ² Hypothesis Test

Assumptions:

Normal population

Population standard deviation is unknown

Simple random sample

Step 1. Hypotheses Set-Up:

H0: σ² = 0.64 (null hypothesis)

Ha: σ² ≠ 0.64 (alternative hypothesis)

Step 2. The significance level α= 0.05

Step 3. Compute the value of the test statistic:

The test statistic is calculated as follows:

χ² = (n - 1) * s² / σ²

where n = 20 is the sample size, s² = 0.73² = 0.5329 is the sample variance, and σ² = 0.64 is the hypothesized population variance.

χ² = (20 - 1) * 0.5329 / 0.64 = 10.93

Step 4. Testing Procedure:

We need to find the critical value(s) for the rejection region and compute the P-value of the test statistic.

Since we are conducting a two-tailed test at a 5% significance level with 19 degrees of freedom (df = n - 1 = 20 - 1 = 19), the critical values are:

χ²(0.025, 19) = 9.24 and χ²(0.975, 19) = 32.85

The P-value for the test statistic of 10.93 with 19 degrees of freedom can be found using a chi-square distribution table or calculator. The P-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. The P-value for a two-tailed test is calculated as follows:

P-value = P(χ² > 10.93) + P(χ² < 0.417)

where P(χ² > 10.93) is the probability of obtaining a chi-square value greater than 10.93, and P(χ² < 0.417) is the probability of obtaining a chi-square value less than 0.417 (the minimum possible value for a chi-square distribution with 19 degrees of freedom).

Using a chi-square distribution table or calculator, we find:

P-value = 0.270 + 0.001 = 0.271

Step 5. Decision:

Since the test statistic of 10.93 is greater than the critical value of 9.24, we reject the null hypothesis. Alternatively, since the P-value of 0.271 is greater than the significance level of 0.05, we fail to reject the null hypothesis.

Conclusion:

At 5% significance level, we do not have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Therefore, we cannot conclude that the standard deviation of capsule weights of the drug is different from 0.8 g.

Complete question: 0.8 mg is acceptable. A sample of 20 capsules was taken and the weights are provided below:

120.3 120.8 120.1 119.7 120.8

119.4 119.1 120.9 118.9 119.5

120.4 121.1 118.6 119.4 119.3

119.8 120.2 119.5 118.9 119.8

(Note: The average and the standard deviation of the data are respectively 119.8 g and 0.73 g.)

At 5% significance level, test the claim that the standard deviation of capsule weights of the drug is different from 0.8 g.

Procedure: Select an answer One mean Z Hypothesis Test One mean T Hypothesis Test One proportion Z Hypothesis Test One variance χ² Hypothesis Test

Assumptions: (select everything that applies)

Normal population

Sample size is greater than 30

The number of positive and negative responses are both greater than 10

Population standard deviation is unknown

Population standard deviation is known

Simple random sample

Step 1. Hypotheses Set-Up:

H0:H0: Select an answer p σ² μ = , where ? p μ σ is the Select an answer population proportion population standard deviation population mean and the units are ? lbs g mg kg

Ha:Ha: Select an answer μ p σ² ? > ≠ < , and the test is Select an answer Right-Tail Left-Tail Two-Tail

Step 2. The significance level α=α= %

Step 3. Compute the value of the test statistic: Select an answer z₀ f₀ χ²₀ t₀ = (Round the answer to 3 decimal places)

Step 4. Testing Procedure: (Round the answers to 3 decimal places)

CVA PVA

Provide the critical value(s) for the Rejection Region: Compute the P-value of the test statistic:

left CV is and right CV is P-value is

Step 5. Decision:

CVA PVA

Is the test statistic in the rejection region? Is the P-value less than the significance level?

? yes no ? yes no

Conclusion: Select an answer Do not reject the null hypothesis in favor of the alternative. Reject the null hypothesis in favor of the alternative.

