Find the solution of the given initial value problem. y'-y=3te^2t , y(0) = 1
y(t)=?

The required exact value of cos(-10π/3) or cos(2π/3) is -1/2.

To find the exact value of the expression cos(-10π/3), we can use the periodicity and symmetry properties of the cosine function.

In this case, we have cos(-10π/3). Since -10π/3 is equivalent to 2π/3 radians (because it completes a full revolution plus an additional 2π), we can rewrite the expression as cos(2π/3).

Now, let's find the exact value of cos(2π/3).

In the unit circle, an angle of 2π/3 (or 120 degrees) is located in the second quadrant. In the second quadrant, the x-coordinate is negative.

Since the cosine function gives the x-coordinate of a point on the unit circle, we can conclude that cos(2π/3) is negative.

Therefore, the exact value of cos(-10π/3) or cos(2π/3) is -1/2.

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Answer:

The equation that models the relationship between the time x (in minutes) and the distance y (in feet) from the ground is:

y = 2000x

This is because for every 2 minutes that go by (i.e. for every increase of x by 1), the balloon rises by 2000ft (i.e. a constant rate of change). So, the equation y = 2000x represents a straight line with a slope (rate of change) of 2000, passing through the origin.

The values of x and y are given as follows:

The geometric mean theorem states that the length of the altitude drawn from the right angle of a triangle to its hypotenuse is equal to the geometric mean of the lengths of the segments formed on the hypotenuse.

The bases for this problem are given as follows:

2 and x.

The altitude is given as follows:

6.

Hence the length x is given as follows:

2x = 6²

2x = 36

x = 18.

Applying the Pythagorean Theorem, the length y is given as follows:

y² = 18² + 6²

y² = 360

[tex]y = \sqrt{36 \times 10}[/tex]

[tex]y = 6\sqrt{10}[/tex]

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The price per inch of the wrap is $0.01 or 100cent

A word problem in math is a math question written as one sentence or more. This statements are interpreted into mathematical equation or expression.

For example, If the age of Roy is x what will be his age in five years time.

His age five years time will be x+5

9 foot roll cost $1.08

1 foot = 12 inches

9 feet = 9 × 12

= 108 inches

Therefore 108 inches = 1.08

1 inch = 1.08/108

= $0.01

= 100cents

therefore the price of the wrap per inch is $0.01 or 100 cent.

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There are three area options for the number of bricks:

The number of bricks that can be painted out is equal to the paint area (A), in square centimeters, divided to the area of a brick (A'), in square centimeters. That is:

n = A / A'

A' = w · h

Where:

There are three possible answers:

Case 1: A = 93750 cm², w = 22.5 cm, h = 10 cm

A' = 22.5 · 10

A' = 225

n = 93750 / 225

n = 416.667

n = 416

Case 2: A = 93750 cm², w = 22.5 cm, h = 7.5 cm

A' = 22.5 · 7.5

A' = 168.75

n = 93750 / 168.75

n = 555.556

n = 555

Case 3: A = 93750 cm², w = 10 cm, h = 7.5 cm

A' = 10 · 7.5

A = 75

n = 93750 / 75

n = 1250

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The graph will look like a circle centered at (2, 1) with radius 3.

To graph the equation [tex]x^2 - 4x + y^2 - 2y - 4 = 0[/tex]  by completing the square, we need to rearrange the terms as follows:

[tex](x^2 - 4x + 4) + (y^2 - 2y + 1) = 9[/tex]

This can be simplified to:

[tex](x - 2)^2 + (y - 1)^2 = 3^2[/tex]

So the equation represents a circle with a center at (2, 1) and a radius 3. To graph the circle, we can plot the center point (2, 1) and then draw a circle with radius 3 around that point.

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The probability that either event will occur is 0.62

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

Event A = 6 + 6 = 12

Event B = 20 + 6 = 26

Both A and B = 6

Other Events = 20

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

Total = A + B + C + Others - Both

So, we have

Total = 12 + 26 - 6 + 20

Evaluate

Total = 52

So, we have

P(A) = 12/52

P(B) = 26/52

Both A and B = 6/52

For either events, we have

P(A or B) = (12 + 26 - 6)/52

Evaluate

P(A or B) = 0.62

Hence, the probability that either event will occur is 0.62

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Based on this graph, a measurement which represent the best estimate for the mean length of bass in the lake include the following: 17 inches.

In Mathematics and Statistics, a mean is sometimes referred to as an average and it can be defined as a ratio of the sum of the total number in a data set (population) to the frequency of the data set.

