Question 2 ( 7 Marks): Use substitution method to evaluate the following integral: I= ∫x √x²+1 dx

The exact values of the cosecant, secant, and cotangent ratios of 5π/6 are 2, -2/√3, and -√3, respectively.

To find the exact values of the cosecant, secant, and cotangent ratios of an angle of 5π/6, we need to use the definitions of these trigonometric functions and the values of the sine, cosine, and tangent of this angle.

First, we can find the sine and cosine of 5π/6 using the unit circle or reference angles:

sin(5π/6) = sin(π/6) = 1/2

cos(5π/6) = -cos(π/6) = -√3/2

Then, we can use the definitions of the cosecant, secant, and cotangent ratios:

cosec(5π/6) = 1/sin(5π/6) = 1/(1/2) = 2

sec(5π/6) = 1/cos(5π/6) = -2/√3

cot(5π/6) = cos(5π/6)/sin(5π/6) = (-√3/2)/(1/2) = -√3

Therefore, the exact values of the cosecant, secant, and cotangent ratios of 5π/6 are 2, -2/√3, and -√3, respectively.

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The cam lift is 3 1/2 inches, which is option D.

To determine the lift of the cam, we need to find the distance from the center of the disc to the highest point of the cam surface.

First, we can find the distance from the center of the disc to the edge of the 1/2" hole. Since the hole is drilled 1 1/2" off-center, this distance is:

(4"/2) - 1 1/2" = 1"

Next, we can find the radius of the cam surface by adding the radius of the shaft (1/2") to the distance from the center of the disc to the edge of the 1/2" hole (1"):

1/2" + 1" = 1 1/2"

Finally, we can find the distance from the center of the disc to the highest point of the cam surface by adding the radius of the disc (4"/2 = 2") to the radius of the cam surface (1 1/2"):

2" + 1 1/2" = 3 1/2"

Therefore, the lift of the cam is 3 1/2 inches, which is option D.

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The simplified expression for this problem is given as follows:

[tex]\sqrt{y^6} = y^3[/tex]

The expression for this problem is defined as follows:

[tex]\sqrt{y^6}[/tex]

The power of a power rule is used when a single base is elevated to multiple exponents, and the simplified expression is obtained keeping the bases and multiplying the exponents.

The square root is equivalent to an exponent of 1/2, while the exponent of y is of 6, hence the exponent f the simplified expression is given as follows

1/2 x 6 = 3.

Hence the simplified expression for this problem is given as follows:

[tex]\sqrt{y^6} = y^3[/tex]

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The amount of money left for gift c is 15 5/10 dollars

Timothy had 30 dollars to spend on three gifts

He spent 10 9/10 dollars on gift A

He spent 3 3/5 dollars on gift B

The amount spent on gift C can be calculated as follows

10 9/10 + 3 3/5

= 109/10 + 18/5

= 545 + 180/50

= 725/50

= 145/10

30 - 145/10

= 30/1 - 145/10

= 300-145/10

= 155/10

= 15 5/10

Hence the amount of Money left for gift C is 15 5/10 dollars

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The population growth rate is given as follows:

k = 0.0213.

The exponential function modeling the population after t years is given as follows:

P(t) = P(0)e^(kt).

In which:

Considering the population in 2007, the initial population is given as follows:

P(0) = 46.8.

21 years later, we have that:

P(21) = 73.2.

Hence the rate is obtained as follows:

46.8e^(21k) = 73.2

e^(21k) = 73.2/46.8

21k = ln(73.2/46.8)

k = ln(73.2/46.8)/21

k = 0.0213.

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The solution to this equation 4x² – 15x = 4 include the following: D. x = 4, -1/4.

In Mathematics and Geometry, the general form of a quadratic function can be modeled and represented by using the following quadratic equation;

y = ax² + bx + c

Where:

Next, we would solve the quadratic function by using the factorization method as follows;

4x² – 15x = 4

Subtract 4 from both sides of the quadratic function:

0 = 4x² – 15x - 4

0 = 4x² - 16x + x - 4

(4x + 1)(x - 4)

x = 4 or x = -1/4.

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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 equation of the line written in general form (4,5) (1,2) is y - x = 1.

The formula to find the line equation from two points is as follows -

(y - [tex] y_{1}[/tex])/(x - [tex] x_{1}[/tex]) = ([tex] y_{2}[/tex] - [tex] y_{1}[/tex])/([tex] x_{2}[/tex] - [tex] x_{1}[/tex]). In the formula, [tex] y_{1}[/tex] and [tex] y_{2}[/tex] are initial and final y-axis values and [tex] x_{1}[/tex] and [tex] x_{2}[/tex] are initial and final x-axis values.

