Three semicircles of radius 1 are constructed on diameter $\overline{AB}$ of a semicircle of radius 2. The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles

Answer:

a) In order to write the ratio h:k in its simplest form, we need to find the greatest common divisor (GCD) of h and k and divide both h and k by that value.

The GCD of 2 and 6 is 2.

So the simplified ratio h:k is 2/2:6/2 = 1:3

b) In order to write the ratio k:l in its simplest form, we need to find the greatest common divisor (GCD) of k and l and divide both k and l by that value.

The GCD of 6 and 9 is 3.

So the simplified ratio k:l is 6/3:9/3 = 2:3

Answer:

the ratio of h and k in simplest form is 1:3

the ratio of k and l is 2:3

The algebraic representation of this dilation is: (x', y') = (-0.9x, -1.5y)

A dilation is a transformation that changes the size of a geometric figure, but not its shape. The center of dilation is a fixed point, and all other points in the figure are expanded or contracted in relation to it. In this case, the center of dilation is the origin, and the original figure is m'n'o'p' and the image figure is mnop.

The algebraic representation of a dilation is given by the equation:

(x', y') = (kx, ky)

Where (x', y') are the coordinates of the image point, (x, y) are the coordinates of the original point and k is the scale factor of the dilation.

We can find the scale factor k by comparing the coordinates of any two corresponding vertices of the figures. For example, if we take the vertex m' and m, we have:

m'(-5, -4) and m(4.5, 6)

Here we can see that the x-coordinate of m' is -5 and the x-coordinate of m is 4.5 and the y-coordinate of m' is -4 and the y-coordinate of m is 6.

So, k = mx/mx' = 4.5/-5 = -0.9 and k = my/my' = 6/-4 = -1.5

We can use this scale factor to find the coordinates of any other point in the image figure.

Therefore, the algebraic representation of this dilation is: (x', y') = (-0.9x, -1.5y)

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A four-regular graph is a  quartic graph.

A quartic graph is an example of a polynomial function with a degree of four. The formula denotes a quartic function's equation: f(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, and e are constants. A quartic function's graph is a straight line with up to four turning points or extrema. The roots of the polynomial are also referred to as the graph's x-intercepts or the answers to the equation f(x) = 0.

The values of the constants a, b, c, d, and e, as well as whether an is positive or negative, determine the shape of the quartic graph. The graph starts out with an upward slope if it is positive and a downward slope if an is negative. Factorization, the square-root approach, or numerical techniques like the bisection method or the Newton-Raphson method can all be used to solve quartic equations.

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The eight term of the geometric sequence 125/9, 25/3,5,3,.... is 243/625.

These steps to answer:

Question above is a geometric sequence consisting of 4 terms to find the 8th term, then we can use the geometric sequence formula below

Tn= a. r^n-1

is known

a= 125/9

r=T4/T2

r=3/5

then to find T4

Tn= a. r^n-1

T8=125/9. (3/5^8-1)

T8=125/9. (3/5)^7)

T8=125/9. (2187/78125)

T8=243/625

A geometric sequence is a sequence that satisfies the quotient of a term with the preceding terms which are of course consecutive. This thing has a constant value. Not only that, geometric sequences are also known as 'measurable sequences' which are still closely related to arithmetic sequences and series.

Examples of geometric sequences are a, b, and c. Then c/b = b/a = constant, this is where the quotient of adjacent terms will be obtained and then it is said to be the ratio of the geometric sequence which is given the symbol "r".

Another example, which is much easier to understand, is for example, if you have a sequence and a series: 2, 4, 8, 16, 32, ….. etc., then from the sequence and series it can be seen between the first and second terms and so on, have the same multiplier.

So, to find the nth term, you can easily find the ratio first. By knowing 'r', then you will easily find Tn.

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Answer:

The total number of questions in the test is 40. This can be calculated by dividing 80% by 32, which equals 0.25. Multiplying 0.25 by 32 gives 8, and then multiplying 8 by 5 gives 40.

