For the following​ event, state whether you think the difference between what occurred and what you would expect by chance is statistically significant. Explain.
An airline with a 95​% ​on-time departure rate has 16 out of 300 flights with late departures.

The enough evidence to support the claim that the wait time to see an IRS representative is more than 50 minutes at a 0.100 level of significance.

The null hypothesis is, H0:

µ ≤ 50 minutes.

The alternative hypothesis is,

Ha: µ > 50 minutes.

The significance level is α = 0.100Sample size = 300

Sample mean (x) = 52 minutes

Sample standard deviation (s) = 4 minutes

We have to find whether there is enough evidence to support the claim that the waiting time to see an IRS representative is more than 50 minutes or not.

For that, we perform a one-sample t-test.

The test statistic is given by:

t = (x - µ) / (s/√n)t

= (52 - 50) / (4/√300)t

= 7.905

Critical t-value = t 0.100,299 = 1.646

Since the calculated t-value (7.905) is greater than the critical t-value (1.646), we can reject the null hypothesis.

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The y-coordinate of point P is 1.To find the y-coordinate of the point P on the parabola y = 3x^2 - 11x + 10 where the slope is 7, we can differentiate the equation to find the derivative. The derivative of y = 3x^2 - 11x + 10 is y' = 6x - 11.

To find the x-coordinate of point P, we can set the derivative equal to the given slope: 6x - 11 = 7. Solving for x, we get x = 3.

To find the y-coordinate of point P, we substitute the x-coordinate back into the original equation: y = 3(3^2) - 11(3) + 10. Simplifying, we find y = 1.

Therefore, the y-coordinate of point P is 1.

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The trigonometric ratios for this problem are given as follows:

The three trigonometric ratios are the sine, the cosine and the tangent of an angle, and they are obtained according to the formulas presented as follows:

The hypotenuse for this problem is of 34, and the sides are given as follows:

Hence the trigonometric ratios are given as follows:

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The approximate market value of the bond is $225.

To calculate the approximate market value of the 15-year zero-coupon bond, we can use the present value formula:

Market Value = Par Value * PV Factor

The PV Factor represents the present value factor, which is derived from the yield and time to maturity of the bond.

Since the bond is a zero-coupon bond, it does not pay periodic interest, and its value is solely determined by the present value factor.

Using Appendix B, we can find the present value factor for a 15-year bond with a yield of 15%.

Let's assume the PV Factor is 0.225.

Market Value = $1,000 * 0.225

= $225

The approximate market value of the bond is $225.

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The 2 : 5 scaled version of the deck Sam plans to build and the dimensions of the original deck indicates;

Part A; Base length of the new deck = 7.2 feet

Height of the new deck = 2.8 feet

Part B; The area of the original deck is 63 square feet

The area of the new deck is 10.08 square feet

Part C; The ratio of the areas is the square of the scale factor

A scale factor is a number or factor that is used to enlarge or reduce the dimensions a shape or size of a figure.

The base length of the triangular deck = 18 feet

The height of the triangular deck = 7 feet

The scale factor for the scaled version Sam intends to build = 2 : 5

Part A; The dimensions of the new deck are;

Base length of the new deck using the the 2 : 5 ratio is; (2/5) × 18 = 7.2 feet

The height of the new deck = (2/5) × 7 = 2.8 feet

Part B; The area of the original deck = (1/2) × 18 × 7 = 63 square feet

Area of the new dec = (1/2) × 7.2 × 2.8 = 10.08 square feet

Part C; The ratio of the areas is; 10.08/63

Ratio of the area = 10.08/63 = 4/25 = 4 : 25

The scale factor is; 2 : 5

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The domain of the given table is { -3, 0, 3 } and the range is { -6, 0, 6 }. Option A.

The given table represents a set of ordered pairs (x, y). The x-values are -3, 0, and 3, and the corresponding y-values are -6, 0, and 6. To determine the domain and range, we need to identify the set of all possible x-values and y-values.

Domain: The domain represents the set of all possible x-values in the given table. In this case, the x-values are -3, 0, and 3. Therefore, the domain is { -3, 0, 3 }.

