2x^2-13x+7=x^2 to the nearest tenth

The simplify value of numeric expression, 3 + 2, after adding 3456 then subtracting 45 and then dividing by 20 is equals the 17.8.

We have an expression of numbers, 3 + 2 we have to apply some arithematic operations on it and determine the final simplfy value. Let the expression be x = 3 + 2, add 3456 in it

=> x = 3 + 2 + 3456

Substracts 45 from above expression

=> x = 3 + 2 + 3456 - 45

Dividing the above expression of x by 20

=>

[tex]\frac{ x } {20} = \frac{ 3 + 2 + 3456 - 45}{20}[/tex]

[tex]= \frac{3416}{20}[/tex]

= 17.8

Hence, required simplify value is 17.8.

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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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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 chart that is typically used when the measure for a sample is weight, volume, number of inches or other variable measurements is the mean chart.

The mean chart is a statistical process control chart that plots the average or mean of the sample against the upper and lower control limits. This chart is useful when the process being measured produces continuous data that is normally distributed.
The range chart is used when the measure for the sample is the range of variation within the sample. This chart shows the difference between the largest and smallest values in the sample, and is useful for detecting changes in variability.

The C chart is used when the measure for the sample is the number of defects or occurrences within a given unit of measurement. This chart is useful for measuring the process capability of a system and identifying areas where improvements can be made.
Finally, the P chart is used when the measure for the sample is the proportion of defective items within a given sample. This chart is useful for measuring the quality of a product or process and identifying areas where defects are occurring.

Overall, the mean chart is the most commonly used chart for variable measurements, but the specific chart chosen will depend on the nature of the data being collected and the goals of the analysis.
To briefly explain each of the chart types:

1. Mean chart: Used for monitoring the central tendency of a variable over time.
2. Range chart: Used for monitoring the variability of a continuous variable, like weight, volume, or number of inches, over time.
3. C chart: Used for monitoring the number of defects in a unit of measure (e.g., per item or per batch) over time.
4. P chart: Used for monitoring the proportion of defective items in a sample over time.

In your case, since you are working with variable measurements like weight, volume, and the number of inches, the most appropriate chart to use is the Range chart (#2). This chart will help you monitor the variability of the measured data over time and allow you to analyze any patterns or trends that may emerge.

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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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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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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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Value of a is $52,427.69

To find the value of "a" in this situation, we can use the formula for the present value of an annuity:

PV = a * [1 - (1 + r)^(-n)] / r

where PV is the present value of the loan, a is the amount of each payment, r is the annual interest rate (expressed as a decimal), and n is the number of payments.

In this case, we know that PV = $335,000, r = 0.14, and n = 15. We want to solve for a.

Substituting these values into the formula, we get:

$335,000 = a * [1 - (1 + 0.14)^(-15)] / 0.14

Simplifying, we get:

a = $52,427.69

Therefore, the person assuming responsibility for this loan would need to make 15 equal payments of $52,427.69 each year for the next 15 years in order to repay the loan. The total amount paid would be $786,415.35, which includes both the principal amount of $335,000 and $451,415.35 in interest.

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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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Juan is 13 years old.

Andrés is 34 years old.

We have,

Let's assume that Juan's age is x.

Then, we know that Andrés' age is x + 21.

We also know that the sum of their ages is 47:

x + (x + 21) = 47

Simplifying the equation:

2x + 21 = 47

Subtracting 21 from both sides:

2x = 26

Dividing by 2:

x = 13

So Juan is 13 years old.

To find Andrés' age, we can substitute Juan's age into the equation we used earlier:

x + 21 = 13 + 21 = 34

Thus,

Juan is 13 years old.

Andrés is 34 years old.

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The complete question.

Juan is 21 years younger than Andrés and we know that the sum of their ages is 47. How old is each of them?

The recurrence relation for the total prize money in the tournament is T(n) = 2T(n/2) + 100n, under the condition that tournament has n contestants, where n is a power of 2.



Let's us consider there are n players in the tournament where n is a power of 2. Each player wins k hundreds of dollars, where k is the number of people in the sub-tournament won by the player.

Let us present T(n) as the total prize money in a tournament with n players. We observe that  T(1) = 100 since there is only one player who loses in the first round and gets $100.

For n > 1, we can divide the tournament into two sub-tournaments each with n/2 players. Let's denote k as the number of people in a sub-tournament won by a player. Then we can see that k = n/2 for each player since each player wins one of two sub-tournaments.

Therefore, each player wins k hundreds of dollars where k = n/2. The total prize money for each sub-tournament is T(n/2). Therefore, we can write:

T(n) = 2T(n/2) + 100n

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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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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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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:

7x + 33 = 10x

3x = 33, so x = 11

These congruent alternate interior angles measure 110°.

The value of x in the parallel line is 11.

When parallel line are crossed by a transversal line, angle relationships are formed such as corresponding angles, alternate exterior angles, alternate interior angles, same side interior angles, vertically opposite angles etc.

Therefore, let's find the value of x using the angle relationship.