Step 6. Interpretation:

At 5% significance level we Select an answer DO DO NOT have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

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The possible values of t for each state are:

(H, H, H, H) or (T, T, T, T): t = 0

(H, H, H, T) or (T, T, T, H): t = 1 or 2

(H, H, T, T) or (T, T, H, H): t = 1, 2, or 3

(H, H, T, H) or (H, T, H, H) or (T, H, H, H) or (T, T, H, T) or (T, H, T, T) or (H, T, T, T): t = 2, 3, or 4

In the 4-period binomial model, there are 16 possible states. We are given the values of the stopping time t for three of these states as follows:

T(HHTT) = 0

T(TTTHH) = 2

T(THTHT) = 2

Using the fact that a stopping time must satisfy the following conditions:

T(H) = 0 and T(T) = 0

For any state s, if T(s) = k, then for any state s' reachable from s, T(s') ≤ k + 1

We can deduce the possible values of t for each of the remaining states. Here are the possible values of t for each type of state:

States of the form (H, H, H, H) or (T, T, T, T): t = 0 (since these are absorbing states)

States of the form (H, H, H, T) or (T, T, T, H): t = 1 or 2 (since the next state can only be (H, H, T, T) or (T, T, H, H) and we already know t for those states)

States of the form (H, H, T, T) or (T, T, H, H): t = 1, 2, or 3 (since the next state can be any of the 4 possible states, and we already know t for some of them)

States of the form (H, H, T, H) or (H, T, H, H) or (T, H, H, H) or (T, T, H, T) or (T, H, T, T) or (H, T, T, T): t = 2, 3, or 4 (since the next state can be any of the 4 possible states, and we already know t for some of them)

Therefore, the possible values of t for each state are:

(H, H, H, H) or (T, T, T, T): t = 0

(H, H, H, T) or (T, T, T, H): t = 1 or 2

(H, H, T, T) or (T, T, H, H): t = 1, 2, or 3

(H, H, T, H) or (H, T, H, H) or (T, H, H, H) or (T, T, H, T) or (T, H, T, T) or (H, T, T, T): t = 2, 3, or 4

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Answer: Whole thing=20.5

First half (17,18,19,20)=18.5

Second half (21,24,25,27) = 24.5

Step-by-step explanation:

eliminate numbers on both sides till you get to the middle if there is an even number add up the two numbers in the middle and divide them by 2

For example, In the whole thing, you are left with 20 and 21 so

20+21 =41/2= 20.5

For example, In the first part, you are left with 18 and 19 so

18+19 =37/2= 18.5

For example, In the first part, you are left with 24 and 25 so

24+25 =49/2= 24.5

This is the contrapositive of the original statement PQ -> P, which allows us to conclude that the argument is valid.

The argument can be validated using the modus tollens rule of inference, also known as the law of contrapositive. This rule states that if we have a conditional statement of the form "If A, then B," and we know that B is false, we can infer that A must also be false.

In the given argument, we have two conditional statements:

(PA -> ST) -> ~(MV -> N) (premise)

PQ -> ~(MV -> N) (premise)

To use modus tollens, we start by assuming the negation of the conclusion we want to prove, which is P. Then, we use the second premise to infer that ~(MV -> N) must be true. Using the logical equivalence ~(p -> q) = p /\ ~q, we can rewrite this as MV /\ ~N.

Next, we can use the first premise to infer that if PA -> ST is true, then MV -> N must be false. Since we have already established that MV /\ ~N is true, we can conclude that PA -> ST must be false as well.

Finally, we use the second premise again to infer that PQ must be false. This is because if PQ were true, then ~(MV -> N) would also be true, which contradicts our previous conclusion.

Therefore, we have shown that if PQ is true, then P must be false. This is the contrapositive of the original statement PQ -> P, which allows us to conclude that the argument is valid.

Complete question: Write the rule of inference that validates the argument.

4.

1. [tex]\frac{P_A-(S \leftrightarrow T)}{\therefore P}$ $(M \vee-N) \rightarrow-P$[/tex]

2. [tex]$\frac{\neg P Q}{\therefore(M \vee-N) \rightarrow Q}$[/tex]

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Therefore, William needs to score a 97 on the third quiz to get an average of 90.

Average: The arithmetic mean is calculated by adding a set of integers, dividing by their count, and then taking the result. For instance, the result of 30 divided by 6 is 5, which is the average of 2, 3, 3, 5, 7, and 10.

The average test score is calculated by dividing the total score on an assessment by the total number of test-takers. As an illustration, if three students each obtained test scores of 69, 87, and 92, their combined scores would be totaled together and divided by three to yield an average of 82.6.

William needs to score "x" on the third quiz to get an average of 90.

