Solve for x...........

The general solution to the homogeneous system of differential equations is:

x(t) = c₁ * \(e^{(-7t)\) * | 4 | + c₂ * \(e^{(-20t)\) * | 2 |

                                                        |-1 |

where c₁ and c₂ are constants.

To find the general solution to the homogeneous system of differential equations -11x' = Ax, where A = [-26 4; 1 1], we first need to find the eigenvalues and eigenvectors of matrix A.

To find the eigenvalues, we solve the characteristic equation:

det(A - λI) = 0

Substituting the values, we get:

| -26-λ 4 |

| 1 1-λ |

Expanding the determinant, we have:

(-26-λ)(1-λ) - 4 = 0

Simplifying and solving the equation, we find the eigenvalues:

λ₁ = -7

λ₂ = -20

Next, let's find the corresponding eigenvectors.

For λ₁ = -7:

(A + 7I)v₁ = 0

| -19 4 |

| 1 8 |

Solving the system of equations, we find the eigenvector corresponding to λ₁:

v₁ = | 4 |

      |-1 |

For λ₂ = -20:

(A + 20I)v₂ = 0

| -6 4 |

| 1 21 |

Solving the system of equations, we find the eigenvector corresponding to λ₂:

v₂ = | 2 |

      |-1 |

Now that we have the eigenvalues and eigenvectors, we can write the general solution to the system of differential equations as:

Substituting the values of the eigenvalues and eigenvectors, we get:

x(t) = c₁ * \(e^{(-7t)\) * | 4 | + c₂ * \(e^{(-20t)\) * | 2 |

                                                        |-1 |

Simplifying this expression, we get:

x(t) = | 4c₁ * \(e^{(-7t)\) + 2c₂ * \(e^{(-20t)\) |

|-c₁ * \(e^{(-7t)\) - c₂ * \(e^{(-20t)\)) |

Therefore, the general solution to the homogeneous system of differential equations is:

x(t) = c₁ * \(e^{(-7t)\) * | 4 | + c₂ * \(e^{(-20t)\) * | 2 |

|-1 |

where c₁ and c₂ are constants.

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In the exponential function y = 61 (0.95)^t, the change represents a decay.

The percentage of decay rate is 5%.

What is an exponential function?

The formula for an exponential function is f(x) = a^x, where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.

The given exponential function is -

y = 61 (0.95)^t

Here, it can be seen that there is a decay rate as the base value 0.95<1.

The percentage rate of decrease is 1-0.95.

Find the percentage decay rate as -

= 1 - 0.95

= 0.05

Convert it into percentage -

= 0.05 × 100

= 5 %

Therefore, the exponential function is a decay function with 5% decay rate.

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Answer:

6a. x = 5/3

6b. x = 2

7a. x = 17

7b. y = 30, x = 3

8. 90-55 = 35

9. 180-30=150

10. 17 - 5 =12

11. opposite and corresponding angles (explanation beneath)

Step-by-step explanation:

6a. 7x-10=x+2

7x=x+10 (add 10 to both sides)

6x=10 (subtract x from both sides)

x = 5/3

6b. -3x+5(3x-1)=2x+15

12x-5=2x+15 (simplify the left side)

12x=2x+20 (add 5 to both sides)

10x=20 (subtract 2x from both sides)

x = 2

7a. (3x-10)+(2x+15) = 90

90+10= 100

100-15= 85

85/5 = 17 (as 3x + 2x = 5x)

x = 17

7b. 5y-45 = 3y +15

5y = 3y +60 (add 45 to both sides)

2y = 60 (subtract 3y from both sides)

y = 30

sub 30 for y into 3y+15: 90+15=105

straight line = 180

180-105=75

25x=75

75/25=3

x=3

8. complementary means 90º total

90-55=35 or 55+35 = 90

9. linear pair mean 180º total

180-30=150 or 30+150 =180

10. LF = 17

as LE is equal to FT, they both are equal to 5

as LE is part of LF, 17 - 5 =12

11. lm and rs are parallel

given that 9 and 12 are opposite angles, and that 12 and 8 are corresponding angles, meaning that 12 and 8 are of the same angle.

and since 12 = 8, and 12 = 9,

therefore you can prove that 8 = 9

Answer:

6. (a)

\(7x - 10 = x + 2\)

Collect like terms

\(7x - x = 2 + 10\)

\(6x = 12\)

Divide both sides with 6

\(x = \frac{12}{6} \)

\(x = 2\)

6. (b)

\( - 3x + 5(3x - 1) = 2x + 15\)

Expand the bracket\( - 3x + 15x - 5 = 2x + 15\)

Collect like terms

\( - 3x + 15x - 2x = 5 + 15\)

\( 10x = 20\)

Divide both sides with 10

\(x = \frac{20}{10} \)

\(x = 2\)

7.(a)

\((2x + 15) + (3x - 10) = 90\)

\(2x + 15 + 3x - 10 = 90\)

\(5x = 90 - 15 + 10\)

\(5x = 85\)

\(x = 17\)

7.(b)

(5y-45)° and (3y+15)° are opposite angles. Thus, they share the same value of angle.

