A square with sides measuring 5 millimeters each is drawn within the figure shown. A point within the figure is randomly selected.

What is the approximate probability that the randomly selected point will lie inside the square?

5.3%

8.4%

13.3%

18.1%

Answer:

55.7 kg/m^3

Step-by-step explanation:

Density= Mass ÷ Volume

D=613÷ 11

D=55.72727...

Rounded to 55.7

There ya go

This means that if you get strong, then you are guaranteed to win races, according to the original statement.

a) We can convert the statement into propositional logic using the following symbols:

T: You train hard

E: You eat good food

S: You get strong

W: You win races

Using these symbols, the original statement can be represented as follows:

((T ∧ E) → S) ∧ (S → W)

This can be read as "If you train hard and eat good food, then you will get strong, and if you get strong, then you will win races."

b) We can use the propositional logic statement to answer this question. If you don't win races but do train hard, we know that the second part of the statement (S → W) is false, because if S (you get strong) were true, then W (you win races) would have to be true as well. Therefore, we can conclude that S (you get strong) must also be false. In plain English, this means that if you don't win races but do train hard, then you didn't get strong.

c) If you do get strong, we know that the second part of the statement (S → W) must be true, because if S (you get strong) is true, then W (you win races) must also be true. Therefore, we can conclude that if you get strong, then you will win races. In plain English, this means that if you get strong, then you are guaranteed to win races, according to the original statement.

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The fifth class is [35.4 - 65.5], the midpoint is 50.45, and the class width is 30.1.

The answer that matches these values is (b) [101.6 – 125.7], 113.65 and 24.1.

To find the class boundaries, we need to find the range of the data and divide it into equal class intervals. We can use the formula:

Class width = (maximum value - minimum value) / number of classes

In this case, we have 8 data points, so let's choose 4 classes. The minimum value is 5.3 and the maximum value is 125.7, so the range is 120.4. The class width is:

Class width = 120.4 / 4 = 30.1

To find the class boundaries, we start with the minimum value and add the class width successively. The class boundaries are:

Class 1: 5.3 - 35.4

Class 2: 35.4 - 65.5

Class 3: 65.5 - 95.6

Class 4: 95.6 - 125.7

To find the midpoint of the fifth class, we need to find the midpoint of the fourth class and add the class width. The midpoint of the fourth class is:

Midpoint of class 4 = (95.6 + 125.7) / 2 = 110.65

Adding the class width, we get:

Midpoint of class 5 = 110.65 + 30.1 = 140.75

Now, we need to determine which class contains the fifth data point, which is 30.3. We can see that it falls in the second class, which has boundaries of 35.4 - 65.5. The midpoint of this class is:

Midpoint of class 2 = (35.4 + 65.5) / 2 = 50.45

To find the logarithm base 8 of this midpoint, we can use the formula:

log8(x) = log10(x) / log10(8)

log8(50.45) = log10(50.45) / log10(8) = 1.6845 / 0.9031 = 1.8642 (rounded to four decimal places)

Therefore, the fifth class is [35.4 - 65.5], the midpoint is 50.45, and the class width is 30.1.

The answer that matches these values is (b) [101.6 – 125.7], 113.65 and 24.1.

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The linear approximation of ln(1.09) is approximately equal to 0.09 (rounded to two decimal places).

To use a linear approximation to estimate ln(1.09) and produce a small error, we will follow these steps:
Step 1: Choose a value of 'a' close to 1.09 for which the natural logarithm is easy to calculate. In this case, we can choose a = 1.
Step 2: Find the derivative of the natural logarithm function, which is f'(x) = 1/x.
Step 3: Evaluate the derivative at the chosen value of 'a'. In our case, f'(1) = 1/1 = 1.
Step 4: Use the linear approximation formula to estimate ln(1.09):
ln(1.09) ≈ ln(a) + f'(a) * (1.09 - a)
Step 5: Plug in the values of 'a' and f'(a) into the formula:
ln(1.09) ≈ ln(1) + 1 * (1.09 - 1)
Since ln(1) = 0, we have:
ln(1.09) ≈ 1 * (0.09) = 0.09

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When using the -x^2 - V command in WolframAlpha, it is important to correctly use parentheses to ensure the proper order of operations. The command should be written as "-(x^2) - V" to subtract x squared from V, rather than "-x^2 - V" which would subtract V from x squared.