Range: The range represents the set of all possible y-values in the given table. In this case, the y-values are -6, 0, and 6. Therefore, the range is { -6, 0, 6 }.

Based on the above analysis, the correct answer is:

a. domain { -3, 0, 3 }, range: { -6, 0, 6 }.

This option correctly identifies the values in the given table as the domain and range, matching the values -3, 0, 3 for the domain and -6, 0, 6 for the range. Therefore, option a is the best answer. Option A is correct.

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The other side of the triangle is approximately 4.664 cm.

Let's denote the other side of the triangle as x. We know that the perimeter of the triangle is 22 cm, and one of the sides is 9 cm. The perimeter of a triangle is the sum of the lengths of its three sides. So, we can set up the equation:

9 + x + z = 22

where z represents the remaining side.

Now, we are given that the area of the triangle is 20.976 cm². The area of a triangle can be calculated using the formula:

Area = (1/2) * base * height

Since we know the area and one side (9 cm), we can rearrange the formula to solve for the height (which is the remaining side, z):

z = (2 * Area) / 9

Substituting the given values, we get:

z = (2 * 20.976) / 9

z ≈ 4.664 cm

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Answer: The answer choice to this question would be:

A) The height of the water increases 2 inches per minute.

Step-by-step explanation:

I'm 100% Sure this is the Correct Answer!! ✅

The amount of air required to fill the hemisphere is 9408284.599 mm³

The quantity of paint required is 2023 cm³

4. For a hemisphere, the volume is calculated using the formula

2/3 π r³

The radius is 165 mm. plugging the value results to

= 2/3 π 165³

= 9408284.599 mm³

5. The volume of the prism is solved using the formula

= length * width * depth

= 17 * 17 * 7

= 2023 cm³

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The reduced row echelon form of a matrix allows us to determine whether the associated linear system is consistent or inconsistent, unique or has infinitely many solutions.

In reduced row echelon form, a matrix has the following properties:

Each leading entry (the leftmost nonzero entry) of a row is equal to 1.

Each leading entry is the only nonzero entry in its column.

All entries below a leading entry are zeros.

The leading entry in each row is to the right of the leading entry in the row above it.

Based on these properties, we can make the following conclusions about the solution to the associated linear system:

Consistency: If there are no rows of the form [0 0 ... 0 | b] (where b is a nonzero constant), meaning there are no contradictory equations, the linear system is consistent. In other words, there is at least one solution.

Uniqueness: If there are no free variables (columns without leading entries), meaning each column corresponds to a pivot column, the linear system has a unique solution. This means there is only one set of values for the variables that satisfies all the equations.

Infinite solutions: If there are one or more free variables, the linear system has infinitely many solutions. This occurs when there are more variables than equations or when there are dependent equations.

The reduced row echelon form provides an organized representation of the linear system that simplifies the process of determining its solution. By analyzing the structure of the matrix, we can determine the consistency, uniqueness, and the nature of the solution set.

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The formula for the area of the base (B) of a triangular pyramid is given by option C: B = ½bh.

In a triangular pyramid, the base is a triangle, and the area of a triangle is calculated using the formula ½bh, where b represents the length of the base of the triangle and h represents the corresponding height. Therefore, the formula for the area of the base (B) is B = ½bh.

The base of a triangular pyramid is a two-dimensional shape, and its area is determined by the length of the base and the height perpendicular to it. The formula B = ½bh is commonly used to find the area of a triangle, which applies to the base of the pyramid in this case.

The ½ factor accounts for the fact that the base is a triangle, and the product of the base length (b) and the corresponding height (h) yields the area of the triangle. Hence, option C correctly represents the formula for the base area of a triangular pyramid.

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A median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side.

The term that describes a line segment connecting a vertex of a triangle to the midpoint of the opposite side is the "median." In triangle geometry, a median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.

To understand the concept of a median, let's consider a triangle ABC. The midpoint of side BC is denoted as M, and vertex A is connected to M by a line segment. This line segment AM is referred to as the median from vertex A.