Mathematically, the mean of a data set can be calculated by using the following formula:

Mean = [F(x)]/n

By critically observing the graph (histogram) shown in the image attached below, we can logically deduce that the distribution is symmetrical. This ultimately implies that, the mean length of bass in the lake would be in the middle, with an estimated value of 17 inches.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Complete Question:

A university professor and his students did a study to determine the mean length of bass in a lake. Each time a bass was caught, its length was measured in inches, and then it was released. The distribution of the lengths is shown in the graph below. Based on this graph, which measurement represents the best estimate for the mean length of bass in the lake?

12 inches

17 inches

20 inches

23 inches

The interior angles of the trapezium is as follows:

The sum of interior angles in a quadrilateral is 360 degrees. Therefore, the diagram above is a trapezium with 4 sides.

Let's find the measure of each interior angles

2x + 2x + 20 + 2x - 10 + 3x - 10 = 360

9x  = 360

divide both sides by 9

x = 360 / 9

x = 40

Therefore, each interior angles is as follows:

2x = 2(40) = 80 degrees

2x + 20 = 2(40) + 20 = 100 degrees

2x - 10 = 2(40) - 10 = 70 degrees

3x - 10 = 3(40) - 10 = 110 degrees

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Answer:

183.89

Step-by-step explanation:

S = square area + area of 4 triangles =

7.1^2 + 4*0.5*9.4*7.1 = 183.89

Answer: your answer is 266.44 square inches

Explanation: Triangles:

6.1x7.6=46.36in

Each triangle is 46.36in

46.36x4=185.44in

Rectangle:

9x9=81in

The rectangle is 81in

All together:

185.44+81=266.44in

So the square is the easiest do 4×4=16

The triangles its just Base×Height/2 in that order so 4×3=12/2=6

Mulitply 6 by 4 because there's 4 triangles and add 16 so 6×4=24+16=40

Hope this helped:)

Rounding to the nearest tenth, the mean lifespan for Kamila's sample is 11.9 years.

To find the mean (average) for Kamila's sample, you need to sum up all the lifespans and then divide by the total number of breeds in the sample.

Sum of lifespans = 13 + 11 + 12 + 14 + 12 + 10 + 11 + 12 = 95

Total number of breeds = 8

Mean = Sum of lifespans / Total number of breeds

= 95 / 8 = 11.875

Rounding to the nearest tenth, the mean lifespan for Kamila's sample is 11.9 years.

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a. The constant of variation is k = 45.

b. The equation for the relation is y = 45/x².

c. x = 5, y is approximately equal to 1.8.

d. The equation provided, y = -52 + (5)/(5x²), seems to be incorrect or incomplete as there are misplaced parentheses and missing values.

The constant of variation for the relation "y varies inversely as the square of x," we can use the general form of an inverse variation equation:

y = k/x², where k represents the constant of variation.

The given information, we have y = 5 when x = -3.

Substituting these values into the equation, we get:

5 = k/(-3)²

5 = k/9

k = 45

The constant of variation is k = 45.

The equation for the relation is y = 45/x².

To find y when x = 5, we can substitute the value of x into the equation:

y = 45/(5)²

y = 45/25

y = 1.8

x = 5, y is approximately equal to 1.8.

The equation provided, y = -52 + (5)/(5x²), seems to be incorrect or incomplete as there are misplaced parentheses and missing values. Please clarify the equation and I'll be happy to assist you further.

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Answer:

3

Step-by-step explanation:

Step 1:  First, we must plug in 7 for x in f(x) and g(x):

f(7) = 2(7) + 1

f(7) = 14 + 1

f(7) = 15

g(7) = 7-2

g(7) = 5

Now, we can divide f(7) by g(7)

f(7) / g(7) = 15 / 5

f(7) / g(7) = 3

Thus, the quantity of f divided by g of 7 is 3

Answer:

x = 3

y = -1

Step-by-step explanation:

             Method:  substitution

x - 3y = 6 --------------(I)

      x  = 6 + 3y -------------(II)

2x + 2y = 4 ----------(III)

Substitute x = 6 +3y in equation (II),

        2*(6+3y) + 2y = 4

Use distributive property to open the parenthesis,

     2*6 + 2*3y + 2y = 4

         12 + 6y + 2y = 4

Combine like terms.