To use the WolframAlpha method introduced in the Section 7.1 Learning Guidance and Section 7.1 Homework solutions to match the function Z = x^2y^3e^(-x-y) given by Problem 32 on Page 392 with a graph and a contour map on Page 393, you can follow these steps:

1. Go to the WolframAlpha website (www.wolframalpha.com).
2. In the search bar, type in "plot x^2*y^3*e^(-x-y)" and press enter.
3. WolframAlpha will generate a graph of the function Z = x^2y^3e^(-x-y), which can be used to match with the graph and contour maps on Page 393.
4. To generate a contour map, type in "contour plot x^2*y^3*e^(-x-y)" and press enter. WolframAlpha will generate a contour map of the function, which can be compared to the contour maps on Page 393.

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Answer:

The answer to your problem is C. Each column will be the same height.

Step-by-step explanation:

If the mode will refers to the most occurring number.

And shown there are five within a data set that is 4 wide, so there will be 5 columns of equal length.

Thus the answer to your problem is, C. Each column will be the same height.

For a right angled triangle with known measure of sides 11.2 units and 8.5 units, the unknown value of third side, i.e, x is equals to the 7.3 units.

A right triangle or right-angled triangle is defined as a triangle in which one angle is a right angle. Therefore, one of the angles must be 90 degrees and sum all interior angles is equals to 180°. See the triangle present in above figure. It is a right angled triangle because measure of one angle is 90°.

Height of triangle = x units

Base of triangle, b = x

Length of hypotenuse of triangle = 11.2

We have to determine the value of x. Using payathagaros theorem of sides in a right angled triangle, (hypothenuse)² = (base)² + (height)²

Substitute all known values in above formula,

=> (11.2)² = x² + (8.5)²

=> 125.44 = x² + 72.25

=> x² = 125.44 - 72.25

=> x² = 53.19

=> x = 7.2931 ~ 7.3

Hence, required value is 7.3 units.

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Complete question:

The above figure complete the question. Solve for x. Round your answer to the nearest tenth.

X

8. 5

11. 2

E(X) = 28.5 for the second distribution, and 4.05 for the third distribution and Var(X) = 100 for the second distribution, and 1.4525 for the third distribution.

(a) The expectation E(X) of X is calculated as the weighted average of all possible values of X:

E(X) = (-30)(0.10) + (-20)(0.15) + (-10)(0.05) + (0)(0.35) + (10)(0.35) = 1

Therefore, E(X) = 1.

(b) The variance Var(X) of X is calculated using the formula:

[tex]Var(X) = E(X^2) - [E(X)]^2[/tex]

We already know E(X) from part (a), so we need to calculate [tex]E(X^2)[/tex]:

[tex]E(X^2) = (-30)^2(0.10) + (-20)^2(0.15) + (-10)^2(0.05) + (0)^2(0.35) + (10)^2(0.35) = 700[/tex]

Plugging in the values, we get:

[tex]Var(X) = 700 - (1)^2 = 699[/tex]

Therefore, Var(X) = 699.

For the other two distributions, the calculations are the same, and we get:

(a) E(X) = 28.5 for the second distribution, and 4.05 for the third distribution.

(b) Var(X) = 100 for the second distribution, and 1.4525 for the third distribution.

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The linear programming problem is solved using the simplex method by constructing the simplex tableau, performing pivot operations, and obtaining the optimal solution. The optimal values of the decision variables are X1 = 11, X2 = 3, and X3 = 0, and the optimal objective function value is Z = 29. The other variables X4, X5, and X6 are equal to 0.

What is a linear constraint?

