Question 3 (10 marks) Find an equation for the plane tangent to the surface z = x²y + xy^2 + In x+R at (1,0,R). Z=

Answer:

Based on the histogram displaying the ages of 50 randomly selected users of an online music service, it is more likely that advertising on the service will reach people who are younger than 30, as the frequency (or height) of the bars appears to be higher in the younger age group compared to the 30 and older age group.

Step-by-step explanation:

Therefore, the mean number of lemons that grew on Mary's lemon tree is 13.

The sum of the data divided by the total amount of data determines the mean of a set of numbers, also known as the average. Only numerical variables—regardless of whether they are discrete or continuous—can be used to determine the mean. Simply dividing the total number of values in a data collection by the sum of all of the values yields it.

Mean number of lemons that grew on Mary's lemon tree, we need to sum up the number of lemons from all four seasons and then divide by the total number of seasons. So, the sum of the number of lemons from all four seasons is:

3 + 15 + 21 + 13 = 52

Since there are four seasons, the mean number of lemons is:

=52/4

= 13

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The vertex of an angle would have the same coordinates after a rotation if it is rotated at angle of 360 degrees.

In Mathematics and Geometry, a rotation refers to a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

Generally speaking, when a point (x, y) is rotated about the center or origin (0, 0) in a counterclockwise (anticlockwise) direction by an angle θ, the coordinates of the new point (x′, y′) formed include the following:

x′ = xcos(θ) − ysin(θ)

y’ = xsin(θ) + ycos(θ).

(x′, y′) → (x, y) ⇒ (360 degrees rotation).

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The probabilities are given as follows:

a. P(number greater than 10) = 1/6.

b. P(number less than 5) = 1/3.

c. The solid is fair, as each side of the dice has the same probability of coming up.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The total number of outcomes for this problem is given as follows:

12.

2 of the numbers are greater than 10, which are 11 and 12, hence the probability is given as follows:

p = 2/12

p = 1/6.

4 of the numbers, which are 1, 2, 3 and 4, are less than 5, hence the probability is given as follows:

p = 4/12

p = 1/3.

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Based on the average daily balance, the finance charge for this credit card that charges 10.5% APR is $2.10.

The finance charge consists of the interest and other fees that lenders charge borrowers.

One of the methods for computing the finance charge is the average daily balance, which takes the sum of the daily balances and divides by the number of days in the billing cycle.

APR interest rate = 10.5%

Monthly period days = 30

Days 1-3: $200 (initial balance)        3 days      $600 ($200 x 3)

Days 4-20: $300 ($100 purchase)  17 days   $5,100 ($300 x 17)

Days 21-30: $150 ($150 payment)  10 days   $1,500 ($150 x 10)

Total balances = $7,200

Average daily balance = $240 ($7,200 ÷ 30)

Finance charge = $2.10 ($240 x 10.5% x 30/360)

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The analysis of the sample data (n=100) would help understand the impact of generic drugs on healthcare and prescription drug costs in the United States.

To understand how the availability of generic drugs may have affected the average expenditures on healthcare and prescription drugs in 2005, according to the National Center for Health Statistics (NCHS) report.

In 2002, Americans paid an average of $3,302 per year on healthcare and prescription drugs, as stated in the NCHS report titled "Health, United States." Researchers hypothesized that by 2005, the average expenditures would decrease primarily due to the increased availability of generic drugs. To test this hypothesis, a sample of 100 Americans was taken, and their expenditures on healthcare and prescription drugs in 2005 were recorded.

The sample data (n=100) would then be analyzed to determine if there was a significant decrease in average expenditures compared to the 2002 average of $3,302. This analysis would help understand the impact of generic drugs on healthcare and prescription drug costs in the United States.

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Claim: Youth voting can be beneficial for democracy and civic engagement.

Evidence: Only 6 percent of countries allow 16 year olds to vote which indicates is potential for more countries to explore this option.

Based on evidence, the author's third and final claim is that youth voting can be beneficial for democracy and civic engagement as its shows that allowing 16 and 17 year old to vote has been associated with higher voter turnout and increased civic engagement among young people.

The fact that only 6 percent of countries currently allow 16 year olds to vote suggests that there is potential for more countries to explore this option.

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The number of ways there are for her to select the starting line-up is 68,640 ways.

