Find dy/dx of i)x, ii)xy, iii) y^2, iv) x^2y

Answer:

[tex]a_{n}=a_{1} r^{n-1}[/tex]

[tex]a_{n}=5 (2)^{n-1}[/tex]

Answer:

3/4

Step-by-step explanation:

NW is 45° counterclockwise (anticlockwise) to North

If she turns 45° anticlockwise she will be facing directly west

If she turns another 180° anti clockwise, she will be facing east

To fact NE she must turn another 45°anticlockwise

Total anti-clockwise turn in degrees = 45 + 180 + 45 = 270°

One complete turn is 360°

So the total turn Miko made as a fraction of a complete turn
= 270/360

= 3/4

The trigonometric relations are solved and equation is A = 3/4

Given data ,

Let the trigonometric relation be represented as A

Now , the value of A is

A = Tan²60° × Sin60° x Tan30° x Cos²45°

On simplifying the equation , we get

Tan²60° = 3

Sin60° = √3/2

Tan30° = 1/√3

And , Cos²45° = 1/2

Now , the equation is A = 3 x √3/2 x 1/√3 x 1/2

A = 3/4

Hence , the trigonometric equation is solved and A = 3/4

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The measure of an interior angle is 140°.

The number of sides that this polygon has is 9 sides.

In Mathematics and Geometry, the measure of the sum of the angles of a convex quadrilateral, pentagon, and a nonagon is equal to 360 degrees. This ultimately implies that, the sum of all the exterior angles of a regular polygon must add up to 360 degrees.

Mathematically, the exterior angle of a regular polygon can be calculated by using this mathematical equation:

Exterior angle = 360/number of sides

40 = 360/number of sides

Number of sides = 360/40

Number of sides = 9 sides.

Additionally, the measure of each interior angle of a regular polygon can be calculated by using this mathematical equation:

Interior angles = [180 × (n - 2)]/n

Interior angles = [180 × (9 - 2)]/9

Interior angles = 1,260/9

Interior angles = 140°

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The type of model that best fits the situation of a $500 raise in salary each year is a linear model.

In a linear model, the dependent variable changes a constant amount for constant increments of the independent variable.

In the given case, the dependent variable is the salary and the independent variable is the year.

You may build a table to show that for increments of 1 year the increments of the salary is $500:

Year         Salary        Change in year         Change in salary

2010           A                           -                                       -

2011           A + 500     2011 - 2010 = 1          A + 500 - 500 = 500

2012          A + 1,000   2012 - 2011 = 1          A + 1,000 - (A + 500) = 500

So, you can see that every year the salary increases the same amount ($500).

In general, a linear model is represented by the general equation y = mx + b, where x is the change of y per unit change of x, and b is the initial value (y-intercept).              

In this case, m = $500 and b is the starting salary: y = 500x + b.

The vertical distance between the two points given are (2, 11/3) and (2, -4/3) is 5 units.

The two points given are (2, 11/3) and (2, -4/3). These points have the same x-coordinate of 2, which means they lie on a vertical line. The vertical distance between the two points is simply the difference between their y-coordinates.

So, the vertical distance between the two points is:

11/3 - (-4/3) = 15/3 = 5

This means that if we were to draw a line segment connecting the two points, the length of the segment would be 5 units, and the segment would be parallel to the y-axis.

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Answer: 216

Step-by-step explanation:

592-376

592-300= 292

292-70= 222

222-6= 216

The slope of the graph f(x) = (1/3)x + 4 is less than the slope of g(x) = 4x-1

f(x) = 1/3 x + 4

g(x) = 4x - 1

The equation of line is given by y = mx +c

On comparing both equation with y = mx + c

m is the slope of the line and c is the intercept of the equation

Let the slope of f(x) is m₁

In f(x) =1/3 x + 4

m₁ = 1/3 = 0.33

Let the slope of g(x) is m₂

In g(x)  = 4x - 1

m₂ = 4

m₁ < m₂

slope of f(x) is less than slope of g(x)

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Answer:

the answer is 54,000

Step-by-step explanation:

(540 × 45) + (540 × 55)

=24300 + 27700

=54000

a) The two expressions do not represent the same value. b) the surface area of a rectangular prism consists of the sum of the areas of all its faces. By doubling the areas of each face

a. To determine if Sofia and Ella's expressions represent the same value for the surface area of a rectangular prism, we can simplify their expressions and compare them.