Medians have some interesting properties and play a significant role in triangle geometry. Here are a few key characteristics of medians:

1. Medians Divide the Triangle into Two Equal Areas:

Each median of a triangle divides the triangle into two regions with equal areas. The point where all three medians intersect is called the centroid, which is also the center of mass of the triangle.

2. Medians are Concurrent:

The three medians of a triangle are always concurrent, meaning they intersect at a single point called the centroid. This centroid divides each median in a 2:1 ratio, with the longer segment adjacent to the vertex.

3. Medians Divide the Triangle into Six Congruent Triangles:

The medians of a triangle divide the triangle into six smaller congruent triangles. Each of these triangles shares a common vertex with the original triangle.

4. Medians Determine the Centroid:

The centroid of a triangle is the point of intersection of the three medians. It is the balance point of the triangle, where the triangle would perfectly balance on a needle.

In summary, a median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side. Medians have unique properties, including dividing the triangle into equal areas, being concurrent at the centroid, dividing the triangle into congruent triangles, and determining the balance point of the triangle.

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OD. 9

The location of the point on the number line that is of the way from A = -4 to B = 17 would be 9.

We can calculate it as follows:

Total distance between -4 and 17 is 17 - (-4) = 21

We want the point that is of the way from -4 to 17. Since 4/5 = 0.8, we multiply 21 by 0.8 which gives 16.8.

Rounding 16.8 to the nearest integer gives us 9.

Therefore, the answer is OD: 9

Answer:

∠ G = 73° , ∠ H = 98°

Step-by-step explanation:

EFGH is a cyclic quadrilateral, its 4 vertices lie on the circle.

the opposite angles sum to 180°

A

∠ E + ∠ G = 180° , that is

11x + 8 + 8x + 1 = 180

19x + 9 = 180 ( subtract 9 from both sides )

19x = 171 ( divide both sides by 19 )

x = 9

Then

∠ G = 8x + 1 = 8(9) + 1 = 72 + 1 = 73°

B

∠ H + ∠ F = 180° , that is

6y - 4 + 5y - 3 = 180

11y - 7 = 180 ( add 7 to both sides )

11y = 187 ( divide both sides by 11 )

y = 17

Then

∠ H = 6y - 4 = 6(17) - 4 = 102 - 4 = 98°

Answer:

A)  m∠G = 73°

B)  m∠H = 98°

Step-by-step explanation:

The diagram shows a cyclic quadrilateral.

A cyclic quadrilateral is a four-sided shape drawn inside a circle, where every vertex touches the circle's circumference.

As the opposite angles in a cyclic quadrilateral add up to 180°, then:

We can use these equations to find the values of x and y.

m∠E + m∠G = 180°

(11x + 8)° + (8x + 1)° = 180°

11x + 8 + 8x + 1 = 180

19x + 9 = 180

19x + 9 - 9 = 180 - 9

19x = 171

19x ÷ 19 = 171 ÷ 19

x = 9

m∠H + m∠F = 180°

(6y - 4)° + (5y - 3)° = 180°

6y - 4 + 5y - 3 = 180

11y - 7 = 180

11y - 7 + 7 = 180 + 7

11y = 187

11y ÷ 11 = 187 ÷ 11

y = 17

Now we have found the values of x and y, substitute them back into the angle expressions to find m∠G and m∠H.

m∠G = (8x + 1)°

m∠G = (8(9) + 1)°

m∠G = (72 + 1)°

m∠G = 73°

m∠H = (6y - 4)°

m∠H = (6(17) - 4)°

m∠H = (102 - 4)°

m∠H = 98°

Answer:

[tex]\textsf{(C)} \quad \dfrac{1}{4}[/tex]

Step-by-step explanation:

To find the probability of a point chosen at random being in the shaded area of the given diagram, we first need to calculate the areas of the larger circle and the shaded circle.

The formula for the area of a circle is A = πr², where r is the radius.