\(5y - 45 = 3y + 15\)

\(5y - 3y = 45 + 15\)

\(2y = 60\)

\(y = 30\)

All four angles shown in the diagram will add up to a total value of 360°. So, minus 360° with the two angles we had found to find the remaining angles or x.

\(360 - (5(30) - 45) - (3(30) + 15) = 2(25x)\)

\(360 - 105 - 105 = 50x\)

\(150 = 50x\)

\(50x = 150\)

\(x = \frac{150}{50} \)

\(x = 3\)

Since we are looking for the value of f(7), we need to find the corresponding output value when the input is 7. From the given function, we see that input 7 corresponds to output 2. f(7) = 2.

To determine the value of f(7), we need to look at the given function f and substitute 7 for the independent variable.

The function f is defined by the ordered pairs (7,2), (9,7), (4,9), and (3,4). The first value in each ordered pair represents the input, while the second value represents the output.

In summary, when we substitute 7 for the independent variable in the given function f = (7,2), (9,7), (4,9), (3,4), we find that f(7) = 2.

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Answer: 30 buttons

35 - 5 = 30

Answer:

30

35-5=30

Step-by-step explanation:

Answer:

D square root of 10 over 5

Step-by-step explanation:

sec means \(\frac{1}{x}\), so if sec of theta is \(\frac{\sqrt{10} }{2}\) in this situation then you can set those equal to one another to get x.

x=\(\frac{\sqrt{10} }{5}\)

cos is just another way to say x so what you got for x is your answer

The secant of theta is equal to the square root of 10 over 2 comma what is, then the value of cos is square root of 10 over 5 .The correct option is d.

For a right angle triangle, secant of the angle formed by the sides opposite to hypotenuse is equal to the ratio of Hypotenuse and the base of triangle,

Given the value of sec of θ

secθ = sqrt (10)/2

Then cos θ = 1/secθ

Plug the value, we get

secθ=  sqrt  (10) / 5

Thus, the correct option is d.

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Answer:

solving for x - 23/4 + 1/2y -3/4z

Step-by-step explanation:

Answer:

X = 8.5, Y = 6.5

Step-by-step explanation:

X = # of children's books

Y = # of puzzle books

Amelie wants to buy 2 more children's books than puzzle books.

15 - 2 = 13 (Save the 2)

13 / 2 = 6.5

6.5 + 2 = 8.5

So; 8.5 + 6.5 = 15

X = 8.5

Y = 6.5

The given series ∑[infinity]n=0x−5n9n converges for all x in the interval (-4,14) in the real number system.

To determine the convergence of the given series, we can use the ratio test. Applying the ratio test, we get:

|((x-5(n+1))/9(n+1)) / ((x-5n)/9n)| = |(x-5)/(9(n+1))|.

For the series to converge, we need the limit of the ratio as n approaches infinity to be less than 1 in absolute value. Hence, we have:

lim(n→∞) |(x-5)/(9(n+1))| < 1

|x-5|/9 < 1

|x-5| < 9

This implies -4 < x-5 < 14, or -4 < x < 14. Therefore, the given series converges for all x in the interval (-4,14) in the real number system.

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The required time that took the car to stop is 0.25 seconds.

Given that,
A 1200 kg car is traveling at a constant velocity of 8 m/s. It hits a wall and comes to a complete stop. If the wall exerts a force of 38000 N on the car, how long would it take the car to come to rest is to be determined.

Distance is defined as the object traveling at a particular speed in time from one point to another.

Here,
u = 8
V = 0
force = 38000
m = 1200
Calculating accelartion, by newton's second law,
force = ma
38000 = 1200a
a = 31.6,
Applying the first equation of motion,
v = u + at
0 = 8 + -31.6t      [took negative because the acceleration is in opposite direction]
8/31.6 = t
t = 0.25

Thus, the required time that took the car to stop is 0.25 seconds.

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It will take 10 years to double the amount.