Keep the values in formula to find the equation -

(y - 5)/(x - 4) = (2 - 5)/(1 - 4)

Simplifying the equation

(y - 5)/(x - 4) = -3/-3

Divide the values

(y - 5)/(x - 4) = 1

Rewriting the equation

(y - 5) = (x - 4)

Rearranging the equation

y - x = 5 - 4

y - x = 1

Hence, the equation of line is y - x = 1.

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The approximate amount of area in the circle is 113.04 square feet.

The area of a circle is given through the expression [tex]A = \pi r^2[/tex], in which π is about equal to 3.14, and r is the radius of the circle.

In this instance, we are given the diameter of the circle, that is 12 feet. The radius of the circle is half of the periphery, so the radius is

r = 12 / 2 = 6 feet

Now, we're suitable to use the methodology for the area of a circle to discover the approximate amount of space within the circle

[tex]A = \pi r^2 = 3.14 * 6^2 = 3.14 * 36 \approx 113.04[/tex] square feet[tex]A = \pi r^2[/tex]

Accordingly, the approximate amount of area in the circle is 113.04 square feet.

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The given absolute value function is f(x) = |x - 3x|.

To evaluate absolute value function f(-2), we substitute -2 in place of x:

f(-2) = |-2 - 3(-2)|

= |-2 + 6|

= |4|

= 4

Therefore, f(-2) = 4.

To evaluate f(0), we substitute 0 in place of x:

f(0) = |0 - 3(0)|

= |0|

= 0

Therefore, f(0) = 0.

To evaluate f(2), we substitute 2 in place of x:

f(2) = |2 - 3(2)|

= |-4|

= 4

Therefore, f(2) = 4.

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The solution to the given set of equations is, No solution.

We have to given that;

A set of equations is given below:

Equation C: y = 6x + 9

Equation D: y = 6x + 2

Now, We can see that;

y = 6x + 9

y = 6x + 2

Equate both equations, we get;

6x + 2 = 6x + 9

2 ≠ 9

Hence, We get;

The solution to the given set of equations is, No solution.

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The graph of the given inequality y ≥ x² + 1 is a shaded region above the downward-facing parabola -x² + 1.

Hence, graph A represents the given inequality.

The inequality  y ≥ x² + 1 represents a region in the coordinate plane where y is greater than or equal to the value of the function -x² + 1 for any given x. The graph of this inequality is a shaded region above the downward-facing parabola -x² + 1. The vertex of this parabola is located at the point (0,1), and as x moves away from 0, the value of the function becomes more negative.

Therefore, the shaded region includes all points (x, y) where y is greater than or equal to the y-value of the parabola at that x-value. The resulting graph is a curve that opens downward and flattens out at y=1 as x moves further away from 0.

Hence, the correct option is A.

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The component form and magnitude of the vector are;

v = 3·i - 2·j

||v|| = √(14)

The component form of a vector is expressed as <x, y>, which indicates the distance of the vector to the right or left, x-component, and the distance up or down of the y-component.

The coordinate points on the vector are; (0, 0), and (3, -2)

The component form of the vector consists of the horizontal and vertical components, which can be found as follows;

Horizontal component = (3 - 0)·i = 3·i

The vertical component = (-2 - 0)·j = -2·j

The component form of the vector is therefore;

v = ⟨3, -2⟩

The magnitude of the vector can be obtained as follows;

||v|| = √((3 - 0)² + (-2 - 0)²) = √(14)

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If a point (x,y) is equidist point from the point (2,3) and (6,1) then the equation of locus is given by 2x -y =6

Using distance formula, the distnace between point (x, y) and (2, 3) is,

d₁ = √[(2 - x)² + (3 - y)²]

And  the distnace between point (x, y) and (6, 1) is,

d₂ = √[(6 - x)² + (1 - y)²]

A point (x,y) is equidist point from the point (2,3) and (6,1)

⇒ d₁ = d₂

⇒  √[(2 - x)² + (3 - y)²] = √[(6 - x)² + (1 - y)²]

⇒ (2 - x)² + (3 - y)² = (6 - x)² + (1 - y)²

⇒ 4 - 4x + x² + 9 - 6y + y² = 36 - 12x + x² + 1 - 2y + y²

⇒ -4x + 12x - 6y + 2y = 37 - 13

⇒ 8x - 4y = 24

⇒ 2x -y = 6

Hence proved.