Sofia's expression: (3 cm x 4 cm) + (3 cm x 5 cm) + (4 cm x 5 cm) + (4 cm x 5 cm)

            = 12 cm² + 15 cm² + 20 cm² + 20 cm²

            = 67 cm²

Ella's expression: 2(3 cm x 4 cm) + 2(3 cm x 5 cm) + 2(4 cm x 5 cm)

          = 2(12 cm²) + 2(15 cm²) + 2(20 cm²)

          = 24 cm² + 30 cm² + 40 cm²

          = 94 cm²

The two expressions do not represent the same value. Sofia's expression calculates the surface area by adding the areas of each face once, while Ella's expression calculates the surface area by doubling the areas of each face and then summing them up.

b. Ella's expression, 2(3 cm x 4 cm) + 2(3 cm x 5 cm) + 2(4 cm x 5 cm), shows more clearly the fact that the surface area of a rectangular prism consists of the sum of the areas of all its faces. By doubling the areas of each face

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The area of the sector in the circle shown above is given as follows:

32.3 in².

The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:

A = πr²

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle, hence it is given as follows:

r = 10 in.

Then the area of the entire circle is given as follows:

A = π x 10²

A = 314 in².

The entire circumference of the circle is of 360º, while the angle is of 37º, hence the area of the sector is given as follows:

37/360 x 314 = 32.3 in².

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Answer: The concept of statistical independence is fundamental to statistical hypothesis testing. In hypothesis testing, we aim to assess whether there is evidence to support a claim or hypothesis about the relationship between variables in a population. The concept of statistical independence allows us to quantify the degree to which variables are related or dependent on each other.

Statistical independence refers to the absence of a relationship between two variables. When two variables are statistically independent, the occurrence or value of one variable provides no information or predictive power about the occurrence or value of the other variable. In other words, knowledge about one variable does not affect our ability to predict or infer the other variable.

Hypothesis testing involves comparing observed data to a null hypothesis, which assumes that there is no relationship or effect between the variables of interest. By assuming statistical independence under the null hypothesis, we establish a baseline against which we can evaluate the observed data and determine whether it provides evidence to reject or accept the null hypothesis.

When conducting statistical analysis, we use various statistical tests and measures to assess the likelihood of observing the data if the null hypothesis were true. If the observed data is highly unlikely under the assumption of independence (i.e., the p-value is below a predetermined significance level), we reject the null hypothesis and conclude that there is evidence of a relationship or dependence between the variables.

However, it's important to note that statistical analysis alone cannot definitively prove or establish causal relationships or dependence between variables. Statistical dependence refers to the presence of a relationship or association between variables, but it does not provide information about the direction or underlying mechanisms of the relationship.

Researchers typically focus on statistical independence rather than statistical dependence because independence is the default assumption when testing hypotheses. By assuming independence, researchers can rigorously evaluate whether the observed data provides evidence to reject the null hypothesis and support the claim of a relationship or effect between variables. Additionally, focusing on independence allows researchers to identify and investigate deviations from independence, which can reveal meaningful patterns, relationships, or dependencies that may exist in the data.

The coefficient matrix A represents a system of linear equations that has a unique solution.

In a system of linear equations, the unique solution exists when the coefficient matrix has full rank and the determinant of the matrix is non-zero.

The other coefficient matrices listed do not have these properties.

1.  [[6, 0, - 2]

     - 2, 0, 6

        1, - 2, 0]

has a determinant of zero, indicating that the system of equations is dependent and does not have a unique solution.