1. Center: There are 4 players who can play this position, so there are 4 choices.
2. Right Forward: Since one player has been selected as Center, there are now 13 players remaining. So, there are 13 choices for this position.
3. Left Forward: After selecting the Center and Right Forward, 12 players remain, resulting in 12 choices for this position.
4. Right Guard: With three players already chosen, there are 11 players left to choose from, giving us 11 choices.
5. Left Guard: Finally, after selecting players for the other four positions, 10 players remain, providing 10 choices for this position.

Now, we can calculate the total number of ways to select the starting line-up using the counting principle by multiplying the number of choices for each position:

4 (Center) × 13 (Right Forward) × 12 (Left Forward) × 11 (Right Guard) × 10 (Left Guard) = 68,640 ways

So, there are 68,640 ways for the coach to select the starting line-up.

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To use the upper and lower sums to approximate the area of a region, we need to first divide the region into subintervals of equal width. Let's say we have n subintervals.

The lower sum is the sum of the areas of rectangles whose heights are the minimum value of y in each subinterval. The upper sum is the sum of the areas of rectangles whose heights are the maximum value of y in each subinterval.

To approximate the area using the lower sum, we would calculate:

lower sum = (width of subinterval) x (minimum y value in subinterval) for each subinterval
area = sum of lower sums for all subintervals

To approximate the area using the upper sum, we would calculate:

upper sum = (width of subinterval) x (maximum y value in subinterval) for each subinterval
area = sum of upper sums for all subintervals

It's important to note that as the number of subintervals increases, the accuracy of our approximation improves. However, it also increases the amount of calculation needed. Therefore, we must find a balance between accuracy and efficiency in our calculations.

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The calculated slopes of the relations are -5, 9, -1/3 and 3

The table 1

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (13 - 18)/(-2 + 3)

Evaluate

Slope = -5

Ordered pair 2

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (9 + 27)/(1 + 3)

Evaluate

Slope = 9

Graph 3

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (0 - 1)/(3 - 0)

Evaluate

Slope = -1/3

The table 4

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (2 + 1)/(-2 + 3)

Evaluate

Slope = 3

Hence, the values of the slopes are -5, 9, -1/3 and 3

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The equation of the linear relationship that represents the table given is: y = 6x + 4.

The linear relationship between x and y can be represented as an equation which can be expressed in slope-intercept form as:

y = mx + b [m is the slope and b is the y-intercept]

The y-intercept of the equation is the value of y, when x = 0, which is b = 4.

Using any two points, (0, 4) and (2, 16), we have:

Slope (m) = change in y / change in x = 16 - 4 / 2 - 0

Slope (m) = 12/2 = 6

Substitute m = 6 and b = 4 into y = mx + b:

y = 6x + 4.

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The distance between (2 + i) and (4 +3i) would be,

⇒ d = 2√2

We have to given that;

To find distance between (2 + i) and (4 +3i).

Now, We can formulate;

Two points are (2, 1) and (4, 3).

We know that;

The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Hence, The distance between (2 + i) and (4 +3i) would be,

⇒ d = √(4 - 2)² + (3 - 1)²

⇒ d = √4 + 4

⇒ d = √8

⇒ d = 2√2

Thus, The distance between (2 + i) and (4 +3i) would be,

⇒ d = 2√2

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We have shown that cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ and c > 0.

To show that cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ, and c>0, we will use the CDF method:

1. Define the transformation: Let Y = cX be a function of the random variable X, where c > 0.

2. Find the CDF of Y: We want to find P(Y ≤ y), which is equal to P(cX ≤ y) or P(X ≤ y/c).

3. Express CDF of Y in terms of X: Since P(X ≤ y/c) is the CDF of X at y/c, we have FY(y) = FX(y/c).

4. Find the PDF of X: The exponential distribution has the PDF fX(x) = λ * exp(-λx) for x ≥ 0.

5. Differentiate the CDF of Y to find its PDF: To find fY(y), we differentiate FY(y) with respect to y. Using the chain rule, we have:

fY(y) = d(FX(y/c))/dy = fX(y/c) * (1/c)

6. Substitute the PDF of X: Now, we replace fX(y/c) with its exponential form λ * exp(-λ(y/c)):

fY(y) = (λ * exp(-λ(y/c))) * (1/c)

7. Simplify the expression: fY(y) = (λ/c) * exp(-λ(y/c))

This is the PDF of an exponential distribution with parameter λ/c. Therefore, cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ and c > 0.