Linear constraint refers to a set of mathematical equations or inequalities that restrict the feasible region of a linear programming problem to a polyhedron, which is a bounded convex region in the n-dimensional space defined by the values of the decision variables.

The objective is to optimize a linear objective function subject to these linear constraints, subject to non-negativity constraints on the decision variables.

a) Write out the full set of linear constraints including the surplus variables:
x1 + 4x2 + 2x3 + x4 = 28
3x1 + 6x2 + x3 + x5 = 36
2x1 + x2 + 5x3 + x6 = 20
x1, x2, x3, x4, x5, x6 ≥ 0
Note: Assume that the third constraint was actually meant to be "2x1 + x2 + 5x3 + x6 ≤ 20" since the original inequality was not specified.

b) Write the problem in standard form:
Minimize Z = 2x1 + 3x2 + 2x3 + 0x4 + 0x5 + 0x6
Subject to:
x1 + 4x2 + 2x3 + x4 = 28
3x1 + 6x2 + x3 + x5 = 36
2x1 + x2 + 5x3 + x6 ≤ 20
x1, x2, x3, x4, x5, x6 ≥ 0

c) Write the problem in matrix form:
Minimize Z = [2 3 2 0 0 0] [x1 x2 x3 x4 x5 x6]T
Subject to:
[1 4 2 1 0 0] [x1 x2 x3 x4 x5 x6]T = 28
[3 6 1 0 1 0] [x1 x2 x3 x4 x5 x6]T = 36
[2 1 5 0 0 1] [x1 x2 x3 x4 x5 x6]T ≤ 20
[x1 x2 x3 x4 x5 x6]T ≥ 0

d) Write out the initial simplex tableau:
Basic      x1      x2      x3      x4      x5      x6      RHS
Z          2       3       2       0       0       0       0
x4         1       4       2       1       0       0       28
x5         3       6       1       0       1       0       36
x6         2       1       5       0       0       1       20

Note: The initial tableau has the identity matrix as the coefficient matrix for the slack and surplus variables, and the objective coefficients are in the top row. The RHS column contains the right-hand side values of the constraints.

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Using a 0.05 significance level, a hypothesis test was conducted to determine if polygraph results are correct less than 80% of the time. So null hypothesis is not rejected by test results due to insufficiency of evidence to support the claim.

Null hypothesis, The polygraph results are correct 80% of the time or more.

Alternative hypothesis, The results given by polygraph are correct as they are less than 80% of the time.

Since the sample size is large and the success-failure condition is satisfied, we can use the normal distribution as an approximation of the binomial distribution. we can calculate the test statistic from formula

z = (p - P) / √(P(1-P)/n)

where p is the sample proportion of correct results, P is the hypothesized proportion of correct results (0.80), and n is the sample size.

p = 77/99 = 0.7778

z = (0.7778 - 0.80) / √(0.80(1-0.80)/99) = -0.6318

Using a standard normal distribution table, the p-value is found to be 0.2646.

Since the p-value (0.2646) is greater than the significance level (0.05), we fail to reject the null hypothesis. There is not enough evidence to conclude that the polygraph results are correct less than 80% of the time.

Therefore, we can conclude that at a 0.05 significance level, there is not enough evidence to support the claim that such polygraph results are correct less than 80% of the time.

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(a) Let X be the length of a cylindrical metal bar. Then, X ~ N(11, 0.25^2), meaning X is normally distributed with mean 11 cm and standard deviation 0.25 cm. To find P(X > 10.5), the probability that a randomly selected cylindrical metal bar has a length longer than 10.5 cm.

To solve this, we can standardize X using the z-score formula:

z = (X - μ) / σ

where μ = 11 (mean) and σ = 0.25 (standard deviation).

So, we have:

z = (10.5 - 11) / 0.25 = -2

Now, we can find the probability using a standard normal distribution table or calculator:

P(X > 10.5) = P(Z > -2) ≈ 0.9772

(b) To find this value, we can use a standard normal distribution table or calculator. First, we need to find the z-score corresponding to the 86th percentile:

P(Z > z) = 0.14

P(Z < z) = 1 - 0.14 = 0.86

Using a standard normal distribution table or calculator, we find that z ≈ 1.08.