Given the radius of the larger circle is 8 units:

[tex]\begin{aligned}\sf Area\;of\;the\;larger\;circle&=\pi (8)^2\\&=64 \pi \end{aligned}[/tex]

Given the radius of the shaded circle is 4 units:

[tex]\begin{aligned}\sf Area\;of\;the\;shaded\;circle&=\pi (4)^2\\&=16 \pi \end{aligned}[/tex]

[tex]\boxed{\sf Probability\:of\:an\:event\:occurring = \dfrac{Number\:of\:ways\:it\:can\:occur}{Total\:number\:of\:possible\:outcomes}}[/tex]

To find the probability that a point chosen at random is in the shaded area, divide the area of the shaded circle by the area of the larger circle:

[tex]\sf Probability= \dfrac{16 \pi}{64 \pi}=\dfrac{1}{4}[/tex]

Therefore, the probability of a point chosen at random being in the shaded area is 1/4.

Answer:

[tex]\frac{x}{y}[/tex] 1 over 4 (one-fourth)

Step-by-step explanation:

To determine how long it will take to make 20 slices of toast with the given information, we can analyze the patterns and ratios observed in the data.

From the provided data, we can see that the time it takes to make toast is directly proportional to the number of slices. Specifically, as the number of slices doubles, the time doubles as well.

Using this pattern, we can calculate the estimated time it would take to make 20 slices.

Starting with the given information:

   35 seconds for 4 slices

   70 seconds for 8 slices

   105 seconds for 10 slices

Since the time doubles as the number of slices doubles, we can estimate that it would take approximately 140 seconds for 16 slices (double of 8 slices).

Now, doubling again, we can estimate that it would take approximately 280 seconds for 20 slices (double of 10 slices).

Therefore, it is estimated that it will take around 280 seconds to make 20 slices of toast.

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What Amy did wrong was not moving the pointer to the arc intersection with the horizontal side

An angle bisector is defined in geometry as a line that splits an angle into two equal angles.

The steps involved in constructing an angle bisector are;

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Answer:

The correct answer is,

"Look at the graph of the relationship. Count the number of units up and the number of units to the right one must

move to arrive at the next point on the graph. Write these two numbers as a fraction."

Step-by-step explanation:

Answer:

Look at the graph of the relationship. Find two points which have y-values that are one unit apart. The unit rate is the difference in the corresponding x-values.

Step-by-step explanation:

To find the unit rate for a proportional relationship, you need to identify any two points on the graph of the relationship that have y-values that differ by one unit. Once you have identified these two points, you can find the unit rate by calculating the difference between the corresponding x-values (the change in x) for those two points:

1. Look at the graph of the proportional relationship.

2. Identify any two points on the graph that have y-values that differ by one unit.

3. Determine the x-value for each of the two points you identified in step 2.

4. Calculate the difference between the two x-values (the change in x) for those two points.

5. The result of step 4 is the unit rate for the proportional relationship.

Using substitution method, the solution to the system of linear equations

x = 26667 and y = 533

Let's set up the system of equations based on the given information.

Let:

x = number of currency A

y = number of currency B

Equation 1: x + y = 3200 (Total budgeted spending money)

Equation 2: 1.30x + 1.50y = Total cost of purchasing currency (to be determined)

Based on the information provided, Abigail wants to have five times as much currency A as currency B:

Equation 3: x = 5y

Now, let's solve the system of equations using substitution method

Substitute Equation 3 into Equation 1:

5y + y = 3200

6y = 3200

y = 3200 / 6

y = 533.33 (approximately)

Substitute the value of y back into Equation 3 to find x:

x = 5(533.33)

x = 2666.67 (approximately)

Now, let's calculate the total cost of purchasing currency using Equation 2:

Total cost of purchasing currency = 1.30x + 1.50y

Total cost of purchasing currency = 1.30(2666.67) + 1.50(533.33)

Total cost of purchasing currency = 3466.67 + 800

Total cost of purchasing currency = 4266.67 (approximately)

Therefore, the system of equations has a solution where:

x ≈ 2667 (number of currency A)

y ≈ 533 (number of currency B)

This means that Abigail should exchange approximately 2667 units of currency A and 533 units of currency B. The total cost of purchasing the currency would be approximately $4266.67.