The average of three quizzes can be calculated using the formula:

average = (sum of scores) / (number of scores)

To get an average of 90, William's total score on all three quizzes needs to be:

90 x 3 = 270

His current total score from the first two quizzes is:

85 + 88 = 173

So, to reach a total score of 270, William needs to score:

270 - 173 = 97

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Answer: b. Solve the equation by factoring. 0=-16+2 -8 +120. 16€²+8E-120=0. 8(2²+E-15)=0. 8(2+-5) (++3)=0. 2=-5=0 =+3=0. + 5 t=-3. 2,5 seconds.

Step-by-step explanation:

We have z ≈ -1.41 as the value such that 92% of the normal curve lies to the right of z.

9. To sketch the areas under the standard normal curve between z=0.32 and z=1.92, follow these steps:

Step 1: Draw a standard normal curve (a bell-shaped curve) with a mean of 0 and a standard deviation of 1.
Step 2: Mark the points z=0.32 and z=1.92 on the horizontal axis.
Step 3: Shade the area between z=0.32 and z=1.92.
To find the specified area between z=0.32 and z=1.92, use a standard normal table or a calculator with a normal distribution function to find the area to the left of z=1.92 and subtract the area to the left of z=0.32.

10. To find the probability that it takes at least eight minutes to find a parking space, follow these steps:
Step 1: Convert the time of 8 minutes to a z-score using the formula

z = (X - μ) / σ, where X is the time, μ is the mean, and σ is the standard deviation.
z = [tex]\frac{(8 - 5) }{2} =1.5[/tex]

Step 2: Use a standard normal table or a calculator with a normal distribution function to find the area to the right of z=1.5, which represents the probability of taking at least 8 minutes.

11. To find the z-score such that 92% of the normal curve lies to the right of z, follow these steps:
Step 1: Since 92% of the curve lies to the right, that means 8% of the curve lies to the left (100% - 92% = 8%).
Step 2: Use a standard normal table or a calculator with a normal distribution function to find the z-score corresponding to an area of 0.08 to the left. You will find that z ≈ -1.41.

So, z ≈ -1.41 is the value such that 92% of the normal curve lies to the right of z.

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The p-value for the hypothesis test is 0.0314.

To find the p-value for this hypothesis test, we can use a t-test since the population standard deviation is unknown.

The test statistic is calculated as:

t = (x - μ) / (s / √n)

where x is the sample mean, μ is the hypothesized population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size.

In this case, we have:

x = 424

μ = 400

s = 83

n = 53

So the test statistic is:

t = (424 - 400) / (83 / √53) ≈ 2.2071

To find the p-value, we need to compare this test statistic to the t-distribution with n-1 degrees of freedom (df = 52, in this case). Using a t-distribution table or calculator, we find that the probability of getting a t-value as extreme or more extreme than 2.2071 (in either direction) is approximately 0.0157.

Since this is a two-tailed test (Ha: u ≠ 400), we need to double this probability to get the p-value:

p-value = 2 * 0.0157 ≈ 0.0314

Therefore, the p-value for the hypothesis test is approximately 0.0314 (rounded to 4 decimal places).

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The domain of a function y = 2(x + 3)2 + 42 is {x € R | x < 3} option d. {x € R | x < 3} means "the set of all real numbers x such that x is less than 3."

To determine the domain of the function, we need to find the set of all possible values of x that will give us a real number for y. The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the value that the function shoots out after we plug an x value in.In this case, there are no restrictions on the value of x except that the expression inside the square root (x + 3)^2 must not be negative, as taking the square root of a negative number would result in an imaginary number. Therefore, the domain of the function is all real numbers less than 3, represented as {x € R | x < 3}. The "€" symbol represents the element of or belonging to a set, so {x € R | x < 3} means "the set of all real numbers x such that x is less than 3."

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The probability of a radiation measurement of 599 or higher from a mountain known to have terrorists, assuming no nuclear danger, is about 0.0668.

We are given that the radiation levels observed by the satellite are normally distributed with a mean of 490 and a variance of 2916. We want to find the probability of a random radiation measurement being 599 or higher, assuming there is no nuclear danger.

First, we need to standardize the radiation level of 599 using the formula:

z = (x - mu) / sigma

where x is the radiation level, mu is the mean, and sigma is the standard deviation. Substituting the values we have:

z = (599 - 490) / √(2916) = 1.5

Now, we can use a standard normal distribution table or calculator to find the probability of a z-score of 1.5 or higher. The table or calculator will give us the area under the standard normal curve to the right of 1.5.

