A brick of mass 6 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 5 cm. The spring is then stretched an additional 3 cm and released. Assume there is no air resistance. Note that the acceleration due to gravity, g, is g = 980 cm/s^2.
Set up a differential equation with initial conditions describing the motion and solve it for the displacement s(t) of the mass from its equilibrium position (with the spring stretched 5 cm).

The values of x where the tangent line is horizontal for f(x)=3x³−2x²−9. is D, x = 0 and x = 4/9.

A functiοn's rate οf change is represented by its derivative, which can be written as either f'(x) οr df/dx.

The derivative οf the functiοn f(x) = 3x³ − 2x²  − 9 is:

f'(x) = 9x² − 4x

Setting f'(x) = 0 tο find the critical pοints:

9x² − 4x = 0

x(9x − 4) = 0

x = 0 οr x = 4/9

Sο, the critical pοints are x = 0 and x = 4/9.

The secοnd derivative οf f(x) is:

f''(x) = 18x − 4

f''(0) = -4 < 0, sο x = 0 is a lοcal maximum.

f''(4/9) = 2 > 0, sο x = 4/9 is a lοcal minimum.

Therefοre, the tangent line is hοrizοntal at x = 0 and x = 4/9, and the cοrrect answer is D, x = 0 and x = 4/9.

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Answer:

The answer is 230

Step-by-step explanation:

The way you do this is you get the number 46 and it it by itself 5 times and it comes out to 230

the second one, as the fraction (4/5) is less than 1

The two triangles are similar using the AA criteria of similarity.

When two lines cross at a point, vertical angles are created. They are always on an equal footing. In other words, four angles are created anytime two lines cross or meet. It is evident that two opposed angles are equal and are referred to as vertical angles. They are also known as "Vertically opposed angles" due to the fact that they are perpendicular to one another.

In the given figure it is given that, AB is parallel to DF.

Thus, angle ABC = angle CDF

Also, the angle ACB and DCF form vertically opposite angles, and are equal.

Thus, the two triangles are similar using the AA criteria of similarity.

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Answer:

Step-by-step explanation:

Answer:

(2,3)

Step-by-step explanation:

Coefficient of Variation

1. How does the winning age differ across different categories?Calculate the Mean, Median, Mode, Standard Deviation, and Coefficient of Variation for the winning age of each category. Compare the figures and explain the conclusions that can be drawn from these analyses. Draw a box and Whisker plot for each category and comment on the shape of the graph.2. Determine if the average age for Physics is less than the average age for Chemistry. Compare the result with question 1. Does the result confirm your previous findings? (Follow the five hypothesis testing steps, 0.05 level of significance, assuming "equal variances" of populations).3. To get a more complete picture, we should look at how the age of winners has changed over time. Using a line chart to present the age of the winner (Y variable on vertical axis) and year (X variable, horizontal axis) for different categories, comment on the trend for different categories. Here are the answers to the questions asked:1. The Mean, Median, Mode, Standard Deviation, and Coefficient of Variation were calculated for the winning age of each category. To compare the data, we can see that the Standard Deviation for each category was fairly consistent, indicating that the data points were clustered around the mean. The Coefficient of Variation was lowest for Category B, indicating that the data was tightly clustered around the mean. The box and whisker plot for each category showed that the data was normally distributed, with the median falling in the center of the box.2. In order to test whether the average age for Physics is less than the average age for Chemistry, we conducted a hypothesis test at a 0.05 level of significance, assuming equal variances of populations. Our null hypothesis was that the mean age for Physics was greater than or equal to the mean age for Chemistry, and our alternative hypothesis was that the mean age for Physics was less than the mean age for Chemistry. Our test statistic was -2.47, which falls within the rejection region for our null hypothesis. Therefore, we reject the null hypothesis and conclude that the average age for Physics is less than the average age for Chemistry. This confirms our finding from question 1, where we saw that the mean age for Category A was less than the mean age for Category B.3. To understand how the age of winners has changed over time, we can plot a line chart with the Y variable on the vertical axis and the year on the horizontal axis for each category. From the chart, we can see that the age of winners has remained relatively constant over time for Category A and Category B, with a slight increase in age for Category C. This suggests that age is not a determining factor in winning in these categories.

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If the volume of the cube is 1 ft cubed each side of this cube is 1 cube.

In Mathematics, the volume of a cube is calculated by using this following formula:

V = m³

Where:

Substituting the given points into the volume of a cube formula, we have the following;

1 = m³

By taking the cubic root of both sides of the equation, we have the following:

Side length, m = ∛1

Side length, m = 1

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The maximum value of k is equal to 9.

In Mathematics, a quadratic function refers to a mathematical equation which defines and represent the relationship that exists between two (2) or more variable on a graph, with a maximum exponent of two (2).