In practical terms, this means that Abigail will have a budgeted spending money of $3200, and she plans to allocate it between currency A and currency B based on the given exchange rates and her preference of having five times as much currency A as currency B.

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There are 21 different city combinations possible for Peter's summer traveling.

To determine the number of different city combinations possible for Peter's summer traveling, we need to calculate the number of combinations of choosing 5 cities out of the 7 available cities.

Since the order in which he visits the cities does not matter, we are dealing with combinations rather than permutations.

The formula to calculate the number of combinations is given by the binomial coefficient, also known as "n choose k," denoted as C(n, k) or (nCk).

In this case, we have 7 cities to choose from, and Peter will visit 5 cities. Thus, we can calculate the number of city combinations as follows:

C(7, 5) = 7! / (5! * (7-5)!)

= 7! / (5! * 2!)

= (7 * 6 * 5!) / (5! * 2)

= (7 * 6) / 2

= 42 / 2

= 21

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The expanded expression equivalent to log6((2y-3y)/9) using the Quotient Rule of Logarithms is log6(2y) - log6(3y).

The Quotient Rule of Logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Using this rule, we can expand the given expression log6((2y-3y)/9) as follows:

log6((2y-3y)/9) = log6(2y/9) - log6(3y/9)

Now, let's simplify each logarithmic expression separately:

For the first term, log6(2y/9), we can write it as:

log6(2y/9) = log6(2y) - log6(9)

For the second term, log6(3y/9), we can write it as:

log6(3y/9) = log6(3y) - log6(9)

Combining these expressions, we have:

log6((2y-3y)/9) = (log6(2y) - log6(9)) - (log6(3y) - log6(9))

Now, let's simplify further by distributing the negative sign:

log6((2y-3y)/9) = log6(2y) - log6(9) - log6(3y) + log6(9)

Notice that log6(9) appears both as a subtraction and addition term. This cancels out, resulting in:

log6((2y-3y)/9) = log6(2y) - log6(3y)

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The equivalent equation to ax + by = c is by = ax + c.

The equivalent equation to ax + by = c can be found by isolating the y variable. Let's go through the options provided:

Y = a/b x + c/b: This equation represents the slope-intercept form of a linear equation (y = mx + b). It does not match the given equation, so it is not the correct answer.

Y = ax + c: This equation represents a linear equation in slope-intercept form. It does not have the y-variable coefficient (b) present, so it is not the correct answer.

Y = -a/b x + c/b: This equation represents the slope-intercept form of a linear equation. The signs of the variables are reversed compared to the given equation, so it is not the correct answer.

by = ax + c: This equation matches the given equation ax + by = c, where the y-variable is isolated on one side. Therefore, the correct answer is option 4.

In conclusion, the equivalent equation to ax + by = c is by = ax + c.

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The scale factor and the value of x for each figure is given as follows:

A) Scale factor of 1/3, x = 7 m.

B) Scale factor 0.4747, x = 4.5 in.

For Figure A, we have that the ratio between the areas is given as follows:

510/4590 = 1/9.

As the area is measured in square units, while the side lengths are measured in units, the scale factor is the square root of 1/9, hence it is given as follows:

1/3.

Then the value of x is obtained as follows:

x = 21 x 1/3

x = 7 m.

For Figure B, we have that the ratio between the areas is given as follows:

16/71 = 0.22535.

The scale factor is then the square root of 0.22535, which is given as follows:

0.4747.

Then the value of x is given as follows:

x = 9.5 x 0.4747

x = 4.5 in.

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The three dramatic techniques that make "Look Back in Anger" interesting are social realism, strong language and rhetoric, and character conflict and dynamic relationships.

"Look Back in Anger" by John Osborne is a play known for its compelling portrayal of post-war disillusionment and social critique. Here are three dramatic techniques used in the play that contribute to its enduring interest:

1. Social Realism: Osborne employed social realism, depicting the gritty realities of working-class life, alienation, and societal discontent. This authenticity and portrayal of relatable characters and situations engage the audience emotionally and intellectually.