Step-by-step explanation:

Answer:

40

Step-by-step explanation:

Based on the given conditions, formulate: 32/80%

Multiply both the numerator and denominator with the same integer:320/8

Cross out the common factor:40

get the result:40

Answer: 40

Answer:

b) [tex]y = -3x - 2[/tex] and [tex]y = -3x + 5[/tex] are parallel because the slope of each line is -3

On solving the provided question we can say that here the probability P(X<3) = 0.7102 and P(X>3) = 0.5801.

Probability theory, a subfield of mathematics, gauges the likelihood of an occurrence or a claim being true. An event's probability is a number between 0 and 1, where approximately 0 indicates how unlikely the event is to occur and 1 indicates certainty. A probability is a numerical representation of the likelihood or likelihood that a particular event will occur. Alternative ways to express probabilities are as percentages from 0% to 100% or from 0 to 1. the percentage of occurrences in a complete set of equally likely possibilities that result in a certain occurrence compared to the total number of outcomes.

P(X = 3) = 0.2903

P(X>3) = 0.5801

P(X<3) = 0.7102

mean=2.8

standard deviation=1.2961

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As you can see from the picture, the probability of getting one head and one tail on the toss of two coins is 0.5. There are two different ways that this can happen.

The likelihood of getting one head and one tail on a coin flip is 0.5, as seen in the image. There are two ways in which this might occur. 50% of each coin is equal to 1/2.

There are four possible results for two coins: heads/heads, tails/tails, tails/tails, etc. Each outcome has a theoretical chance of 1/4, or.25. Half of the time, you'll receive heads.

Since the occurrence is complementary and equally likely to occur, it is already known that the probability is half/half, or 50%. As a result, there is a 50% chance of receiving either heads or tails. Take the four-coin experiment into consideration. 

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Please share the correct question !!!

(x−2)2^=y   // this is incorrect mathematical expression

The correct question should be:

[tex](x-2)^{2} =y[/tex] , Find the value of "x"

step-by-step solution:

[tex](x-2)^{2} =y[/tex]

[tex](x-2)=y^{\frac{1}{2} }[/tex]

[tex]x=y^{\frac{1}{2} } + 2[/tex]   OR   [tex]x=\sqrt{y}+ 2[/tex]

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Answer: Dilation by 1/4 and translation

Step-by-step explanation:

Answer:

Hope This Helps <3

Step-by-step explanation:

4-8 +2 = -11

6-3-7 = -4

1 +4-6 =  -1

2-7+6 = 1

The value of 0/infinity is Zero but not undetermined value.

Generally zero in numerator and any other number in the denominator gives the result zero always. i.e 0/ x where x is any other number and the number can be real, imaginary or complex. But the case with infinity is different as zero upon infinity gives the result as zero only it's because infinity is never can be a member of set of number. Because infinity is  just an expression that is greater than any assignable quantity or countable number. In mathematics, the concept of infinity describes something  larger than the natural number. It usually describes something without a limit. This concept is not only used in the mathematics but also in physics. In mathematics and physics, this concept is utilized in a wide range of fields. The term infinity can also be employed for an extended number system.

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The solution to the linear system is [tex]|A_x|=\left[\begin{array}{cc}17&3\\9&-3\end{array}\right]{,|A|=\left[\begin{array}{cc}5&3\\-8&-3\end{array}\right]} ,|A_y|=\left[\begin{array}{cc}5&17\\-8&9\end{array}\right]\\[/tex]

An equation is a expression that shows the relationship between numbers and variables.

A linear system of equation:

Ax + By = C

Dx + Ey = F

Can be solved using:

[tex]|A_x|=\left[\begin{array}{cc}C&B\\F&E\end{array}\right]{,|A|=\left[\begin{array}{cc}A&B\\D&E\end{array}\right]} ,|A_y|=\left[\begin{array}{cc}A&C\\D&F\end{array}\right]\\\\Hence:\\\\x=\frac{|A_x|}{|A|} ,y=\frac{|A_y|}{|A|}[/tex]

Hence, for:

5x + 3y = 17

-8x - 3y = 9

[tex]|A_x|=\left[\begin{array}{cc}17&3\\9&-3\end{array}\right]{,|A|=\left[\begin{array}{cc}5&3\\-8&-3\end{array}\right]} ,|A_y|=\left[\begin{array}{cc}5&17\\-8&9\end{array}\right]\\[/tex]

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I do not think that eleri can cover a distance of 1 km.

Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.

The Distance Formula itself is actually derived from the Pythagorean Theorem which is a 2 + b 2 = c 2 {a^2} + {b^2} = {c^2} a2+b2=c2 where c is the longest side of a right triangle (also known as the hypotenuse) and a and b are the other shorter sides (known as the legs of a right triangle).

In the abov equestion,

for 1 lap eleri cover a distance of 400m

for 2 laps she will cover a distance of 800m

since

800m<1km

So eleri cannot cover a distance more than 800 m

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Answer:

B) [tex]2^2 \times 5 \times 37[/tex]

Step-by-step explanation:

[tex]740=2 \times 370=2^2 \times 185=2^2 \times 5 \times 37[/tex]

Factorise means to carry over the common number or letter to the outside of a set of brackets, whilst keeping the rest of the equation inside the brackets.

Answer:

Carry f to the outside, meaning that you would be left with f(4f +1).

An expression using subtraction to find the distance between the two divers is |B - A| or |A - B| or B - A as B > A and the distance using additive inverses is 4 unit.

The number line displays the separation in meters between two divers A and B and the shipwreck at position X:

A = -2.5

X = 0

B = 1.5

We have to write an expression using subtraction to find the distance between the two divers and the distance using additive inverses.

Distance between the two divers = |B - A| or |A - B| or B - A as B > A

Distance between the two divers = |1.5 - (-2.5)| or |-2.5 - 1.5| or  1.5 - (-2.5)

Additive inverse of a number "x" is -x. So Additive inverse of -2.5 = 2.5

Distance between the two divers = 1.5 + 2.5

Distance between the two divers = 4

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BD bisects angle ABC for the value of x = 4

An angle is a shape created by two rays that share a terminus and are referred to as the angle's sides and vertices, respectively. Angles created by two rays are on the plane where the rays are located. The meeting of two planes also creates angles. Dihedral angles are these.

An angle is a shape created by two rays or lines that meet at the same terminal. The Latin word "angulus," which meaning "corner," is the source of the English term "angle." The shared terminus of two rays is known as the vertex, and the two rays are referred to as sides of an angle.

In geometry, an angle bisector is a line that divides an angle into two equal angles. The term "bisector" refers to a device that divides an item or a form into two equal halves. An angle bisector is a ray that divides an angle into two identical segments of the same length.

Since 3x + 1 and 4x - 3 are angle bisectors so,

3x + 1 = 4x - 3

or, x = 4

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If a triangle has a perimeter of 48 kilometers. Two of the sides measure 12 kilometers and 20 kilometers. The length of the third side is 16.

Let x km represent  the length of the third side

Let Length of other two sides is  12 km  and  20 km.

Hence,

Perimeter of triangle = Sum of all sides =  48 km

12 + 20 + x = 48

32 +x = 48

x = 48 - 32

x = 16

Therefore we can conclude that the length of the third side is 16.

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Therefore , the solution of the given problem of integer comes out to be 1/100 = 0.01

Zero, a positive integer, or a negative integer denoted by a minus sign are all examples of integers. A negative number is the additive reciprocal of a positive number that it corresponds to. A bold Z or a bold "mathbb Z" is frequently used in mathematical notation to denote a group of integers. A positive, negative, or zero integer—not a fraction—is referred to as an integer (pronounced IN-tuh-jer). The numbers -5, 1, 5, 8, 97, and 3,043 are examples of integers. 1.43, 1 3/4, 3.14, and other numbers are non-integer examples. Integers are a collection of integers and their antipodes. Decimals and fractions are not part of the set of integers.

Here,

=> 1/4 =0.25

=> 1/20 = 0.05

=> 1/25 = 0.04

=> 1/50 = 0.02

=> 1/100 = 0.01

Therefore , the solution of the given problem of integer comes out to be 1/100 = 0.01

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In order for the number to form a terminating decimal, 2 and 5 must be its only prime factors. Obviously any number less than [tex]0.01[/tex] or any value of [tex]n > 100[/tex] will not have a nonzero hundredths digit.