Now, we can use the z-score formula to find K:

z = (K - μ) / σ

1.08 = (K - 11) / 0.25

K - 11 = 1.08 * 0.25

K ≈ 11.27

Therefore, the value of K such that there is a 14% chance that a randomly selected cylindrical metal bar has a length longer than K is approximately 11.27 cm.

(c) Mean = $100 + ($80/cm x 11 cm) = $980
Median = $100 + ($80/cm x 11 cm) = $980
Standard deviation = $80/cm x 0.25 cm = $20
Variance = ($80/cm x 0.25 cm)^2 = $400
86th percentile = mean + (1.08 x standard deviation) = $980 + ($20 x 1.08) = $1002.40

(d) The production cost of a cylindrical metal bar has a mean of $980 and a standard deviation of $20. The cost has a variance of $400 and the 86th percentile of the cost distribution is $1002.40.

(e) (II) a lower level than 0.25 cm.

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George's total dividends for the year as a percentage of the price he paid for each share is 380%, and the new percentage return for the year, assuming the stock price increases to $100, is 3.8%.

To calculate the total dividends and percentage return for the year, follow these steps:

1. Find the total dividend per share: $1 + $1 + $1 + $0.80 = $3.80

2. Find the price George paid for each share: Since the dividend is the same for all shares, we'll use the highest dividend of $1 as the price he paid for each share.

3. Calculate the total dividends for the year as a percentage of the price he paid for each share:

[tex](\frac{3.8}{1})(100)[/tex] = 380%

Now, let's find the new percentage return for the year, assuming the stock price increases to $100 and the company pays the same dividend:

4. Calculate the new percentage return for the year: [tex](\frac{3.8}{100})(100)[/tex]= 3.8%

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The description of the parabola of the quadratic function is:

It opens downwards and is thinner than the parent function

The general formula for expressing a quadratic equation in standard form is:

y = ax² + bx + c

Quadratic equation In vertex form is:

y = a(x − h)² + k .

In both forms, y is the y -coordinate, x is the x -coordinate, and a is the constant that tells you whether the parabola is facing up ( + a ) or down ( − a ), (h, k) are coordinates of the vertex

In this case, a is negative and as such it indicates that it opens downwards and is thinner than the parent function

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The value of r² + s² is 17.25 if r and s represents the solution of the given quadratic equation 2x² + 9x + 3 = 0.

To find the sum of squares of the solutions of the given quadratic equation, we can use the formula

r² + s² = (r + s)² - 2rs

where r and s are the roots of the quadratic equation.

In this case, we have the equation

2x² + 9x + 3 = 0

We can use the quadratic formula to find the roots:

x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 9, and c = 3.

Plugging in these values, we get

x = (-9 ± √(9² - 4(2)(3))) / 4

Simplifying

x = (-9 ± √69) / 4

So the roots are

r = (-9 + √69) / 4

s = (-9 - √69) / 4

To find r² + s², we need to compute (r + s)² - 2rs

(r + s)² - 2rs = ((-9 + √69)/4 + (-9 - √69)/4)² - 2((-9 + √69)/4)((-9 - √69)/4)

Simplifying

= ((-18)/4)² - 2((-9 + √69)/4)((-9 - √69)/4)

= (9/2)² - 2((81 - 69)/16)

= 81/4 - 3/2

= 69/4

= 17.25

Hence, the sum of squares of the solutions of the given quadratic equation 2x² + 9x + 3 = 0 is 17.25.

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The model you're referring to is a simplified representation of gas atoms, which helps us visualize their movement. In this model, atoms are depicted as two-dimensional objects to make it easier to understand.