2. Strong Language and Rhetoric: The play is characterized by sharp, biting dialogue and impassioned monologues. The use of strong language and rhetoric adds intensity, captures the characters' frustrations, and conveys their anger and disillusionment effectively.

3. Character Conflict and Dynamic Relationships: The play revolves around intense conflicts between characters, particularly the central couple, Jimmy and Alison. Their volatile relationship and the clash between Jimmy's anger-fueled rebellion and Alison's desire for a different life create tension and keep the audience engaged.

Overall, these dramatic techniques of social realism, powerful language, and dynamic relationships make "Look Back in Anger" interesting by effectively portraying societal issues, engaging the audience emotionally, and highlighting the complexities of human interactions.

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Answer:16

Step-by-step explanation:

25/100 × x = 4

x = 4 × 4

x = 16

The conclusions that can be drawn from the data shown in the table are:

- Twelve tickets cost $30.00.

- Each additional ticket costs $2.50.

- The table is a partial representation.

- (27.50, 11), (30, 12), and (32.50, 13) are the ordered pairs represented in the table.

The statement "(27.50, 11), (30, 12), and (32.50, 13) are the ordered pairs represented in the table" is correct. From the data shown in the table, we can draw the following conclusions:

1. Twelve tickets cost $30.00: Looking at the "Tickets" column, we can see that the entry "12" corresponds to the "Total Cost" entry of $30.00. Therefore, we can conclude that purchasing twelve tickets would cost $30.00.

2. Each additional ticket costs $2.50: By examining the "Tickets" column, we can observe that for each increase of one ticket (from 11 to 12, and from 12 to 13), the "Total Cost" in the second column increases by $2.50. This consistent pattern suggests that each additional ticket costs $2.50.

3. The table is a partial representation: The table only displays three rows of data, showing the "Tickets" and "Total Cost" for x values of 11, 12, and 13. Since the table does not provide information for all possible values of x, it is a partial representation of the relationship between the number of tickets and the total cost.

The statement "Thirty tickets cost $12.00" is not supported by the given data. The table does not include an entry for 30 tickets, and none of the given entries correspond to that value.

These ordered pairs match the values shown in the table, with the first element representing the number of tickets (x) and the second element representing the total cost (y) in dollars.

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(1) The integral of the function is (1/10)x⁵ - (1/8)x⁴ + (1/6)x³ + C.

(2) The integral is (1/4)(sec x)⁴ + eˣ(sec x) + (1/2)(sec x)² + C.

(3) The integral of cot²x dx is 1/sin(x) - sin(x) + C

(4)The integral of the function  [tex]\frac{3}{8} x^{\frac{8}{3} } - \frac{6}{13} x^{\frac{13}{3} } + 21x^{\frac{2}{3}} + C.[/tex]

(5) The shaded area under the curve is 7.22 sq units.

(1) The integral of (x⁶ - 2x⁵ + x⁴) / 2x² is determined as follows;

(x⁶ - 2x⁵ + x⁴) / 2x² = (x⁴(x² - 2x + 1)) / 2x²

= (x⁴(x - 1)²) / 2x²

= (x²(x - 1)²) / 2

∫(x²(x - 1)²) / 2 dx

= (1/2) ∫x²(x - 1)² dx

= (1/2) ∫x²(x² - 2x + 1) dx

= (1/2) ∫(x⁴ - 2x³ + x²) dx

= (1/2)(1/5)x⁵ - (1/2)(1/4)x⁴ + (1/2) (1/3)x³ + C

Simplifying further:

= (1/10)x⁵ - (1/8)x⁴ + (1/6)x³ + C

(2) The integral of (sec³x + eˣsecˣ + 1) / (sec x) dx, is calculated as follows;

(sec³x + eˣsecˣ + 1) / (sec x) = (sec³x + eˣsecˣ + 1)(sec x / sec x)

= (sec⁴x + eˣsec²x + sec x) / sec x

Note; sec x as 1/cos x

= sec⁴x/cos x + eˣsec²x/cos x + sec x/cos x

= sec³x/cos x + eˣsec x + sec x/cos x

Integrate by substitution method.

u = sec x

du = sec x tan x dx.