            12 + 8y      = 4

Subtract 12 from both sides,

                        8y  = 4 - 12

                        8y = -8

Divide both sides by 8,

                          y = -8÷ 8

                            [tex]\boxed{\bf y = (-1)}\\[/tex]

Substitute y = -1 in equation (II) and find the value of x.

               x = 6 + 3*(-1)

                  = 6 - 3

               [tex]\sf \boxed{\bf x = 3}[/tex]

Answer:

(3, - 1 )

Step-by-step explanation:

x - 3y = 6 ( add 3y to both sides )

x = 3y + 6 → (1)

2x + 2y = 4 → (2)

substitute x = 3y + 6 into (2)

2(3y + 6) + 2y = 4

6y + 12 + 2y = 4

8y + 12 = 4 ( subtract 12 from both sides )

8y = - 8 ( divide both sides by 8 )

y = - 1

substitute y = - 1 into (1)

x = 3(- 1) + 6 = - 3 + 6 = 3

solution is (3, - 1 )

The general format of the solution to the system of equations in this problem is given as follows:

(x, -1/5x).

Considering the augmented matrix for the system, the system of equations in the context of this problem is defined as follows:

The second equation is true for any ordered pair (x,y), hence the system has an infinite number of solutions.

The infinite number of solutions has the format given as follows:

x + 5y =  0

5y = -x

y = -x/5.

Hence the ordered pair is:

(x, -1/5x).

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The multiplication of the expression (s - 3)³ is determined as s³ - 9s³ + 27s - 27.

The product of the expression given as (s - 3)³ is calculated  as follows;

The given expression;

(s - 3)³, this can also be written as;

(s - 3)³ =  (s - 3) (s - 3) x (s - 3)

We will simplify the expression as follows;

(s - 3)(s - 3) x (s - 3)

= (s - 3)(s - 3) x (s - 3)

= (s² - 3s -3s + 9 )(s - 3)

= ( s² - 6s + 9 ) (s - 3)

= s³ - 3s² - 6s² + 18s + 9s - 27

= s³ - 9s³ + 27s - 27

Thus, the given expression can be simplified using binomial theorem or using simple multiplication method as shown above.

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The ratio of technicians to all the helpers at the Harper Middle School fall play is 9 : 24.

The ratio of technicians to all helpers can be calculated by dividing the number of technicians by the total number of helpers.

Number of technicians = 9

Total number of helpers = 15 ushers + 9 technicians = 24

The ratio of technicians to all helpers = 9 / 24

The ratio will then be:

The ratio of technicians to all helpers =  9 : 24

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Complete question:

There are 15 ushers and 9 technicians helping at the Harper Middle School fall play. What is the ratio of technicians to all helpers?

Answer:

60°

Step-by-step explanation:

In similar triangles, the corresponding angles are congruent.

          ∠O = R

          O = 70°

In ΔOMN,

      ∠O + ∠M + ∠N = 180     {Angle sum property of triangle}

       70 + 50 + Ф    = 180

               120 + Ф  = 180

Subtract 120 from both sides,

                        Ф  = 180 - 120

                         Ф = 60°  

It is possible for a sample to follow a normal distribution, depending on the nature of the data being accumulated.

The regular distribution is a bell-shaped curve that is symmetric across the mean, and it's far typically used to model various natural phenomena and measurements in fields such as information, engineering, and social sciences.

If the data being gathered is continuous and the sample length is large sufficient, it is regularly assumed that the pattern follows a normal distribution.

This assumption is based at the critical limit Theorem, which states that the sampling distribution of the suggest of any independent, random variable will tend toward a ordinary distribution because the sample size increases.

But, it's far critical to notice that not all samples will observe a ordinary distribution, and in a few cases, different distributions may be greater suitable for the records being amassed.

It's far consequently essential to analyze the information and evaluate the assumptions being made before using the regular distribution as a model.

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Answer:

[tex]\sqrt{9}[/tex] is rational and [tex]\sqrt{10}[/tex] is irrational

Step-by-step explanation:

The square root of 9 is 3 and the square root of 10 is a decimal that never stops.

The square root of 9 (√9) is rational because it results in an integer, which can be expressed as a fraction (3/1). However, the square root of 10 (√10), yields an approximate decimal that cannot be precisely expressed as a fraction, hence is an irrational number. Thus the statement √9 is rational and √10 is irrational is true.

In Mathematics, a rational number is a number that can be expressed as a fraction where the numerator and denominator are integers and the denominator is not zero. Irrational numbers are numbers that cannot be expressed as fractions.

The square root of 9 is √9, which equals 3. This can be expressed as a fraction (3/1), so √9 is rational.

However, the square root of 10 is √10, which equals to approximately 3.16227766017. This cannot be accurately expressed as a fraction, so √10 is irrational.