We now count the possibilities for [tex]n[/tex]:

[tex]5^0 : 2^0, 2^1,..., 2^6 \rightarrow 7 \text{ values}\\5^1 : 2^0, 2^1,..., 2^4 \rightarrow 5 \text{ values}\\5^2 : 2^0, 2^1, 2^2 \rightarrow 3 \text{ values}[/tex]

for a subtotal of [tex]15[/tex] values.

However, some of these have a nonzero tenths digit and a zero hundredths digit. In other words, we need to remove all of the single-digit terminating decimals from this list. These are [tex]\dfrac{1}{1} = 1.00,\dfrac{1}{2} = 0.50,\dfrac{1}{5}=0.20, \text{ and }\dfrac{1}{10} = 0.10 \implies 4 \text{ values}[/tex]

which means the final answer is [tex]15-4=11[/tex].

Answer:

8.1

Step-by-step explanation:

12/x = tan56

12/x =1.48

so x = 12/1.48 =8.1

The hexagon's side length is 4.5 inches whereas the equilateral triangle's side length is 9.

While an exterior angles are equivalent to the sum of the twin interior angles that become not directly adjacent to it, the inside angles necessarily add up to 180°. By lowering the inclination of the target vertex from 180°, one may also get a triangle's exterior angle.

The equilateral triangle's side length is 9 inches.

The hexagon's sides measure 4.5 inches long.

The equilateral triangle's side length is equal to 1.4x plus 2.

Perimeter = 3(1.4x + 2) = 4.2x + 6 as a result.

The hexagon's side length is equal to 0.5x plus 2.

The perimeter is 6(0.5x + 2) = 3x + 12 as a result.

the perimeters being equal.

So,\s4.2x + 6 = 3x + 12

4.2x - 3x = 12-6

1.2x = 6\sx = 5

Hence,

The equilateral triangle's side length is equal to 1.4x + 2 = 1.4(5) + 2 = 9 inches.

The hexagon's side length is equal to 0.5x + 2 (or 0.5(5) + 2; or 4.5 inches).

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The complete question is-

An equilateral triangle has a side length of 1.4x + 2 inches. A regular hexagon has a side length of 0.5x + 2 inches. The perimeters are equal. What is the side length of the triangle? What is the side length of the hexagon? Show your work.

As per the formula of surface area of cube, the length of the cube is 5.45 meters.

The general formula to calculate the surface area of the cube is calculated as,

=> SA = 6a²

here a represents the length of cube.

Here we know that  the side of a cube with a surface area of 180 square meters than a cube with the surface area of 120 square meters.

When we apply the value on the formula, then we get the expression like the following,

=> 180 = 6a²

where a refers the length of the cube.

=> a² = 30

=> a = 5.45

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Answer:

47

Step-by-step explanation:

Answer:

40m + 40

Step-by-step explanation:

8(5m+5)

40m + 40

The rate in centimeters per second, at which the radius is increasing at the instant when the radius is 5 centimeters is 0.16 cm/s.

Using a known value of a function at a certain point along with its rate of change at that moment, one application of derivatives is to estimate an unknown value of a function at a particular point.

Given:

Both radius r and area A is a function of time, so in fact:

A(t) = π [r(t)]²

Deriving this we get that the area changes and the radius changes as:

[tex]\frac{dA(t)}{dt} = 2\pi r(t)\frac{dr(t)}{dt}[/tex]

But,

[tex]\frac{dA(t)}{dt} = 5 \frac{cm^{2} }{s}[/tex]

Considering when r(t) = 5cm at the instant we get,

[tex]5 = 2\pi . 5\frac{dr(t)}{dt}[/tex]

Simplify 5 and take 2π to the other side and divide,

The rate of change of radius is given by:

[tex]\frac{dr(t)}{dt} =\frac{1}{2\pi } = 0.16 cm/s[/tex]

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Answer:

a) 4 cosx^4 - cos²2x=0

We can start by splitting the equation into two parts:

4 cosx^4 = cos²2x

cosx^4 = (1/4)cos²2x

From here, we can use the identity:

cos²2x = (cos 2x)² = (2cos²x -1)²

So, we have:

cosx^4 = (1/4)(2cos²x -1)²

Now we can square root both sides:

cosx^2 = (1/2) (2cos²x -1)

Now we square both sides again:

cos²x = (1/4) (2cos²x -1)

We can now subtract (1/4) from both sides and add cos²x to both sides:

(3/4)cos²x = (1/4)

We can now divide both sides by (3/4) to find the value of cosx:

cosx = sqrt(1/3)

So the solutions of the equation are x = (2n+1)π/3, where n is an integer.

b) sec x tan x =sqrt(2)

We know that sec x = 1/cosx and tan x = sinx/cosx.