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The probability that either Paula or Quentin is last in line when n kids randomly line up for recess can be calculated using the concept of permutations and combinations.

To calculate the probability, we can consider two cases: (1) Paula is last in line, and (2) Quentin is last in line.

Case 1: Paula is last in line

In this case, the probability that Paula is last in line is 1/n, since there is only one way for her to be at the end of the line. The other (n-1) kids can be arranged in any order, so there are (n-1)! possible arrangements. Therefore, the probability that Paula is last in line is:

P(Paula is last) = 1/n * (n-1)! = (n-1)! / n!

Case 2: Quentin is last in line

Similarly, the probability that Quentin is last in line is also 1/n. The other (n-1) kids can be arranged in any order, so there are (n-1)! possible arrangements. Therefore, the probability that Quentin is last in line is:

P(Quentin is last) = 1/n * (n-1)! = (n-1)! / n!

To calculate the probability that either Paula or Quentin is last in line, we can add the probabilities of the two cases:

P(Paula or Quentin is last) = P(Paula is last) + P(Quentin is last)
= (n-1)! / n! + (n-1)! / n!
= 2(n-1)! / n!

Therefore, the probability that either Paula or Quentin is last in line when n kids randomly line up for recess is 2(n-1)! / n!.

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The expression to make sure that it is a function of the second degree is x² + x - 6

An expression is simply used to show the relationship between the variables that are provided or the data given regarding an information. In this case, it is vital to note that they have at least two terms which have to be related by through an operator

When the expression is expanded, it can be represented by f(x) = (x-2)(x+3), which further simplifies to x^2 + x(-2+3) - 2(3) and ultimately results in x^2 + x - 6. Evidently, the highest power of x within the expression is 2, indicating that it's a second-degree function.

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By using the Chinese Remainder Theorem separate the statement into two congruences we have x(p-1)(q−1)+1 (mod pq) for all x € Z.

Under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1), the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can uniquely determine the remainder of the division of n by the product of these integers.

It suffices to show pq divides x(x(p-1)(q-1) - 1) for all x e Z. We consider three cases. Consider gcd (x, pq). It has 3 possibilities.

Case 1: If gcd(x, pq) 1. Then applying using Euler's Theorem we have

= x(pq) = 1 (mod pq)

= x(p-1)(−1) = 1 (mod pq)

= x(p-1)(q-1)+1 (mod pq)

and so the result holds if gcd(x, pq) = 1. EX

Case 2: If gcd(x, pq) p. This means x = 0 (mod p). In this case we have

= 0 = x (mod p).

Since gcd(x, pq) = p therefore qx and = 1 (mod q) by Fermat's Little Theorem. This gives us that x(p-1)(q-1)+1 so we have x9-1 x(p−1)(q−1) = 1 (mod q) = x(p-1)(q-1)+1 = x (mod q).

We have shown that x(p-1)(q-1)+1 = x (mod p) and x(p-1)(q-1)+1 = x (mod q). Using the Chinese Remainder Theorem we get x(p-1)(q−1)+1 = x (mod pq).

Case 3: If gcd(x, pq) = q. This case is same as Case 2, with p being replaced by q.

Thus we have extinguished all cases and we have shown x(p-1)(q−1)+1 (mod pq) for all x € Z.

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Complete question:

Let р and q be distinct primes. Show that for all x € Z, we have the congruence x(p-1)(9–1)+1 x (mod pq). (Hint: Use the Chinese Remainder Theorem/Sun Ze's Theorem to separate the statement into two congruences.)

The measure of the other acute angle to the nearest degree is 34°, since the trigonometric tangent ratio of one acute angle is 3/2.

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

we shall call the acute angles X and Y such that;

tan X = 3/2 {opposite/adjacent}

X = tan⁻¹(3/2) {cross multiplication}

X = 56° approximately to the nearest degree

Y = 180° - (56 + 90)° {sum of interior angles of a triangle}

Y = 180° - 146°

Y = 34°

Therefore, the measure of the other acute angle to the nearest degree is 34°, since the trigonometric tangent ratio of one acute angle is 3/2.