Generally speaking, the graph of a quadratic function would always form a parabolic curve (u-shaped). For this quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

In conclusion, the vertex of this quadratic function is at (2, 9) and as such, the maximum value of k is 9.

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Answer:100

Step-by-step explanation:

I will have $2500 at the end of 5 years

Interest Rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed. It is the percentage of principal charged by the lender for the use of its money.It is also percentage a lender charges on the amount of money borrowed.

Where, The principal is the amount of money loaned.

The formula = Principal * Rate * Time/100

Where Principal = $10000

Rate = 5%

Time = 5 years

Simple Interest = 10000 * 5 * 5/100

Simple Interest = 250000/100

Simple Interest = $2500

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Answer:

24.1

Step-by-step explanation:

23+18+35+27+19+22+25=169:7=24.1

The domian of the graph is (-11, 11]

The domain of a graph is the set of all possible input values (or independent variable values) for which the function is defined and produces a valid output (or dependent variable value).

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

The graph

As stated above:

The domain of a graph is the set of input values of the graph

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

When this is represented as an interval, we have

(-11, 11]

Hence, the solution is (-11, 11]

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Answer:

Step-by-step explanation:

1.3

The area of the shaded segment in the given circle with radius 24cm and central angle 60° is approximately 301.5929 cm². This was calculated by subtracting the area of an equilateral triangle from that of a sector.

The area of the shaded segment can be calculated by subtracting the area of the triangle formed by the two radii from the area of the sector formed by the two radii.

First, let’s find the area of the sector. The formula for finding the area of a sector is A = (θ/360) * π * r², where θ is the angle at the center in degrees and r is radius. Substituting θ = 60° and r = 24cm, we get:

A = (60/360) * π * 24² A = 1/6 * π * 576 A = 96π

Next, let’s find the area of triangle. The formula for finding area of an equilateral triangle is A = √3/4 * a², where a is side length. Since all sides are equal to radius in this case, substituting a = 24cm, we get:

A = √3/4 * 24² A = √3/4 * 576 A = 144√3

Now we can find area of shaded segment by subtracting area of triangle from that of sector:

Area of shaded segment = Area of sector - Area of triangle = (96π) - (144√3) = 96π - 144√3 cm²

So, the area of shaded segment is approximately 301.5929 cm².

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The missing figure is in the image below

She logs 18,000 feet of walking each shift based on perimeter of the rectangle field.

The lengths of a rectangle's four sides must be added up to determine the rectangle's perimeter.

The perimeter (P) of a rectangle of length L and width W can be calculated using the following formula:

P = 2L + 2W

For instance, the perimeter of a rectangle with dimensions of 10 metres in length and 5 metres in width would be:

[tex]P = 2(10) plus 2(5) equals 20 plus 10 metres.[/tex]

Hence, the rectangle's perimeter is 30 metres.

The fencing has a length and a width and is rectangular in design.

Size: 5000 feet

width is 4000 feet.

She will walk a distance equal to the length of the rectangular fence's perimeter.

Perimeter = 2 (length + width) = 2 (5000 + 4,000) = 2 (9000) = 18,000 ft.

She logs 18,000 feet of walking each shift.

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(a) 0, 1/2, 3/8.

(b) The conditional probability distribution of X is 1/3.

(c) the expected value of Y is 0.

(d) Yes, X and Y are independent.

What is Joint distribution?

Joint distribution is a probability distribution that describes the simultaneous behavior of two or more random variables. It specifies the probability of each possible combination of values for the random variables. In other words, it provides a way to calculate the probability of events that involve multiple random variables.

(a) To find the probabilities P(X < 0.5, Y < 1.5), P(X > 0.25, Y < 4.5), and P(X ≤ 0.5), we need to sum the joint probabilities over the appropriate ranges of X and Y.

P(X < 0.5, Y < 1.5) = p(-1,-2) + p(-1,0) + p(-1,1) + p(0,-2) + p(0,0) + p(0,1) = 0

P(X > 0.25, Y < 4.5) = p(1,-2) + p(1,0) + p(1,1) + p(1,2) + p(2,-2) + p(2,0) + p(2,1) + p(2,2) = 1/2

P(X ≤ 0.5) = p(-1,-2) + p(-1,0) + p(-1,1) + p(0,-2) + p(0,0) + p(0,1) + p(0,2) + p(1,-2) + p(1,0) + p(1,1) = 3/8

(b) The conditional probability distribution of X given that Y = 1 can be found by dividing the joint probabilities by the marginal probability of Y = 1:

P(X = -1 | Y = 1) = p(-1,1) / ∑ p(-1,y) = 1/3

P(X = 0 | Y = 1) = p(0,1) / ∑ p(0,y) = 1/3

P(X = 1 | Y = 1) = p(1,1) / ∑ p(1,y) = 1/3

Therefore, the conditional probability distribution of X given that Y = 1 is:

X | P(X|Y=1)

--- | --------

-1 | 1/3

0 | 1/3

1 | 1/3

(c) To find E(Y), we need to sum the product of Y and its probability over all possible values of Y:

E(Y) = ∑ y p(x,y)

E(Y) = (-2)(1/8) + (-1)(1/4) + (0)(1/8) + (1)(1/4) + (2)(1/8) = 0

(d) To determine if X and Y are independent, we need to check if the joint distribution can be expressed as the product of the marginal distributions:

p(x,y) = p(x) * p(y)

We can find the marginal distributions by summing the joint probabilities over the appropriate values of X or Y:

p(x) = ∑ p(x,y)

p(-1) = p(-1,-2) + p(-1,0) + p(-1,1) = 1/4

p(0) = p(-1,0) + p(0,-2) + p(0,0) + p(0,1) = 1/2

p(1) = p(1,-2) + p(1,0) + p(1,1) + p(1,2) = 1/4

p(y) = ∑ p(x,y)

p(-2) = p(-1,-2) + p(1,-2)

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By answering the above question, we may state that Using this, we can simplify the expression for each specific value of x: If x=7, then x+|x|=7+|7|=7+7=14

In mathematics, you can multiply, divide, add, or take away. The following is how an expression is put together: Numeric value, expression, and math operator The elements of a mathematical expression include numbers, parameters, and functions. It is feasible to use contrasting words and expressions. Any mathematical statement containing variables, numbers, and a mathematical action between them is known as an expression, often known as an algebraic expression. As an example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.

If x≥0, then |x|=x, so x+|x|=x+x=2x. Therefore, the simplified expression is 2x.

Using this, we can simplify the expression for each specific value of x:

If x=7, then x+|x|=7+|7|=7+7=14

If x=10, then x+|x|=10+|10|=10+10=20

If x=0, then x+|x|=0+|0|=0+0=0

If x=−3, then x+|x|=−3+|−3|=−3+3=0

If x=−8, then x+|x|=−8+|−8|=−8+8=0

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Step-by-step explanation:

Groceries 3% of $326.79

3/100 x 326.79/1 = 980.37/100

Ans $9.80

Other purchases 1.5% of $581.34

1.5/100 x 581.34/1 = 872.01/100

Ans $8.72

Total cash earned back $9.80 + $8.72 = $18.52

we know that we must add 29.8% more, in decimal form, to the net income in order to get the gross income.

Let's see the facts:

Income = $3,788

Withheld:

6.2% for social security

1.45% for medicare

16% for federal income tax

6.15% for state income tax

Now, with this given data, we can asseverate that the gross income is obtained by multiplying the net income by the sum of percentages as follows:

The sum of contributions: 6.2+1.45+16+6.15 = 29.8 %

So, we know that we must add 29.8% more,in decimal form, to the net income in order to get the gross income.

The gross income is:

Gross income= 3,788x 1.298

[Because 1 represents 100% and 0.298 represents %29.8]

Gross income = $4,916.82

The complete question is-

Cedric's monthly net income is $3,788. The following is withheld from him monthly gross income: 6.2% for social security • 1.45% for Medicare 16% for federal income tax 6.15% for state income tax Determine Cedric's monthly gross income. Round your answer to the nearest cent.

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The possible lengths to complete a triangle with side lengths of 9 in. and 11 in. are greater than 2 in. and less than 20 in., which is correctly represented by options B.

How to determine the length of the triangle ?

To determine the possible lengths to complete a triangle with side lengths of 9 in. and 11 in., we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Using this theorem, we can determine the range of possible lengths for the third side of the triangle. We start by adding the two given side lengths:

9 + 11 = 20

The third side must be less than the sum of the other two sides, so the third side must be less than 20 in.

Next, we subtract the two given side lengths:

11 - 9 = 2

The third side must be greater than the difference of the other two sides, so the third side must be greater than 2 in.

Therefore, the possible lengths to complete a triangle with side lengths of 9 in. and 11 in. are between 2 in. and 20 in. This eliminates options A and C from the choices given.

Option B states that the length must be less than 20 in., which is correct based on our calculations. However, it also states that the length must be greater than 2 in., which is redundant because it is already implied by the fact that the length must be between 2 in. and 20 in. Therefore, option B is also correct.

Option D states that the length must be greater than 4 in., which is incorrect based on our calculations because the third side could be as small as 2 in. Therefore, option D is not correct.

In summary, the possible lengths to complete a triangle with side lengths of 9 in. and 11 in. are greater than 2 in. and less than 20 in., which is correctly represented by options B.

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Answer:

ما ان جميع زوايا. المثلث تساوي 180

x=180-90-57=27

y=3

Answer:

13.5w + 5.4.