So we can substitute these values into the equation:

1/cosx * sinx/cosx = sqrt(2)

This simplifies to:

sinx = sqrt(2)cosx

We can now divide both sides by cosx:

tan x = sqrt(2)

So the solutions of the equation are x = pi/4 + nπ, where n is an integer.

Note that the solutions are restricted to the domain where cos(x) is not equal to 0.

It is important to note that this solution is only a small part of the full solution set and that there might be more solutions in specific intervals.

The general solution of the system x' (t) is[tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex]

The given system of equation is X'A=X

where

[tex]$$A=\left[\begin{array}{cc}-15 & -39 \\6 & 15\end{array}\right]$$[/tex]

Eigenvector of a square matrix is defined as a non-vector in which when a given matrix is multiplied,  it is equal to a scalar multiple of that vector. Let us suppose that A is an n x n square matrix, and if v be a non-zero vector, then the product of matrix A, and vector v is defined as the product of a scalar quantity λ and the given vector, such that:

Where

v = Eigenvector and λ be the scalar quantity that is termed as eigenvalue associated with given matrix A

The values of A are given by

[tex]$|A-\lambda I|=0 \\\left|\begin{array}{cc}-15-\lambda & -39 \\6 & 15-\lambda\end{array}\right|=0 \\[/tex]

[tex](-15-\lambda)(15-\lambda)-(6)(-39)=0 \\[/tex]

[tex]\Rightarrow-(15+\lambda)(15-\lambda)+(6)(39)=0 \\[/tex]

[tex]\\\Rightarrow(15+\lambda)(15-\lambda)-(6)(39)=0 \\\\\Rightarrow 225-\lambda^2-234=0 \\[/tex]

[tex]\Rightarrow-\lambda^2-9=0 \\[/tex]

[tex]\Rightarrow \lambda^2=-9 \\[/tex]

[tex]\Rightarrow \lambda=\pm 3 i[/tex]

Now, eigen vector u corresponding to [tex]$\lambda=3 i$[/tex] is given by

[tex]$$\begin{gathered}{[A-3 i I] u=O} \\{\left[\begin{array}{cc}-15-3 i & -39 \\6 & 15-3 i\end{array}\right]\left[\begin{array}{l}u_1 \\u_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\\\\end{gathered}$$[/tex]

Applying [tex]R_2 \rightarrow R_2-6 R_1[/tex]

[tex]& {\left[\begin{array}{cc}1 & \frac{-39}{-15-3 i} \\0 & 0\end{array}\right]\left[\begin{array}{l}u_1 \\u_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

[tex]& \Rightarrow u_1+\frac{39}{15+3 i} u_2=0 \\[/tex]

[tex]& \Rightarrow u_1=-\frac{13}{5+i} u_2=-\frac{13}{5+i} * \frac{5-i}{5-i} u_2=-\frac{13(5-i)}{26} u_2=-\frac{5-i}{2} u_2 \\[/tex]

Thus, by choosing [tex]$u_2[/tex]=1 eigenvector corresponding to [tex]$\lambda=3 i$[/tex] is [tex]$$u=\left[\begin{array}{c}-\frac{5-i}{2} \\1\end{array}\right]$$[/tex]

[tex]R_1 \rightarrow \frac{1}{-15+3 i} R_1 \\[/tex]

[tex]{\left[\begin{array}{cc}1 & \frac{-39}{-15+3 i} \\6 & 15+3 i\end{array}\right]\left[\begin{array}{l}v_1 \\v_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

Applying [tex]R_2 \rightarrow R_2-6 R_1 \\[/tex]

[tex]{\left[\begin{array}{cc}1 & \frac{-39}{-15+3 i} \\0 & 0\end{array}\right]\left[\begin{array}{l}v_1 \\v_2\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\[/tex]