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The complete question is:

if a point (x,y) is equidist point from the point (2,3) and (6,1) show that the equation of locus is given by 2x -y = 6

The probability that the answer is correct given that the

a) Let's use Bayes' theorem to calculate the probability that the answer is correct given that the person you asked said "East". Let H be the event that the person is honest, L be the event that the person is a liar, E be the event that Julia resides in the East and W be the event that Julia resides in the West. Then we have:

P(E|H) = 2/3 (the probability that an honest person gives the correct answer)

P(E|L) = 1/2 (the probability that a liar gives the correct answer)

P(H) = 1/4 (the probability that the person is honest)

P(L) = 3/4 (the probability that the person is a liar)

By the law of total probability, we have:

P(E) = P(E|H)P(H) + P(E|L)P(L) = (2/3)(1/4) + (1/2)(3/4) = 5/12

Then, using Bayes' theorem, we have:

P(H|E) = P(E|H)P(H)/P(E) = (2/3)(1/4)/(5/12) = 2/5

So the probability that the answer is correct given that the person said "East" is 2/5.

b) The probability that the same person gives the same answer twice in a row is:

P(E∩E) = P(E)P(E|H)P(H) + P(E)P(E|L)P(L) = (5/12)(2/3)(1/4) + (5/12)(1/2)(3/4) = 5/24

Using Bayes' theorem again, we have:

P(H|EE) = P(EE|H)P(H)/P(EE) = (2/3)^2(1/4)/(5/24) = 8/15

So the probability that the answer is correct given that the person said "East" twice in a row is 8/15.

c) The probability that the same person gives the same answer three times in a row is:

P(E∩E∩E) = P(E)P(E|H)^2P(H) + P(E)P(E|L)^2P(L) = (5/12)(2/3)^2(1/4) + (5/12)(1/2)^2(3/4) = 5/32

Using Bayes' theorem again, we have:

P(H|EEE) = P(EEE|H)P(H)/P(EEE) = (2/3)^3(1/4)/(5/32) = 4/5

So the probability that the answer is correct given that the person said "East" three times in a row is 4/5.

d) The probability that the same person gives the same answer four times in a row is:

P(E∩E∩E∩E) = P(E)P(E|H)^3P(H) + P(E)P(E|L)^3P(L) = (5/12)(2/3)^3(1/4) + (5/12)(1/2)^3(3/4) = 5/48

Using Bayes' theorem again, we have:

P(H|EEEE) = P(EEEE|H)P(H)/P(EEEE) = (2/3)^4(1/4)/(5/48) = 16/25

So the probability that the answer is correct given that the

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Answer: C

Step-by-step explanation: When did the calendar change from 13 months?

The 1752 Calendar Change

Today, Americans are used to a calendar with a "year" based the earth's rotation around the sun, with "months" having no relationship to the cycles of the moon and New Years Day falling on January 1. However, that system was not adopted in England and its colonies until 1752.

To find the probability of the event "the first die is 6 or the sum is 8," we need to count the number of outcomes that satisfy this event and divide it by the total number of possible outcomes.

To find the probability of the event "the first die is 6 or the sum is 8," we need to consider the total possible outcomes when rolling two dice and the favorable outcomes for this event.

Total possible outcomes: 6 sides on each die, so there are 6 x 6 = 36 possible outcomes.

Favorable outcomes:
1. First die is 6: There are 6 possible outcomes (6,1), (6,2), (6,3), (6,4), (6,5), and (6,6).
2. Sum is 8: There are 5 possible outcomes (2,6), (3,5), (4,4), (5,3), and (6,2).

However, (6,2) is counted twice, so we should subtract 1 from the total favorable outcomes: 6 + 5 - 1 = 10.

Probability = Favorable outcomes / Total possible outcomes = 10/36. Simplifying the fraction gives 5/18.

So, the probability of the event "the first die is 6 or the sum is 8" is 5/18.

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Mathematical induction can be used to prove that for every integer k ≥ 5, k^2 - 3k ≥ 10.

Base Case: Let k = 5,

Then,  k^2 - 3k = 5^2 - 3(5) = 10

Since 10 >= 10 is true, the base case holds.

Inductive Step: Assume that for some integer n >= 5, n^2 - 3n >= 10 is true.

We want to prove that (n + 1)^2 - 3(n + 1) >= 10 is also true.

Expanding the left-hand side of the inequality, we get:
(n + 1)^2 - 3(n + 1) = n^2 + 2n + 1 - 3n - 3

On simplifying ,we get:
n^2 - n - 2 >= 0
On factoring,we get:

(n - 2)(n + 1) >= 0

Since n >= 5, n - 2 >= 3, and n + 1 >= 6, so both factors are positive. Therefore, the inequality is true for all n >= 5.

By mathematical induction, we have proved that for every integer k >= 5, k^2 - 3k >= 10.