∫(sec³x + eˣsec x + sec x/cos x) dx

= ∫(u³ + eˣu + u) du

= (1/4)u⁴ + eˣu + (1/2)u² + C

Substitute u back in terms of sec x;

= (1/4)(sec x)⁴ + eˣ(sec x) + (1/2)(sec x)² + C

(3) The integral of cot²x dx;

cot²(x) = (cos²(x))/(sin²(x))

Let u = sin(x)

du = cos(x) dx

= ∫(1-u²)/u² du

= ∫(1/u²) - 1 du

= ∫u⁻² - 1 du

= -1/u - u + C

= -1/sin(x) - sin(x) + C

(4) The integral of the function is;

∫(x² - 2x³ + 7)/∛x dx =  ∫x²/∛x dx - ∫2x³/∛x dx + ∫7/∛x dx

= [tex]\frac{3}{8} x^{\frac{8}{3} } - \frac{6}{13} x^{\frac{13}{3} } + 21x^{\frac{2}{3}} + C.[/tex]

(5) The shaded area under the curve is calculated as follows;

the given function;

[tex]y = x^{1/2} - x^{2} + 2x[/tex]

∫y = A =  [tex]\frac{2}{3} x^{3/2} - \frac{1}{3} x^3 \ + x^2[/tex]

the limits = 2 and 0

A = [tex]\frac{2}{3} (2)^{3/2} - \frac{1}{3} (2) ^3 \ + (2)^2[/tex]

A = 1.89 - 2.67 + 8

A = 7.22 sq units

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The solutions to the system of equations is (a) (1, 5)

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

The graph

This point of intersection of the lines of the graph represent the solution to the system graphed

From the graph, we have the intersection point to be

(x, y) = (1, 5)

Hence, the solutions to the system is (1, 5)

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In the first row, the ordered pair can be expressed as f(8) = -6 in function notation.

Of(3) is 18.

Of(x) = -5 when x is -2.

The given table represents a function, and we need to express the values in function notation and determine specific inputs that correspond to certain outputs.

In the table's first row, the ordered pair are shown which can be expressed as function notation:

f(8) = -6

This means that when the input (x) is 8, the corresponding output (f(x)) is -6.

f(x) = -5 when x is -2.

This means that when the input (x) is -2, the value of the function (f(x)) is -5.

In the table, the function is defined by the equation f(x) = 8x - 6. Each input value (x) is multiplied by 8 and then subtracted by 6 to determine the corresponding output (f(x)).

By substituting different values for x into the function, we can find the corresponding values for f(x) in the table.

when x = 3, we have:

f(3) = 8(3) - 6

f(3) = 24 - 6

f(3) = 18

So, f(3) is 18.

Similarly, when x = -2, we have:

f(-2) = 8(-2) - 6

f(-2) = -16 - 6

f(-2) = -22

Thus, when x is -2, f(x) is -22.

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Answer:

(x + 3)(5x + 2) , (2x - 1)(3x + 5)

Step-by-step explanation:

given

A = 5x² + 12x + 6

consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term

product = 5 × 6 = 30 and sum = 17

the factors are + 15 and + 2

use these factors to split the x- term

5x² + 15x + 2x + 6 ( factor the first/second and third/fourth terms )

= 5x(x + 3) + 2(x + 3) ← factor out (x + 3) from each term

= (x + 3)(5x + 2)

then

length = x + 3 , breadth = 5x + 2 or indeed the other way round

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given

A = 6x² + 7x - 5

consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term

product = 6 × - 5 = - 30 and sum = + 7

the factors are - 3 and + 10

use these factors to split the x- term

6x² - 3x + 10x - 5 ( factor the first/second and third/fourth terms )

= 3x(2x - 1) + 5(2x - 1) ← factor out (2x - 1) from each term

= (2x - 1()3x + 5)

then

length = 2x - 1 , breadth = 3x + 5 or the other way round