Given that, the amount $1400 to double if it is invested at 7% interest compounded monthly, we need to calculate the time,

\(A = P(1+r/n)^{nt}\)

\(2800 = 1400(1+0.0058)^{12t}\)

\(2= (1.0058)^{12t\)

㏒ 2 = 12t ㏒ (1.0058)

0.03 = 12t (0.0025)

12t = 120

t = 10

Hence, it will take 10 years to double the amount.

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To find the intersection point between the curve y = tan(x) and the line y = 7x, we can use Newton's method. Newton's method is an iterative numerical method used to approximate the root of a function.

We need to find the x-value where the curves intersect, so we can set up the equation tan(x) - 7x = 0. We want to find a solution between x = 0 and some unknown value denoted as X.

Using Newton's method, we start with an initial guess x_0 for the solution and iterate using the formula:

x_(n+1) = x_n - f(x_n) / f'(x_n),

where f(x) = tan(x) - 7x and f'(x) is the derivative of f(x).

We continue this iteration until we reach a desired level of accuracy or convergence. The resulting value of x will be the approximate intersection point between the two curves.

Please note that without specific values or range for X or an initial guess x_0, it is not possible to provide a specific numerical answer. However, you can apply Newton's method using an initial guess and the given function to find the approximate intersection point.

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The power of substitutes is one of the five forces in Porter's Five Forces framework and it is a measure of how easy it is for customers to switch to alternative products or services. The higher the power of substitutes, the more competitive the industry and the lower the profitability.

The power of substitutes is based on the premise that when there are readily available alternatives to a product or service, customers can easily switch to those alternatives if they offer better value or meet their needs more effectively. This poses a threat to the industry as it reduces customer loyalty and puts pressure on pricing and differentiation strategies.

The availability and quality of substitutes influence the degree to which customers are likely to switch. If substitutes are abundant and offer comparable or superior features, the power of substitutes is strong, increasing the competitive intensity within the industry. On the other hand, if substitutes are limited or inferior, the power of substitutes is weak, providing more stability and protection to the industry.

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Answer:

−5

Step-by-step explanation:

6−4−5−6−2+6

Subtract 4 from 6 to get 2.

2−5−6−2+6

Subtract 5 from 2 to get −3.

−3−6−2+6

Subtract 6 from −3 to get −9.

−9−2+6

Subtract 2 from −9 to get −11.

−11+6

Add −11 and 6 to get −5.

−5

Answer:

-5

Step-by-step explanation:

6 - 4 = 2

2 - 5 = -3

-3 - 6 = -9

-9 - 2 = -11

-11 + 6 = -5

Answer:

312cm2

Step-by-step explanation:

Area of the triangles:

A =1/2bh

A=1/2(10*12) = 1/2*120=60.

There are two triangles. 60*2=120. 120 is the area of both triangles.

Area of the rectangle:

16*12=192.

120+192=312cm2

Answer:

.8

Step-by-step explanation:

Given the volume of the sphere as 26667 cm³, we calculated the radius to be approximately 17.7 cm using the formula for the volume of a sphere. By multiplying the radius by 2, we found that the diameter of the sphere is approximately 35.4 cm.

To calculate the diameter of a sphere when given its volume, we can use the formula for the volume of a sphere:

V = (4/3) * π * r³

Where V is the volume and r is the radius of the sphere. Since we are given the volume, we can rearrange the formula to solve for the radius:

r = (\(\sqrt[3]{(3V / (4\pi )}\)))

Substituting the given volume V = 26667 cm³ into the formula, we have:

r = (\(\sqrt[3]{(3 * 26667 / (4\pi )))}\)

Calculating this expression, we find:

r ≈ (\(\sqrt[3]{80001 / \pi ))}\) ≈ 17.7 cm

Now that we have the radius, we can calculate the diameter by multiplying the radius by 2:

d = 2 * r ≈ 2 * 17.7 ≈ 35.4 cm

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Equivalent expression for the expression ( log 2 - log 6 ) is,

⇒ - log 3

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ log 2 - log 6

Now, Simplify the expression by using logarithmic rule as;

⇒ log 2 - log 6

⇒ log 2 / 6

⇒ log 1/3

⇒ log 3⁻¹

⇒ - log 3

Therefore, We get;

Equivalent expression for the expression ( log 2 - log 6 ) is,

⇒ - log 3

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Answer:

hyp=12.76

Step-by-step explanation:

sin36=opp/hyp

hyp=opp/sin36

hyp=7.5/sin36

hyp=12.76 rounded to the nearest hundredth

Now, from equation (5)

Answer:

As x to negative infinity, y to negative infinity

Step-by-step explanation:

Answer: 21

Step-by-step explanation:

Let the consecutive odd numbers be represented by x and x+2. Therefore,

x × (x+2) = 483

x² + 2x = 483

x² + 2x - 483 = 0

x² + 23x - 21x - 483 = 0

x(x + 23) - 21(x + 23) = 0

(x - 21) = 0

x = 0 + 21

x = 21

The smaller integer is 21

The larger integer is 21 + 2 = 23.