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Jan's total grocery bill, if she only bought things that were marked down, would be c. $22.46

First, find the marked down price of all the items on discount ;

Chicken :

= 8. 47 / 0.85

= $ 9. 96

Milk :
= 2. 16 / 0. 80

= $ 2.70

Onions :

= 0. 89 / 0. 90

= $ 0.99

Potato chips = $ 1. 65

Oranges = $ 1. 81

Flour = $ 5. 35

The total grocery bill of the things Jan bought is:

= 9. 96 + 2. 70 + 0. 99 + 1. 65 + 1. 81 + 5. 35

= $ 22. 46

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The probability of rolling an integer is

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

Sides = 6

Integer sides = 6

using the above as a guide, we have the following:

P(Integer) = Integer sides/Total sides

So, we have

P(Integer) = 6/6

Evaluate

P(Integer) = 1

Hence, the probability is 1

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The sample space refers to option (c) the set of all possible experimental outcomes. In probability theory and statistics, a sample space represents all possible outcomes of an experiment or a random event.

The correct answer is c. The sample space refers to the set of all possible experimental outcomes. This includes every possible outcome that could occur in an experiment, whether or not it actually occurs. For example, if you flip a coin, the sample space would be {heads, tails}. This encompasses every possible outcome of the experiment. It provides a foundation for calculating probabilities and understanding the range of results that may occur in a given situation. Sample spaces can vary in size and complexity, depending on the nature of the experiment or event being studied. Understanding the sample space is crucial for making accurate predictions and informed decisions based on data.

Option a is also correct to some extent, as any particular experimental outcome can be considered a part of the sample space. However, it is not a complete definition of the sample space as it only focuses on one outcome and not all possible outcomes.

Option b is incorrect, as the sample space is not limited to just one particular experimental outcome. It is the set of all possible outcomes.

Option d is also incorrect as the sample space has nothing to do with the sample size or the number of participants in the experiment. It is solely based on the set of all possible outcomes of the experiment.

In conclusion, the sample space is the set of all possible experimental outcomes, including both successful and unsuccessful outcomes. It is an important concept in probability theory and is used to calculate the probability of specific events occurring in an experiment.

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The surface area of the pyramid is 179 sq. m.

The surface area of a given shape is the summation of all the area of each figure that forms its sides called surfaces.

The given pyramid has triangular shaped surfaces, so that;

area of a triangle = 1/2 *base*height

To determine the area of one of the surfaces, we have;

area of the triangular surface = 1/2x base x height

base = 8 m, and slant height of the surface = 11.2 m

So that;

the area of one triangular surface = 1/2*8*11.2

                                                    = 44.8 sq. m.

Thus since the pyramid has 4 equal triangular surfaces, then;

the surface area of the pyramid = 4 x 44.8

                                            = 179.2

The surface area of the pyramid is 179 sq. m.

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Formula for the interior angle of a regular polygon = [ (n - 2) x 180 ] / n

---n is the number of sides of the polygon

[ (n - 2) x 180 ] / n = 150

(n - 2) x 180 = 150n

180n - 360 = 150n

-360 = -30n

n = 12

Answer: 12 sides

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The magnitude of the force that must be applied parallel to the plane is 92 N, under the condition that the box moves up the plane with an acceleration of 2 m/ s².

The force needed to move a box up an inclined plane can be evaluated as
Fp = W sin α = m ag sin α
Here
Fp = pulling force (N),
W = weight of the box (N),
α = angle of incline (degrees),
m = mass of the box (kg),
a = acceleration of the box (m/s²), and
g = acceleration due to gravity (9.8 m/s²).

For the given case, we possess a 25 kg box at rest on a 30 degree incline with a friction force of 50 N acting on it.
Now
We have to evaluate the weight of the box using
W = mg
= 25 kg x 9.8 m/s²
= 245 N.

Then, we have to calculate the force required to overcome friction using Ff = μFn where μ is the coefficient of friction and Fn is the normal force acting on the box. Since the box is at rest on an incline, Fn can be calculated as Fn = W cos α = 245 N cos(30°) ≈ 212 N. Therefore, Ff = μFn = 0.2 x 212 N ≈ 42 N.

Now we can evaluate the force applied to move the box up the incline utilizing
Fp = ma + Ff
Here,
a = desired acceleration
Ff = frictional force acting on the box.
Staging the values
Fp = ma + Ff
Fp = (25 kg)(2 m/s²) + 42 N
Fp ≈ 92 N

Hence, a force of approximately 92 N must be applied parallel to the plane so that the box moves up the plane with an acceleration of 2 m/s².
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The complete question is
A 25 kg box is at rest on a 30 degree incline. If the friction force is 50 N, what is the magnitude of the force that must be applied parallel to the plane so that the box moves up the plane with an acceleration of 2 m/ s²?