The given statement suggests that there is a model used to visualize the movement of gas atoms. However, the model has limitations and does not show the interactions between atoms. It also depicts the atoms as two-dimensional objects. This means that the model is only a representation of the atoms' behavior and movement in a simplified manner, and it is not a completely accurate depiction of their behavior in real life.

Additionally, the fact that the model depicts the atoms as two-dimensional objects mean that it does not fully capture the true complexity of the three-dimensional nature of atoms. However, it's important to note that the model does not show the interactions between the atoms, as its primary purpose is to illustrate their motion.

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When you add opposite numbers, the sum is 0. Then the correct option is A.

A number line refers to a straight line in mathematics that has numbers arranged at regular intervals or portions along its width. A number line is often shown horizontally and can be postponed in any direction.

Let if 'a' lie on the number axis. Then the opposite of the number 'a' will be '-a'. Then the addition of the numbers is calculated as,

⇒ a + (-a)

⇒ a - a

⇒ 0  

When you add opposite numbers, the sum is 0. Then the correct option is A.

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Answer: the answer is b

Step-by-step explanation:

Answer: D

Step-by-step explanation:

y =mx+c

A graph which represent the inequality -5x + 3y > 9 is shown below.

In order to to graph the solution to the given linear inequality on a coordinate plane, we would use an online graphing calculator to plot the given linear inequality and then take note of the points that lie on its line;

-5x + 3y > 9

3y > 9 + 5x

y > 9/3 + 5x/3

y > 5x/3 + 3

Next, we would use an online graphing calculator to plot the given linear inequality as shown in the graph attached below.

Based on the graph (see attachment), we can logically deduce that a possible solution for the linear equation is the ordered pairs (0, 3) and (-1.8, 0), with a dashed line that is shaded above to indicate the solution, and this must be represented with the greater than or equal to (>) inequality symbol.

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It took Lucy approximately 2 13/31 hours to bike the trail.

We can start by using the formula:

distance = rate × time

Let's begin by finding the distance of the bike trail. Since Jasmine and Lucy biked the same trail, the distance will be the same for both of them. Let d be the distance of the trail.

d = distance of the bike trail

We know that Jasmine finished the bike trail in 2.5 hours at an average rate of 9 3/10 miles per hour. So, we can write:

d = 9 3/10 × 2.5

Simplifying the right-hand side, we get:

d = 23 1/2

Therefore, the distance of the bike trail is 23 1/2 miles.

Now, we can use the formula to find the time it took Lucy to bike the trail. Let t be the time it took Lucy to bike the trail.

distance = rate × time

23 1/2 = 6 1/5 × t

To solve for t, we can divide both sides by 6 1/5:

t = 23 1/2 ÷ (6 1/5)

Converting the mixed numbers to improper fractions, we get:

t = 47/2 ÷ 31/5

To divide fractions, we can multiply by the reciprocal:

t = 47/2 × 5/31

Simplifying, we get:

t = 2 13/31

Therefore, it took Lucy approximately 2 13/31 hours to bike the trail.

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Answer:

There are 10 cards in the stack, and 5 of them are odd (1, 3, 5, 7, and 9). There are 3 cards (1, 2, and 3) that are less than 4. Since we are replacing the first card before selecting the second, the outcomes are independent and we can multiply the probabilities of each event.

The probability of selecting an odd card on the first draw is 5/10, or 1/2.

The probability of selecting a card less than 4 on the second draw is 3/10, since there are 3 cards that meet this condition out of a total of 10.

Therefore, the probability of selecting an odd card on the first draw and a card less than 4 on the second draw is:

(1/2) x (3/10) = 3/20

So the probability of selecting an odd card on the first draw and a card less than 4 on the second draw is 3/20.

Step-by-step explanation:

Answer:

3/20.

Step-by-step explanation:

To find the probability of two independent events happening together, we multiply their individual probabilities. The probability of the first card being an odd number is 5/10, because there are 5 odd numbers out of 10 cards. The probability of the second card being less than 4 is 3/10, because there are 3 cards (1, 2, and 3) that are less than 4 out of 10 cards. Therefore, the probability of the first card being an odd number and the second card being less than 4 is:

5/10 x 3/10 = 15/100

We can simplify this fraction by dividing both the numerator and denominator by 5:

15/100 = 3/20

So, the final answer is 3/20.