Using a calculator, we can find this probability as follows:

P(Z > 1.5) = 0.0668 (rounded to four decimal places)

Therefore, the probability of a random radiation measurement being 599 or higher is approximately 0.0668, assuming there is no nuclear danger.

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The likelihood of randomly selecting a terrier from the shelter is (g) unlikely

From the question, we have the following parameters that can be used in our computation:

P(terriers) = 15%

When a probability is at 15% or less than 50%, it means that

The probability is unlikely or less likely

Hence, the true statement is (b) unlikely

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The number of unemployed people in Apexico in February 2008 was 10,080,000. So, correct option is A.

To calculate the number of unemployed people in Apexico in February 2008, we can use the following formula:

Number of Unemployed = Total Population x Unemployment Rate

Substituting the given values:

Number of Unemployed = 168,000,000 x 0.06 = 10,080,000

Option A is the correct answer.

This calculation is based on the assumption that the unemployment rate is constant across the entire population of Apexico.

This calculation provides a rough estimate of the number of unemployed people in Apexico in February 2008, but it should be interpreted with caution.

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Answer:

3x-5y=0

The standard form for linear equations in two variables is Ax+By=C, so just plug it in.

Answer:

Step-by-step explanation:

Standard form is ax+by=c

[tex]\frac{x}{5} =\frac{y}{3} \\[/tex]      cross multiply to get rid of fractions

3x=5y     now bring the 5y to the other side, by subtracting both sides by 5y

3x-5y=0

The observed proportion of men who said they had not been screened is 0.8 or 80%.

The observed proportion of men who had not been screened can be calculated by dividing the number of men who had not been screened (160) by the total sample size (200):

Observed proportion of yet-to-be-screened men = number of unscreened men/ total sample size.

Observed proportion = 160/200 = 0.8

Therefore, the observed proportion of men who said they had not been screened is 0.8 or 80%.

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Answer:Garret had 1/10 of the pizza left.

Step-by-step explanation:thats good

The probabilities are:
- Answer the first question incorrectly: 0.750
- Answer the first 2 questions incorrectly: 0.5625
- Answer the first 5 questions incorrectly: 0.237
- Answer at least 1 of the first 5 questions correctly: 0.763

1. The probability of answering the first question incorrectly: Since there are 4 choices and only 1 is correct, the probability of choosing an incorrect answer is 3 incorrect choices out of 4 total choices. So, the probability is 3/4 or 0.750.

2. The probability of answering the first 2 questions incorrectly: For each question, the probability of answering incorrectly is 3/4. To find the combined probability, multiply the individual probabilities: (3/4) * (3/4) = 9/16 or 0.5625.

3. The probability of answering the first 5 questions incorrectly: Similarly, the combined probability is (3/4) * (3/4) * (3/4) * (3/4) * (3/4) = 243/1024 or approximately 0.237.

4. The probability of answering at least 1 of the first 5 questions correctly: Instead of calculating the probability of each possible correct scenario, it's easier to calculate the probability of answering all 5 questions incorrectly (which we've already done in step 3) and subtract that from 1. So, the probability is 1 - 0.237 = 0.763 or 763/1000.

So, the probabilities are:
- Answer the first question incorrectly: 0.750
- Answer the first 2 questions incorrectly: 0.5625
- Answer the first 5 questions incorrectly: 0.237
- Answer at least 1 of the first 5 questions correctly: 0.763

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dx/dy = d/dy(-e^(-y)) = -(-e^(-y)) * 1 = e^(-y). Therefore, dx/dy is equal to e^(-y).

The derivative of a function tells you the rate at which the function is changing with respect to its independent variable. In the case of dx/dy = -e^(-y), x is a function of y, and you're trying to find the derivative of x with respect to y.

To find dx/dy, you need to use the chain rule of differentiation. The chain rule states that if y is a function of t and x is a function of y, then dx/dt = dx/dy * dy/dt.

In this case, you have x as a function of y given by x = -e^(-y). So, you can rewrite dx/dy as d/dy(-e^(-y)).

To differentiate -e^(-y) with respect to y, you can use the chain rule again. The derivative of e^(-y) with respect to y is -e^(-y) (since the derivative of e^u with respect to u is e^u), and then you need to multiply by the derivative of -y with respect to y, which is -1.

So, dx/dy = d/dy(-e^(-y)) = -(-e^(-y)) * 1 = e^(-y). Therefore, dx/dy is equal to e^(-y).