We can drag the particles in mass/grams measurement to the corresponding descriptions as follows:

1.  1.6744 × 10⁻²⁴: Two electrons and 0ne proton

2. 5.021 × 10⁻²⁴: Two protons and one neutron

3. 5.0224 × 10⁻²⁴: One proton and two neutrons

4. 3.3484  × 10⁻²⁴: One electron, one proton, and one neutron

To match the measurements to the descriptions first note that one neutron is 1.6749 × 10⁻²⁴. One proton is equal to  1.6726 × 10⁻²⁴ and one electron is equal to  9.108 × 10⁻²⁸.

To obtain the right combinations, we have to add up the particles to arrive at the constituents. So, for the figure;

1.6744 × 10⁻²⁴, we would

Add 2 electrons and one proton

= 2(9.108 × 10⁻²⁸) + 1.6726 × 10⁻²⁴

= 1.6744 × 10⁻²⁴

The same applies to the other combinations.

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The within-group estimate of variance is the estimate of the variance of the population of individuals based on the variation among the scores in each of the actual groups studied.

The within-groups estimate of variance is the estimate of the variance of the population of individuals based on the variation among the:
Scores in each of the actual groups studied.
This estimate represents the variation within each group and helps in understanding the population's variance by looking at individual differences within the groups.

The estimated within-group variance is the sum of the within-group variances for each group in the model. Effectively, this is the sum of the variance of each value (j) from its group (i) divided by the sample size minus one.

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The payment amount for the loan is approximately $832.54.

To calculate the payment amount for a loan, we can use the formula for the present value of an annuity. The formula is as follows:

\[ P = \frac{A \times r}{1 - (1 + r)^{-n}} \]

Where:

- P is the loan principal (initial amount borrowed)

- A is the payment amount

- r is the interest rate per period (expressed as a decimal)

- n is the total number of periods

In this case, the loan principal (P) is $3500, the interest rate (r) is 9% (or 0.09 as a decimal), and the number of periods (n) is 4 (since payments are made annually for 4 years). We need to solve for A, the payment amount.

Plugging in the given values into the formula, we get:

\[ 3500 = \frac{A \times 0.09}{1 - (1 + 0.09)^{-4}} \]

To solve for A, we can rearrange the equation:

\[ A = \frac{3500 \times 0.09}{1 - (1 + 0.09)^{-4}} \]

Let's calculate the value of A using this equation:

\[ A = \frac{3500 \times 0.09}{1 - (1.09)^{-4}} \]

\[ A \approx \frac{315}{0.3781} \]

\[ A \approx \$832.54 \]

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Answer:

d. 360°

Hope this help!

Answer:

\(\sf r^6\)

Step-by-step explanation:

         \(\boxed{\bf \dfrac{a^m}{a^n}=a^{m-n}}\)

In exponent division, if bases are same, subtract the powers.

 \(\sf \dfrac{r^9}{r^3}=r^{9-3}=r^{6}\)

So, the percentage of data values represented on a box plot that falls between the lower quartile and the upper quartile is 50%.

In a box plot, the lower quartile (Q1) represents the 25th percentile, and the upper quartile (Q3) represents the 75th percentile. The interquartile range (IQR) is the range between the lower quartile and the upper quartile. To determine the percentage of data values that fall between the lower quartile and the upper quartile, we need to consider the IQR.

The IQR represents the middle 50% of the data. Therefore, the percentage of data values between the lower quartile and the upper quartile is 50%. In other words, half of the data values are within the IQR range, while the remaining 50% are outside this range, including the lower 25% below the lower quartile and the upper 25% above the upper quartile.

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Answer:

x= 62

y= 54

Step-by-step explanation:

Step one:

given data

let the numbers be x and y and the larger be x the smaller be y

The difference between two numbers is 8

x-y= 8-----------1

If the larger is  subtracted from three times the smaller, the difference is 100

3y-x=100------------2

from eqn 1, x= 8+y

put this in eqn 2

3y-(8+y)=100

3y-8-y=100

collect liker terms

3y-y-8=100

2y=108

y= 54

put y= 54 in eqn 1

x-y=8

x-54= 8

x= 8+54

x=62