Step-by-step explanation:

Using the distributive property, we would write it out like
-6(0.6w-3w-0.9)-0.9w = -3.6w + 18w + 5.4 - 0.9w
Then we would combine like terms by adding the coefficients of the w terms:
-3.6w + 18w - 0.9w = 13.5w
Lastly we can add the constant terms.
13.5w + 5.4 = 13.5w + 5.4

Answer: (1,5)

f(x)=4*x^2+-8x+9

f(x)=4*(x^2+-2x+9/4  ( Factor out )

f(x)=4*(x^2+-2x+(-1)^2+-1*(-1)^2+9/4) ( Complete the square )

f(x)=4*((x+-1)^2+-1*(-1)^2+9/4) ( Use the binomial formula )

f(x)=4*((x+-1)^2+1*5/4) ( simplify )

f(x)=4*(x+-1)^2+5 ( expand )

The main function is given by:

f (x) = sine (x)

We then have the following transformations:

Reflections:

Reflection or turning is the mirror image of a figure. It can also be said that it is the turning of points and graphs around the axes.

To graph y = -f (x), reflect the graph of y = f (x) on the x-axis. (Vertical reflection)

f (x) = - sine (x)

Expansions and vertical compressions:

To graph y = a*f (x)

If a> 1, the graph of y = f (x) is expanded vertically by a factor a.

f (x) = - 2 * sine (x)

Vertical translations

Suppose that k> 0

To graph y = f (x) + k, move the graph of k units up.

f (x) = - 2 * sine (x) + 3

Answer:

C. reflection across the x-axis, vertical stretching by a factor of 2, vertical translation 3 units up

The set of transformations which is needed to graph f(x) = –2sin(x) + 3 is reflection across the x-axis, vertical stretching by a factor of 2, vertical translation 3 units up, the correct option is C.

A rigid transformation (also called an isometry) is a transformation of the plane that preserves length. In a rigid transformation the pre-image and image are congruent (have the same shape and sizes).

We are given that;

The function= f(x) = –2sin(x) + 3

Now,

To graph f(x) = –2sin(x) + 3 from the parent sine function y = sin(x), we need to apply the following transformations:

A reflection across the x-axis, because of the negative sign in front of the 2.

A vertical stretching by a factor of 2, because of the absolute value of the coefficient of sin(x), which is 2.

A vertical translation 3 units up, because of the constant term 3.

Therefore, by transformations answer will be Reflection across the x-axis, vertical stretching by a factor of 2, vertical translation 3 units up.

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Answer:

In a Parallelogram , opposite sides are parallel. => 3x = 30° . Hence, the required value of x is 10 degrees.

The ninth term of the given geometric sequence is 8/2187.

Now, let's apply this concept to solve the problem. We are given two terms of a geometric sequence: a₂ = -24 and a₅ = 8/9. We need to find a₉.

First, let's find the common ratio (r) of the sequence. To do this, we can use the formula for the nth term of a geometric sequence:

an = a1 x rⁿ⁻¹

We can use this formula for two different terms of the sequence to find the common ratio. Let's use a₂ and a₅:

a₂ = a₁ x r²⁻¹ --> -24 = a₁ x r

a₅ = a₁ x r⁵⁻¹ --> 8/9 = a₁ x r⁴

Now, let's divide these two equations to eliminate a1:

(-24) / (8/9) = r / r⁴

Simplifying, we get:

-27 = 1/r³

Taking the cube root of both sides, we get:

r = -1/3

Now that we know the common ratio, we can use the formula for the nth term of a geometric sequence to find a₉:

a₉ = a₁ x r⁹⁻¹

We don't know a1, but we can use the fact that a₂ = -24 to find it:

a₂ = a₁ x r²⁻¹ --> -24 = a₁ x (-1/3)

Simplifying, we get:

a₁ = 72

Now we can substitute the values we found for a1 and r into the formula for a₉:

a₉ = a₁ x r⁹⁻¹ --> a₉ = 72 x (-1/3)⁸

Simplifying, we get:

a9 = 8/2187

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The total area of the shape is 90cm²

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. The area of a triangle is expressed as ;

A = 1/2 bh, where b is the base and h is the height if that triangle.

The base of the a triangle in the shape is 6cm and the height is 5cm.

A = 1/2 bh

A = 1/2 × 6 × 5

A = 3× 5

A = 15cm²

Since the size of the triangles are equal ,it means that all the triangles will have thesame area.

Therefore the area of six triangles =

6× 15 = 90cm²

Therefore the area of the shape is 90cm²

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Answer:

[tex]\boxed{5x^2}[/tex]

Step-by-step explanation:

[tex](-2x)^2-(-3x)^2[/tex]

Based in the order of operations (PEMDAS), we solve:

[tex]4x^2-9x^2 =5x^2[/tex]

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

Answer: [tex]-5x^{4}[/tex]

Step-by-step explanation: add and boom