Similarly, the matrices

[5, 10, 5

 4, 1, 4

- 1, - 2, - 1]

and, [2, 0, - 2

        - 7, 1, 5

          4, - 2, 0]

also do not satisfy the conditions for a unique solution.

Therefore, the coefficient matrix A represents a system of linear equations that has a unique solution.

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Answer:

$280

Step-by-step explanation:

7•40=$280

The total amount to each person end up paying $20.54.

To find the total amount each person will pay, first calculate the 20% tip on the total bill and then divide the sum by the number of people.

To split the bill evenly among Raphael and his four friends, we first need to find the total cost including the 20% tip.

The tip is 20% of the original bill, which is equivalent to 0.20 x $85.60 = $17.12.

Therefore, the total cost of the bill with the tip is $85.60 + $17.12 = $102.72.

To split this evenly among the five people, we divide by 5:

$102.72 ÷ 5 = $20.54

So each person will end up paying $20.54.

20% of $85.60 is ($85.60 * 0.20) = $17.12

Add the tip to the total bill:

$85.60 + $17.12 = $102.72

Divide the total amount by the number of people (5): $102.72 / 5 = $20.54

Therefore, the answer is B. $20.54.

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Circle B is dilation of 6 unit left and 4 unit down to the center of circle A.

We have to given that;

Circle A is transformed into Circle B using a sequence of two transformations.

And, The first transformation is a dilation centered at (6,0).

Here, Center of circle A is,

⇒ A = (6, 0)

And, Center for circle B is,

⇒ B = (0, - 4)

Hence, We get;

⇒ B = (6 - 6, 0 - 4)

⇒ B = (0, - 4)

Hence, Circle B is dilation of 6 unit left and 4 unit down to the center of circle A.

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The measure of the indicated angles using available arc angles which subtends angles at the circumference are:

(5). ? = 103, (6). ? = 80°, (7). ? = 32°, and (8). ? = 110°

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.

So;

(5). m∠JBL + 92° = 180° {sum of opposite interior angles of a cyclic quadrilateral}

m∠JBL = 88°

arc angle (JK + LK) = 2(88)

arc angle JK = 176 - 70 = 106°

arc angle (JK + JB) = 2(?)

? = (106 + 100)/2

? = 103

(6). arc angle QR = 360° - are. QSR

arc angle QR = 360 - (80 + 120) = 160°

arc angle QR = 2(?)

? = 160/2

? = 80°

(7). arc angle GH = 2(m∠GFH)

m∠GFH = 116/2

m∠GFH = 58

m∠FGH = 90° {angle in a semi circle is a right angle}

? = 180 - (58 + 90)

? = 32°

(8). arc angle DCF = 360 - arc angle DEF

arc DCF = 360 - (68 + 72)

arc DCF = 220

arc DCF = 2(?)

? = 220/2

? = 110°

Therefore, the measure of the indicated angles using available arc angles which subtends angles at the circumference are: (5). ? = 103, (6). ? = 80°, (7). ? = 32°, and (8). ? = 110°

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The required reference angle of 13π/8 is 3π/8.

The reference angle of an angle in standard position is the positive acute angle formed between the terminal side of the angle and the x-axis.

Given an angle of 13π/8, we can determine its reference angle as follows:

Angle: 13π/8

Since the angle is in the fourth quadrant, we subtract it from 2π (one full revolution) to find the reference angle:

Reference angle: 2π - 13π/8 = 3π/8

Therefore, the reference angle of 13π/8 is 3π/8.

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The length of the side of the square in simplest form is 2sqrt(14).

The area of a square is given by the formula [tex]A = s^2[/tex], where A is the area and s is the length of a side.

We are given that the area of the square is 56 units, so we can set up the equation:

[tex]56 = s^2[/tex]

To solve for s, we can take the square root of both sides of the equation:

sqrt(56) = [tex]sqrt(s^2)[/tex]

We can simplify the square root of 56 by factoring it:

sqrt(56) = sqrt(222*7) = 2sqrt(14)

So, we have:

2sqrt(14) = s

This is an improper fraction because the numerator is larger than the denominator. Therefore, the length of the side of the square in simplest form is 2sqrt(14).