Therefore, the true statement among the options provided is: √9 is rational and √10 is irrational.

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The length of the midsegment of the trapezoid with bases 18 and 28 is gvien as follows:

23.

The trapezoid midsegment theorem, states that the length of the midsegment of the trapezoid is equals to the mean of the length of the bases of the trapezoid.

The bases of the trapezoid are given as follows:

18 and 28.

Hence the length of the midsegment of the trapezoid is given as follows:

(18 + 28)/2 = 23.

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Answer:

Set your calculator to Degree mode.

(1/2)(11.3)(5.2)(sin A) = 17.8

29.38sinA = 17.8

sin A = 890/1,469

A = sin^-1 (890/1,469) = 37.3°

Answer:

The slope is 5.

Hope this helps!

Step-by-step explanation:

( x, y )

( 6, 50 ) and ( 12, 80 )

[tex]\frac{80-50}{12 - 6} < = \frac{y_{2} -y_{1} }{x_{2}-x_{1} }[/tex]

[tex]\frac{30}{6} = \frac{5}{1} = 5[/tex]

The slope is 5.

150 degrees is the measure of the angle x from the figure

The given diagram is a circle and the angle x and 30 degrees can be seen to be on a straight line.

The sum of the measure of angle on a straight line is 180 degrees, hence:

x + 130 = 180

Subtract 130 degrees from both sides to have:

x + 30 - 30 = 180 - 30
x = 180 - 30
x = 150 degrees

Hence the measure of the angle x from the given equation is 150 degrees

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Answer:

A. 4ab² + 2x

Step-by-step explanation:

If you want to get rid of those pesky parentheses, you have to use the superpower of distributive property. It lets you multiply everything inside the parentheses by the number outside. For example:

4(ab²+1/₂c + x) = 4ab² + 2c + 4x

Boom! No more parentheses!

To combine like terms, you have to add or subtract the numbers in front of the terms that have the same variable and exponent. For example:

2x - 4x = -2x

Bam! Only one x left!

Here are the steps to simplify the expression:

4(ab²+1/₂c + x) - 2(c+x)

= 4ab² + 2c + 4x - 2c - 2x (use superpower)

= 4ab² + 4x - 2x (combine like terms)

= 4ab² + 2x (simplify)

So, the answer is A. 4ab² + 2x.

After using ruler and following the steps described below

We get the standard position of 35 degree.

The given angle is = 35 degree

To draw standard position of angle:

We proceed it to draw by ruler

So, in order to get position,

Follow the following steps:

1: Draw a straight line with your ruler.

2: Place your protractor on the line and align the base of the protractor with the line.

3: Find the 35 degree mark on the protractor and make a small mark on the line at that point.

4: Move your protractor to the mark you just made.

5: Align the base of the protractor with the line and make another small mark on the line at the 35 degree mark.

6: Draw a straight line connecting the two points you just marked.

And there you have it! A 35 degree angle drawn on paper.

And after following these steps we get the standard potion angle 35 degree.

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The products of the matrices A and B are [tex]AB = \left[\begin{array}{cc}-50 &7\\44 &-10\end{array}\right][/tex] and [tex]BA = \left[\begin{array}{cc}-12&-24 \\-16&-48\end{array}\right][/tex]

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

Matrix A = [tex]\left[\begin{array}{cc}-3&-8\\2&8\end{array}\right][/tex]

Matrix B = [tex]\left[\begin{array}{cc}6&3\\4&-2\end{array}\right][/tex]

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

[tex]AB = \left[\begin{array}{cc}-3 * 6 + -8 * 4 &-3 * 3 -8 * -2\\2 * 6 + 8 * 4 &2 * 3 + 8 * -2\end{array}\right][/tex]

Evaluate

[tex]AB = \left[\begin{array}{cc}-50 &7\\44 &-10\end{array}\right][/tex]

Next, we have

[tex]BA = \left[\begin{array}{cc}6 * -3 + 3 * 2&6 * -8 + 3 * 8 \\4 * -3 + -2 * 2&4 * -8 + -2 * 8\end{array}\right][/tex]

Evaluate

[tex]BA = \left[\begin{array}{cc}-12&-24 \\-16&-48\end{array}\right][/tex]

Hence, the values are [tex]AB = \left[\begin{array}{cc}-50 &7\\44 &-10\end{array}\right][/tex] and [tex]BA = \left[\begin{array}{cc}-12&-24 \\-16&-48\end{array}\right][/tex]

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