[tex]\Rightarrow v_1+\frac{39}{15-3 i} v_2=0 \\\Rightarrow v_1=-\frac{39}{15-3 i}[/tex]

[tex]\begin{gathered}\Rightarrow v_1+\frac{39}{15-3 i} v_2=0 \\\v_2=-\frac{13(5+i)}{26} \\\Rightarrow v_1==-\frac{5+i}{2}\end{gathered}$$[/tex]

Thus, by choosing [tex]$v_2=1$[/tex] eigenvector corresponding to [tex]$\lambda=-3 i$[/tex] is

[tex]$$v=\left[\begin{array}{c}-\frac{5+i}{2} \\1\end{array}\right]$$[/tex]

Hence, the general solution is given by

[tex]X(t)=c_1 e^{3 i t} u+c_2 e^{-3 i t} v \text { [using } e^{i t}=\cos t+i \sin t \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5-i}{2}(\cos 3 t+i \sin 3 t) \\(\cos 3 t+i \sin 3 t)\end{array}\right]+c_2\left[\begin{array}{c}-\frac{5+i}{2}(\cos 3 t-i \sin 3 t) \\(\cos 3 t-i \sin 3 t)\end{array}\right] \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5}{2}(\cos 3 t+i \sin 3 t)+\frac{i}{2}(\cos 3 t+i \sin 3 t) \\(\cos 3 t+i \sin 3 t)\end{array}\right][/tex][tex]+c_2\left[\begin{array}{c}-\frac{5}{2}(\cos 3 t-i \sin 3 t)-\frac{i}{2}(\cos 3 t-i \sin 3 t) \\(\cos 3 t-i \sin 3 t)\end{array}\right] \\[/tex]

[tex]X(t)=c_1\left[\begin{array}{c}-\frac{5}{2} \cos 3 t-\frac{5 i}{2} \sin 3 t+\frac{i}{2} \cos 3 t-\frac{1}{2} \sin 3 t \\\cos 3 t+i \sin 3 t\end{array}\right][/tex][tex]+c_2\left[\begin{array}{c}-\frac{5}{2} \cos 3 t+\frac{5}{2} i \sin 3 t-\frac{i}{2} \cos 3 t-\frac{1}{2} \sin 3 t \\\cos 3 t-i \sin 3 t\end{array}\right ][/tex]

[tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex]

Therefore, the general solution of the system X(t) is [tex]X(t)=c_1\left(\left[\begin{array}{c}-5 \\2\end{array}\right] \cos 3 t-\left[\begin{array}{l}1 \\0\end{array}\right] \sin 3 t\right)[/tex][tex]+c_2\left(\left[\begin{array}{c}1 \\0\end{array}\right] \cos 3 t+\left[\begin{array}{c}-5 \\2\end{array}\right] \sin 3 t\right)\end{gathered}[/tex].

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The first partial derivatives of the function z = x sin(xy) is x²cos(xy)

The term partial derivatives is defined as the rate of change of a function with respect to a variable and the derivatives are fundamental to the solution of problems in calculus and differential equations.

Here we have given that the function z = x sin(xy).

And as per the definition of partial derivative the value is calculated as,

Here we have given that

=> f(x, y) = x sin(xy)

And then here we need to find fx​ we treat y as constant and differentiate with respect to x, then we get

=> fx = sin(x y) + xy cos(xy)

Similarly now we have to find fy​ we treat x as constant and differentiate with respect to y

=> fy​ = x²cos(xy)

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The number of pages that the student will read in 7 hours, given the number of pages read in 4 hours, is

The constant variation is 14 pages.

This constant variation is direct.

The quantity that connects two variables that are directly or inversely proportional to one another is known as the constant of variation.

Variation in mathematics demonstrates how one variable fluctuates in respect to another. Typically, a ratio is used to illustrate this relationship. When we remark that a variation is continuous, we are referring to how consistently the ratio changes.

The constant of variation in this instance therefore, is:

= Number of pages read / Number of hours

= 56 / 4

= 14 pages

This constant of variation is direct.

The number of pages read in 7 hours is:

= Constant of variation x number of hours

= 14 x 7

= 98 pages

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