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The length of the rectangular prisms is L = 2ft

We know that the volume of a rectangular prism of length L, width W, and height H is:

V = L*W*H

We know that:

V = 98 ft³

W = 7ft

H = 7ft

Replacing all that we will get:

98 ft³ = L*7ft*7ft

Solving this for L we will get:

(98 ft³)/(7ft*7ft) = L

2ft = L

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Answer:

2 1/4kms = how many meters?

Step-by-step explanation:

You're welcome.

Answer:

M = 2250

Step-by-step explanation:

First of all solve the mixed number which is 9/4 and as a decimal it is 2.25

Now as the meters it is....

2250!!!!

Hope this helps, have a great day!!!!!!

(Multiply 2.25 times 1000 and that gives you 2250)

The solution to the initial value problem is: y = (-1/9)e^(-3/2 t) + (1/3)(t – 4) + 10/9

To solve this initial value problem, we first need to find the homogeneous solution by setting H(t – 4) to 0. So we have:

2y' + 3y = 0

This is a first-order linear homogeneous differential equation, which we can solve using the separation of variables:

2y' = -3y

dy/y = -3/2 dt

ln|y| = -3/2 t + C

y = Ce^(-3/2 t)

Now we need to find the particular solution for H(t – 4) = 1. We can use the method of undetermined coefficients, guessing that the particular solution has the form y_p = A(t – 4) + B. Substituting this into the differential equation, we get:

2A + 3(A(t – 4) + B) = 1

Simplifying and equating coefficients, we get:

3A = 1

A = 1/3

Plugging this back into the equation and solving for B, we get:

2(1/3) + 3(1/3)(-4) + B = 0

B = 10/9

So the particular solution is y_p = (1/3)(t – 4) + 10/9.

The general solution is the sum of the homogeneous and particular solutions:

y = Ce^(-3/2 t) + (1/3)(t – 4) + 10/9

To find the value of C, we use the initial condition y(0) = 1:

1 = C + 10/9

C = -1/9

Therefore, the solution to the initial value problem is:

y = (-1/9)e^(-3/2 t) + (1/3)(t – 4) + 10/9

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The upper Darboux integral as [tex]$\frac{1}{4}b^4$[/tex]and the lower Darboux integral is 0.

The upper Darboux integral of a function f(x) on the interval [a,b] is defined as the supremum of the sums of the form

[tex]$\sum_{i=1}^n M_i(x_i - x_{i-1})$[/tex]where[tex]$M_i$[/tex] is the supremum of f(x) over the ith subinterval[tex]$[x_{i-1}, x_i]$[/tex]

Similarly, the lower Darboux integral is defined as the infimum of the same sums with the infimum of f(x) over each subinterval. For the function f(x) =[tex] x^3[/tex]

On the interval [0, b), we can see that the function is increasing and therefore its maximum value on each subinterval is achieved at the right endpoint. Thus, the upper Darboux integral is given by

[tex]$\int_0^b f(x)dx[/tex]  \sup\limits_{\mathcal{P}} \sum_=

[tex]{i=1}^n[/tex][tex]M_i(x_i - x_{i-1})[/tex] = [tex]lim_{|\mathcal{P}|\rightarrow 0} \sum_{i=1}^n f(x_i^)(x_i - x_{i-1}) [/tex][tex]{i=1}^n[/tex]

where $\mathcal{P}$ is a partition of [0,b] and $|\mathcal{P}|$ is the norm of the partition. Since $f(x) = [tex]x^3$[/tex]

is continuous on [0,b), we can apply Exercise 1.3 and Example 1 from chapter 1 to show that the limit above equals

f(x)= [tex]lim_{|\mathcal{P}|\rightarrow 0}[/tex][tex]sum_{i=1}^n (x_i^*)^3(x_i - x_{i-1})[/tex] = [tex]\frac{1}{4}b^4$[/tex]

Similarly, the lower Darboux integral can be computed using the left endpoint of each subinterval to get[tex]$\int_0^b [/tex]f(x)dx = [tex] \inf\limits_{\mathcal{P}} \sum_{i=1}^n[/tex][tex]m_i(x_i - x_{i-1})[/tex] =[tex] \lim_{|\mathcal{P}|\rightarrow 0} \sum_{i=1}^n (x_{i-1}^*)^3(x_i - x_{i-1}) = 0$[/tex]

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