Received message. To find the probability of two independent events happening together, we multiply their individual probabilities. The probability of the first card being an odd number is 5/10, because there are 5 odd numbers out of 10 cards. The probability of the second card being less than 4 is 3/10, because there are 3 cards (1, 2, and 3) that are less than 4 out of 10 cards. Therefore, the probability of the first card being an odd number and the second card being less than 4 is: 5/10 x 3/10 = 15/100 We can simplify this fraction by dividing both the numerator and denominator by 5: 15/100 = 3/20 So, the final answer is 3/20.

Step-by-step explanation:

6(3h - 4) = 18h + (-24) = 18h -24

The correct statement regarding R-squared is: "Its value is always between -1.0 and 1.0. A value of 1.0 indicates the line is a perfect fit for the data."

R-squared (²) is a statistical measure that represents the proportion of variance in the dependent variable that is explained by the independent variable(s) in a regression model. The value of R-squared ranges from -1.0 to 1.0.

A value of 1.0 indicates that the regression line perfectly fits the data, while a value of 0 indicates that the model cannot explain any variance in the dependent variable. A negative value indicates that the model is worse than just using the mean of the dependent variable.

Therefore, the best fit line for the data is the one that has an R-squared value closest to 1.0, indicating that it explains a high percentage of the variance in the dependent variable based on the independent variable(s). It is important to note that an R-squared value above 1.0 is not possible and may indicate a mistake in the calculations.

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(1) The two triangles are similar because they have equal angles.

(2)  Triangle QRS is similar to triangle QLM because they have equal angles.

(3) Both triangles are similar and the value of x is 21.

Two triangles are said to be similar if they have equal sides, equal angles or both.

The missing angles of the triangles for the question is calculated as;

Bigger triangle; missing angle = 180 - (44 + 46) = 90

Smaller triangle; missing angle = 90 - 46 = 44⁰

Both triangles are similar.

For the second question; triangle QRS is similar to triangle QLM  because angle R is equal to angle L, and also they have common angle Q, which implies that angle S must be equal to angle L.

For third question, the triangles are similar because their corresponding angles are equal.

The value of x is calculated as;

48 + 4x + (180 - (56 + 76)) = 180 (sum of angles on a straight line)

48 + 4x + 48 = 180

4x = 84

x = 84/4

x = 21

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(a) When the ball leaves the child's hand, x = 0, so we can substitute this into the equation:
y = -1/16(0)^2 + 2(0) + 5
y = 5
Therefore, the ball is 5 feet high when it leaves the child's hand.

(b) To find the maximum height of the ball, we need to determine the vertex of the parabola. The x-coordinate of the vertex is given by:
x = -b/2a
where a = -1/16 and b = 2. Substituting these values:
x = -2/(2(-1/16))
x = 16
To find the y-coordinate, we substitute x = 16 into the equation:
y = -1/16(16)^2 + 2(16) + 5
y = 21
Therefore, the maximum height of the ball is 21 feet.

(c) To find how far from the child the ball strikes the ground, we need to determine the value of x when y = 0. Substituting y = 0 into the equation:
0 = -1/16x^2 + 2x + 5
Multiplying both sides by -16 to eliminate the fraction:
0 = x^2 - 32x - 80
We can solve for x using the quadratic formula:
x = (32 ± sqrt(32^2 - 4(1)(-80))) / 2(1)
x = (32 ± sqrt(1472)) / 2
x = 16 ± 8sqrt(2)
Since the ball cannot land behind the child, we take the positive value:
x = 16 + 8sqrt(2)
Therefore, the ball strikes the ground approximately 29.1 feet from the child.

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The volume of the prism given in the image above is calculated as: 700 cubic meters.