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The circle graph percentage is solved and the measure of ∠KL = 54° and the measure of ∠LMJ = 198°

Given data ,

Let the circle graph gives the percentage of students who favor the different lunch menus offered by the school cafeteria

Now , the measures of the angles are given by

The measure of ∠KL = 15 %

15 % ( 360 ) = 54°

Therefore , the measure of ∠KL = 54°

And , the measure of ∠LMJ = ( 24 % + 21 % ) of 360

The measure of ∠LMJ = 198°

Hence , the circle graph percentage is solved

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The probability of a student chosen at random playing an instrument other than the guitar is 29/45.

To calculate the probability of a student playing an instrument other than the guitar, we need to determine the total number of students who play instruments other than the guitar and the total number of students in the middle school.

In the seventh grade, the number of students playing instruments other than the guitar is 3 (bass) + 10 (drums) + 5 (keyboard) = 18.

In the eighth grade, the number of students playing instruments other than the guitar is 3 (bass) + 6 (drums) + 2 (keyboard) = 11.

The total number of students playing instruments other than the guitar is 18 + 11 = 29.

Now, let's calculate the total number of students in the middle school by summing up the number of students in each grade:

Seventh grade: 12 (guitar) + 3 (bass) + 10 (drums) + 5 (keyboard) = 30

Eighth grade: 4 (guitar) + 3 (bass) + 6 (drums) + 2 (keyboard) = 15

The total number of students in the middle school is 30 + 15 = 45.

Therefore, the probability of a student chosen at random playing an instrument other than the guitar is 29/45.

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The correct option is A, the relative frequency is 0.03

To find the relative frequency for a given outcome, we need to take the quotient between the number of times that we got that outcome, and the total number of times that the experiment is done.

Here the outcome  is "wants fruit smooties"

And for the total "number of times that the experiment is done" we need to count the number of students

We know:

Total number = 500

Number of 8th grades who want a fruit smoothies = 17

Relative frquency = 17/500 = 0.03

The correct option is the first one.

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Based on a system of equaitons, the initial opening fee (one-time payment) is $5.00, the monthly maintenance fee is $11.00, while an equation that models Adam's fees over time in months, x is y = 11x + 5.

A system of equations or simultaneous equations are two or more equations solved concurrently or at the same time.

An equation is a mathematical statement showing the equality or equivalence of two or more algebraic expressions.

While algebraic expressions combine variables with numbers and mathematical operands, equations use the equal symbol (=).

Total Fees after x months

x = # of months    y = total fees ($)

2                                      27

3                                      38 (11 x 3 + 5)

4                                      49 (11 x 4 + 5)

5                                     60

Let w = the monthly maintenance fee

Let z = the one-time opening fee

5w + z = 60 ... Equation 1

2w + z = 27 ... Equation 2

Subtract Equation 2 from Equation 1:

3w = 33

w = 11

b) The monthly maintenance fee is $11.00.

a) The initial (one-time) opening fee is $5.00.

z = 60 - 55

= 5

c) The total fees after x months is y = 11x + 5.

Thus, the one-time opening fee and the monthly maintenance fee can be computed using a system of equations.

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Answer:

Perimeter of a rectangle

Step-by-step explanation:

You have to combine all of the sides which results in 2 lengths and 2 widths being combined (2l+2w)

Answer:

The simplified expression for g(a+5) - g(5) is 2a^2 + 20a.

Step-by-step explanation:

To evaluate and simplify the expression g(a+5) - g(5), we need to substitute the function g(t) = 2t^2 into the given expression.

Let's start by evaluating g(a+5):

g(a+5) = 2(a+5)^2

Expanding the expression:

g(a+5) = 2(a^2 + 10a + 25)

g(a+5) = 2a^2 + 20a + 50

Next, let's evaluate g(5):

g(5) = 2(5)^2

g(5) = 2(25)

g(5) = 50

Now we can substitute these values back into the expression g(a+5) - g(5):

g(a+5) - g(5) = (2a^2 + 20a + 50) - 50

Simplifying:

g(a+5) - g(5) = 2a^2 + 20a

Therefore, the simplified expression for g(a+5) - g(5) is 2a^2 + 20a.