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Answer:

To calculate the probability that Kaelyn was selected out of 30 people chosen at random from a group of 480 people, we can use the concept of probability.

Given:

Total number of people in the group (N) = 480

Number of people chosen (n) = 30

We want to find the probability of Kaelyn being selected, which is the probability of selecting 1 person out of 480 people, or 1/480.

So, the probability that Kaelyn was selected is:

P(Kaelyn) = 1/480

This is the final answer, as it represents the probability of Kaelyn being selected out of 30 people chosen at random from a group of 480 people.

Step-by-step explanation:

Answer:

To calculate the probability that Kaelyn was selected out of 30 people chosen

Step-by-step explanation:

Answer:

11

Step-by-step explanation:

Volume of right cone = (1/3) · π · r² · h

V = 968π

h = 24 units

Let's solve

968π = (1/3) · π · r² · 24

2904π =  π · r² · 24

121π = π · r²

121 = r²

r = 11

So, the radius is 11 units.

In the statistical argument provided, Emily concludes that everyone in the class must have received a high grade on the quiz based on the information from her study group. The fallacy being committed in this argument is a combination of A. Biased Sample Fallacy and B. Hasty Generalization Fallacy.

A. Biased Sample Fallacy occurs when the sample is used to make a conclusion that is not representative of the entire population. In this case, Emily's study group consists of 16 students out of a total of 30 students in her class. The study group may not be representative of the whole class, as it is a smaller sample and could be composed of more diligent or prepared students.

B. Hasty Generalization Fallacy is when a conclusion is made based on insufficient evidence. In this argument, Emily concludes that everyone in the class must have received a high grade based on the performance of her study group alone. This is a hasty generalization as she has not considered the performance of the other students in the class.

To sum up, the argument commits both A. Biased Sample Fallacy and B. Hasty Generalization Fallacy, as it bases its conclusion on a potentially unrepresentative sample and insufficient evidence.

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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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Answer:

73 1/2

Step-by-step explanation:

Answer:

Inches (fraction) : 73 1/2

Inches: 73.5

Feet: 6.13

Meters: 1.87

Centimeters: 186.69

Millimeters: 1,866.9

Step-by-step explanation:

The area of the given rectangle is 322 square miles.

We know that the area of a rectangle is equal to the product between the dimensions. In this case we know that the dimensions of the rectangle are:

Length  = 23 miles.

Width = 14 miles.

Then the area of this rectangle will be a product between these two values, we will get:

Area = (23 mi)*(14 mi)

Area = 322 mi ²

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Answer

The result of 48 - 36 is 12. Then, if you divide 12 by 36, the result is 0.3333 or 1/3.

Step-by-step explanation:

(a) 40,475,358.

(b) 330
(c) 0.0008%.

(a) To find |S|, the total number of sets of 7 distinct numbers a player can choose, we need to find the combinations of choosing 7 numbers from the 80 available options. This can be calculated using the combination formula:

C(n, k) = n! / (k! * (n - k)!)

In this case, n = 80 (total numbers) and k = 7 (numbers to choose). So, |S| = C(80, 7):

|S| = 80! / (7! * (80 - 7)!)
|S| = 80! / (7! * 73!)

(b) To find |E|, the number of sets where all 7 numbers chosen by the player are in the winning set of 11 numbers chosen by the organizers, we need to find the combinations of choosing 7 numbers from the 11 available options in the winning set:

|E| = C(11, 7)
|E| = 11! / (7! * (11 - 7)!)
|E| = 11! / (7! * 4!)

(c) To find the probability of winning, we need to calculate the ratio of the favorable outcomes (|E|) to the total possible outcomes (|S|):

P(winning) = |E| / |S|

P(winning) = (11! / (7! * 4!)) / (80! / (7! * 73!))

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The sample space for rolling a fair six-sided die is {1, 2, 3, 4, 5, 6}.

option C.

A sample space describes the possible outcome of an event.

The sample space for rolling a fair six-sided die consists of the six possible outcomes and we can list them as follows;

S = {1, 2, 3, 4, 5, 6}

Each outcome represents the number that appears on the top face of the die after it is rolled. So this sample space contains all the likely outcome each time we roll the die.

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The decimal tax rate to four decimal places is 0.0519.

To find the decimal tax rate, we want to divide the amount of money to be raised by the means of property tax by the assessed cost of taxable assets, and then convert it to a decimal place as it is requested and needed .

Decimal tax charge = (amount of money raised by assets tax / Assessed value of taxable assets)

Decimal tax price = ($28,000,000 / $540,000,000)

Decimal tax fee = zero.0519 (rounded to four decimal places)

Consequently, the decimal tax rate to four decimal places is 0.0519.

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