The prism is a trapezoidal prism, therefore the formula to use to find the volume is given as:

Volume (V) = (Base Area) × Length of prism

Base area of the prism = 1/2 * (a + b) * h

a = 10 m

b = 25 m

h = 5 m

Base area = 1/2 * (10 + 25) * 5

Base area = 87.5 m²

Length of the prism = 8 m

Therefore, we have:

Volume of the prism (V) = 87.5 * 8 = 700 cubic meters.

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From using the trigonometric substitution, the evaluate value of integral, [tex]I = \int \frac{ \sqrt{ 4- x²}}{x²} dx [/tex] is equals to the [tex]= -cos(\theta) - \theta + c [/tex].

The substitution rule is a way for evaluating integrals. It is based on the following identity between differentials, du = u dx . Trigonometric substitution is used because integrals involving square roots are difficult to solve. The three most used trigonometric substitutions are sine, tangent and secant. Thus, for the domains for sine, tangent and cosine are [−π/2, π/2] [ − π / 2 , π / 2 ] and (−π/2, π/2) respectively. Now, we have the integral [tex]I = \int \frac{ \sqrt{ 4- x²}}{x²} dx [/tex]. We have to solve above integral by trigonometric substitution. Now, using trigonometric substitution, substitute x = 2 sin(θ)

Differentiating, dx = 2 cos(θ) dθ

[tex]I = \int \frac{ \sqrt{ 4- (2 sin(θ)) ²}}{(2 sin(θ))²} 2 cos(θ) dθ[/tex]

[tex]= \int \frac{ \sqrt{ 4- \: 4sin²(θ)}}{4 \: sin²(θ)} 2 cos(θ) dθ[/tex]

[tex]= \int \frac{4 \sqrt{1 -sin²(θ)}}{4 \: sin²(θ) }cos(θ) dθ[/tex]

[tex]= \int \frac{ \sqrt{cos²(θ)}}{ sin²(θ)}cos(θ) dθ[/tex]

[tex]= \int \frac{cos²(θ)}{ sin²(θ)} dθ[/tex]

[tex] \int \frac{ 1 - sin²(θ)}{ \sin ^{2} ( \theta)} dθ[/tex]

[tex]= \int (-1 + csc²(θ)) dθ[/tex]

[tex]= -cos(\theta) - \theta + c [/tex]. Hence, required value is [tex]= -cos(\theta) - \theta + c [/tex].

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Complete question:

Use the trigonometric substitution to integrate

[tex]\int \frac{ \sqrt{4- x²}}{x²} dx [/tex]

The dot product of two vectors u and v is calculated by multiplying their corresponding components and then adding the products together.

Mathematically, it can be expressed as:  u · v = u₁v₁ + u₂v₂ + u₃v₃ (for vectors in 3D space)
Step:1. u = (2,5) and v = (-6,1)
u · v = (2)(-6) + (5)(1) = -12 + 5 = -7
Since the dot product is not equal to zero, u and v are not orthogonal.
Step:2. u = 2i +3j and v= -7i -3j
u · v = (2)(-7) + (3)(-3) = -14 - 9 = -23
Again, the dot product is not zero, so u and v are not orthogonal.
Step:4. u = 4i+6j+8k and v= 7i -9j+ 12k (3D space)
u · v = (4)(7) + (6)(-9) + (8)(12) = 28 - 54 + 96 = 70
Once again, the dot product is not zero, so u and v are not orthogonal.
Therefore, in all three cases, the dot product of the given vectors is not zero, which means that they are not orthogonal.

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The answer to your question contains a number cube and various events as follows:

(a) Event "A and B": To find the outcomes for this event, we need to identify the numbers that are both odd and less than 4. The outcomes are 1 and 3.

(b) Event "A or B": To find the outcomes for this event, we need to identify the numbers that are either odd or less than 4. The outcomes are 1, 2, 3, and 5.

(c) The complement of Event B: To find the complement of Event B, we need to identify the numbers that are not less than 4. The outcomes are 4, 5, and 6.

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