Hence,

7x + 33 = 10x (alternate interior angles)

33 = 10x - 7x

3x = 33

divide both sides by 3

x = 33 / 3

Therefore,

x = 11

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we can look at this as an exponential growth, and if something is P today and next year is 2P, hell it doubled and then 4P and so on, so doubling is implying that, whatever P is, will be twice that much in a year, or we can word it as, it'll be 100% more than what it's today, that said, we can just write a Growth equation for "t" years with an annual rate of 100%.

[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &50\\ r=rate\to 100\%\to \frac{100}{100}\dotfill &1\\ t=years \end{cases} \\\\\\ A = 50(1 + 1)^{t} \implies A = 50(2)^t[/tex]

Answer:

To find the point that is halfway between A(4, -2) and B(12, 10), we can find the average of the x-coordinates and the average of the y-coordinates.

average x-coordinate = (4 + 12)/2 = 8 average y-coordinate = (-2 + 10)/2 = 4

Therefore, the point that is halfway between A and B has the coordinates (8, 4), which is answer choice B.

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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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:

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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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 difference of three times a number x and six is greater than or equal to the sum of fifteen and twenty-four times the number is 3x-6≥15+24x

The difference of three times a number and six is greater than or equal to the sum of fifteen and twenty four times a number

Difference is subtraction and sum is nothing but addition

Let the number be x.

The given sentence is changed to the expression or inequality as given below.

3x-6≥15+24x

Hence, the difference of three times a number x and six is greater than or equal to the sum of fifteen and twenty-four times the number is 3x-6≥15+24x

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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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⊂ Hey, islandstay ⊃

Answer:

Slope = -2

Step-by-step explanation:

Formula for Slope(m):

(y₂ - y₁) / (x₂ - x₁)

Solve:

(x₁, y₁) and (x₂, y₂)

(2₁, -8₁) and (-4₂, 4₂)

Now put it in the slope formula;

4 - (-8) / -4-2

12/-6

Slope(m) = -2

xcookiex12

4/20/2023

If you have just arrived at the downtown bus stop, you should expect to wait about 26 minutes for a bus to arrive.

To calculate the expected waiting time, we need to find the weighted average of the waiting times for each interval, where the weights are the probabilities of each interval occurring.

Let t1, t2, and t3 be the waiting times for intervals of 20 minutes, 40 minutes, and 2 hours, respectively.

Then, we have:

t1 = 10 minutes (half the interval time)

t2 = 20 minutes (half the interval time)

t3 = 60 minutes (half the interval time)

The probabilities of each interval are 0.2, 0.4, and 0.4, respectively.

Therefore, the expected waiting time is:

E(waiting time) = 0.2 * t1 + 0.4 * t2 + 0.4 * t3

= 0.2 * 10 + 0.4 * 20 + 0.4 * 60

= 26 minutes

So, on average, you should expect to wait about 26 minutes for a bus to arrive.

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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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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 solution to the initial boundary value problem is u(x,t) = 1/π.

To solve the initial boundary value problem ut = 2uxx for x ∈ (-π, π], t ∈ [0, + [infinity]), ux(0, t) = ux,(1,t) = 0, for t ∈ [0, + [infinity]), u(x,0) = π^2 – π^2 for r ∈ [ -π, π], we can use the method of separation of variables.

Assume u(x,t) = X(x)T(t), then we have:

X''(x) + λX(x) = 0, T'(t) + 2λT(t) = 0

where λ is a separation constant. The general solution for the spatial equation is X(x) = A sin(nx) + B cos(nx), where n = sqrt(λ) and A, B are constants. Since u(0,t) = u(1,t) = 0, we have A = 0 and B cos(nπ) = 0, which implies n = kπ for k = 1, 2, 3, ... Thus, the spatial eigenfunctions are X_k(x) = cos(kπx), and the corresponding eigenvalues are λ_k = -(kπ)^2.

The time equation can be solved as T(t) = Ce^(-2λ_k t), where C is a constant. Therefore, the general solution for the initial boundary value problem is:

u(x,t) = Σ C_k cos(kπx) e^(-2(kπ)^2 t)

where the sum is taken over all k = 1, 2, 3, .... To determine the constants C_k, we use the initial condition u(x,0) = π^2 – π^2 = 0. This gives:

Σ C_k cos(kπx) = 0

Since the eigenfunctions form an orthogonal set on [-π, π], we can multiply both sides by cos(mπx) and integrate over [-π, π] to obtain:

C_m = 0 for m = 1, 2, 3, ...

Thus, the only non-zero constant is C_0, which can be determined using the normalization condition:

1 = ∫_(-π)^π (u(x,t))^2 dx = C_0^2 π^2

Therefore, C_0 = 1/π. Thus, the solution to the initial boundary value problem is:

u(x,t) = (1/π) cos(0πx) e^(-2(0π)^2 t) = 1/π e^0 = 1/π

In conclusion, the solution to the initial boundary value problem is u(x,t) = 1/π.

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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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