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The binomials represents the length of each of it sides = a + 5.

The given expression is;

a² + 10a + 25

It represents the area of square.

Since area of square = (side)²

Therefore,

(side)² = a² + 10a + 25

           =  a² + 5a + 5a + 25

           =  a(a+5) + 5(a+5)

           =   (a+5)(a+5)

           =   (a+5)²

⇒(side)² = (a+5)²

Taking square root both sides

Hence

side of square = a + 5

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Answer:18

Step-by-step explanation:

It will take approximately 7 weeks for the insect population to surpass 16,000. Option D.

To determine the number of weeks it will take for the population to surpass 16,000, we need to find the value of t when P exceeds 16,000. Let's set up the equation:

16,000 = 15,000 + 2500 x sin(πt/52)

Subtracting 15,000 from both sides:

1,000 = 2500 x sin(πt/52)

Dividing both sides by 2500:

0.4 = sin(πt/52)

To solve for t, we need to find the inverse sine of both sides of the equation:

πt/52 = arcsin(0.4)

t = (52/π) x arcsin(0.4)

t ≈ (52/π) x 0.4115

t ≈ 6.6 weeks

Therefore, it will take approximately 7 weeks for the insect population to surpass 16,000.

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A graph of the image of the figure after a reflection across the x-axis is shown below.

In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y).

This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.

By applying a reflection over or across the x-axis to triangle GLQ, we have;

(x, y)                →      (x, -y)

G (3, 4)             →      (3, -(4)) = G' (3, -4)

L (1, 2)                →     (1, -(2)) = L' (1, -2).

Q (4, -1)              →      (4, -(-1)) = Q' (4, 1)

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Answer:

y is 0,3

Step-by-step explanation:

To find the x-intercept, substitute in

0

for

y

and solve for

x

. To find the y-intercept, substitute in

0

for

x

Answer:

(0,3)

Step-by-step explanation:

Two methods:

Method 1:  General method for any equation

Method 2: Method specific for parabolas in standard form

Method 1:  General method for any equation

For any two-variable equation to be graphed, the y-intercept is the point where the graph crosses the y-axis.  The y-axis is a vertical line through the origin (0,0).

Any y-intercept is on that line, and to get to that point starting from the origin, one can't travel left or right to get to the y-intercept point (without moving back to the y-axis).  The only movement would be up or down.

Since no left-right movement will happen, the x-coordinate is zero.

For any two-variable equation, the x and y coordinates of any point on the graph are linked by the equation.  If it is known that the x-value is zero, the y-value associated with that x-value is given by substituting zero into the equation everywhere there is an "x", and solving for "y".

[tex]y=-4x^2-x+3[/tex]

[tex]y=-4(0)^2-(0)+3[/tex]

Order of operations requires exponents before multiplication, or addition & subtraction...

[tex]y=-4(0)-(0)+3[/tex]

multiplication...

[tex]y=0-0+3[/tex]

addition & subtraction, from left to right...

[tex]y=3[/tex]

So, when the x-value is zero, the y-value is three.  Therefore, the ordered pair representing that point is (0,3).

Method 2: Method specific for parabolas in standard form

The given equation is the equation for a parabola (as stated in the question), and it is given in "standard form": [tex]y=ax^2+bx+c[/tex], where a, b, and c are real numbers (and a isn't equal to zero, because then the x-squared term would be zero, and the equation would really just be a linear equation).

Note that for our equation, it is in standard form if we rewrite the equation to only use addition, [tex]y=-4x^2+-1x+3[/tex], where [tex]a=-4, ~b=-1 ~ \text{and}~c=3[/tex]

For a parabola in standard form, the y-intercept is always at a height of "c".

So, the y-intercept would be (0,3).