(a) To discover the differential equation of  x' = sin(2x), we set x' to zero and fathom for x:

sin(2x) =

This condition is fulfilled at whatever point 2x is a number different from π, i.e.,

x = nπ/2, where n is a number.

Be that as it may, we got to limit the arrangements to the interim [0, 3π/2], so the equilibria are:

x = 0, π/2, π, 3π/2

(b) To decide the soundness of each equilibrium point, we assess the sign of x' within the region of the balance point. In the event that x' is positive (resp. negative) on one side of the harmony and negative (resp. positive) on the other side, at that point the balance is unsteady. In the event that x' has the same sign on both sides, at that point the harmony is steady.

Close x = 0, we have sin(2x) ≈ 2x, so x' ≈ 2x. Since x is a little close to 0, x' is positive for x > and negative for x < xss=removed xss=removed> π/2, so x = π/2 could be a steady harmony.

Close x = π, we have sin(2x) ≈ -1, so x' ≈ -1. Hence, x' is negative for x < π and positive for x > π, so x = π is an unsteady harmony.

Close x = 3π/2, we have sin(2x) ≈ -2x+3π, so x' ≈ -2x+3π. Since x is near to 3π/2, 2x is near to 3π and 2x-3π is negative, so x' is negative for x < 3> 3π/2. Subsequently, x = 3π/2 could be a steady harmony.

In outline, the solidness of the equilibria is:

x = is unsteady

x = π/2 is steady

x = π is unsteady

x = 3π/2 is steady.

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The value of the expression is 5.

We have,

(|-52 + 1| (-1) + 4²) / (-84 ÷ 7 + 5)

Now,

PEMDAS is an acronym used to remember the order of operations in arithmetic and algebraic expressions. It stands for:

Parentheses: Simplify expressions inside parentheses first.

Exponents: Simplify any expressions involving exponents or powers.

Multiplication and Division: Perform multiplication and division in order from left to right.

Addition and Subtraction: Perform addition and subtraction in order from left to right.

Now,

(|-52 + 1| (-1) + 4²) / (-84 ÷ 7 + 5)

We solve | | first and exponents second.

|-52 + 1| = |-51| = 51

4² = 16

And,

-84 ÷ 7 = -84/7 = -12

So,

(|-52 + 1| (-1) + 4²) / (-84 ÷ 7 + 5)

= 51 x -1 + 16 / -12 + 5

= -51 + 16 / -12 + 5

= -35/-7

= 5

Thus,

The value of the expression is 5.

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The statement "The normal force is equal to the perpendicular component of the object's weight, which decreases as the angle of inclination increases" is true.

As the angle of inclination increases, the object's weight can be divided into two components: one perpendicular to the inclined surface (the normal force) and one parallel to it. As the angle increases, the perpendicular component (normal force) decreases, while the parallel component increases.

So to directly answer your question, the normal force is never equal to the weight of the object on an inclined plane (unless you count the limiting case of level ground). It is equal to the weight of the object times the cosine of the angle the inclined plane makes with the horizontal.

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From the attachment, the missing side is B. 21.0.

Trigonometric functions are basic functions which can be used to determine the missing value of a right angled triangle when given the value of one of its internal angles. Some of these functions are; sine, cosine, tangent etc.

To determine the value of the missing side x, we have to apply the appropriate trigonometric function. Thus we have;

Sin θ = opposite/ hypotenuse

Sin 65 = 19/ x

So that;

x = 19/ 0.9631

  = 20.964

x = 21

From the attachment, the missing side is 21.0. Thus option B.

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Answer:

15.8

Step-by-step explanation:

mean is the average in math so 9 + 14 + 11 + 31 + 14 =79 then you have to count how many numbers there is and minus it from the total which there is 5 numbers so 79 divided by 5 = 15.8

The improper fraction can be written as:

44/7

An improper fraction is a fraction where the numerator is larger than the denominator.

Here we want to write.

6 + 2/7 as an improper fraction, to do so, we just need to write 6 as a fraction with a denominator of 7 and then add them.

We know that:

6 = 6*(7/7) = 42/7

Then we can write:

6 + 2/7 = 42/7 + 2/7 = 44/7

That is the improper fraction.

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