Factorise 9x³y²(3x -24) ÷3xy (x-8) ​

1) General solution for second order differential equation, y" – 2y' + y = 0, is y = (c₁x + c₂)eˣ .

2) General solution for differential equation, 9y" + 6y' + y = 0, is y =(c₁x + c₂)e⁻³ˣ.

3) General solution for differential equation, 4y"- 4y'- 3y = 0, is y = c₁ e⁶ˣ+ c₂e⁻⁴ˣ.

4) General solution for differential equation, 4y" + 12y' +9y = 0, is y = (c₁x + c₂)e⁻⁶ˣ.

5) General solution for differential equation, y" – 2y' + 10y = 0, is y = eˣ (c₁cos(6x) + c₂sin(6x)).

6) General solution for differential equation, y" – 6y' +9y = 0 is y = (c₁x + c₂)e³ˣ.

7) General solution for differential equation, 4y" + 17y' + 4y = 0, is y = c₁e⁻ˣ + c₂e⁻¹⁶ˣ.

8) General solution for differential equation, 16y" + 24y' +9y = 0, is y = (c₁x + c₂)e⁻¹²ˣ.

9) General solution for differential equation, 25y" – 20y' + 4y = 0, is y = (c₁x + c₂)e¹⁰ˣ.

10) General solution for differential equation, 2y" + 2y' + y = 0, is y = e⁻ˣ (c₁cos(2x) + c₂sin(2x)).

General solution is also called complete solution and complete solution = complemantory function + particular Solution

Here right hand side is zero so particular solution is equals to zero. Therefore, evaluating the complementary function will be sufficient to determine the general solution to the differential equation.

1) y"-2y' + y = 0, --(1)

put D = d/dx, so (D² - 2D + 1)y =0

Auxiliary equation for (1) can be written as, m² - 2m + 1 = 0 , a quadratic equation solving it by using quadratic formula,

\(m =\frac{-(- 2) ± \sqrt { 4 - 4}}{2}\)

=> m = 1 , 1

The roots of equation are real and equal. So, general solution is y = (c₁x + c₂)eˣ .

2) 9y" + 6y' + y = 0 or (9D² + 6D + 1)y =0 Auxiliary equation can be written as, 9m² + 6m + 1 = 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{ - (6) ± \sqrt {36 - 4×4}}{2}\)

=> m = - 6/2

=> m = -3 , -3

The roots of equation are real and equal. So, general solution is y = (c₁x + c₂)e⁻³ˣ.

3) 4y"- 4y'- 3y = 0

put D = d/dx, so (4D² - 4D - 3)y = 0

Auxiliary equation can be written as, 4m² - 4m - 3 = 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{-(-4) ± \sqrt {16 - 4×4×(-3)}}{2}\)

=> m = (4 ± 8)/2

=> m = -4 , 6

The roots of equation are real and equal. So, general solution is y = c₁ e⁶ˣ + c₂e⁻⁴ˣ.

4) 4y" + 12y' +9y = 0 or (4D² + 12D + 9)y= 0

Auxiliary equation can be written as, 4m² + 12m + 9= 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{-(12) ± \sqrt{144 - 4×4×9}}{2}\)

=> m = -12/2

=> m = -6 , -6

The roots of equation are real and equal. So, general solution is y = (c₁x + c₂)e⁻⁶ˣ.

5) y" – 2y' + 10y = 0 or (D² - 2D + 10)y = 0 Auxiliary equation can be written as, m² - 2m + 10 = 0 , a quadratic equation

solving it by using quadratic formula,

\(m =\frac{ - (-2) ± \sqrt {4 - 4×1×10}}{2}\)

=> m = (2 ± 6i)/2 ( since, √-1 = i)

=> m = 1 + 6i , 1-6i

The roots of equation are imaginary and unequal. So, general solution is y =eˣ (c₁cos(6x) + c₂sin(6x)).

6) y" – 6y' +9y = 0 or (D²- 6D + 9)y =0

Auxiliary equation can be written as, m² - 6m + 9 = 0 , a quadratic equation

solving it by using quadratic formula,

\(m =\frac{ - (-6) ± \sqrt {36 - 4×1×9}}{2}\)

=> m = 6/2 = 3,3

The roots of equation are real and equal. So, general solution is y = (c₁x + c₂)e³ˣ.

7) 4y" + 17y' + 4y = 0 or (4D²+ 17D + 4)y=0

Auxiliary equation can be written as, 4m² + 17m + 4 = 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{- (-17) ± \sqrt {16 - 4×4×17}}{2}\)

=> m = ( -17 ± 15)/2

=> m = (-17 + 15)/2, (- 17 - 15)/2= -1, -16

The roots of equation are real and unequal. So, general solution is y = c₁e⁻ˣ + c₂e⁻¹⁶ˣ.

8) 16y"+24y'+9y =0 or (16D²+ 24D + 9)y= 0

Auxiliary equation can be written as, 16m² + 24m + 9 = 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{ - (24) ± \sqrt {576 - 4×9×16}}{2}\)

=> m = (-24 ± 0)/2

=> m = -12,-12

The roots of equation are real and equal. So, general solution is y = (c₁x + c₂)e⁻¹²ˣ.

9) 25y"- 20y' +4y =0 or (25D²-20D + 4)y = 0

Auxiliary equation can be written as, 25m²- 20m + 4 = 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{ - (-20) ± \sqrt {400 - 4×4×25}}{2}\)

=> m = 20/2

=> m = 10 , 10

The roots of equation are real and equal. So, general solution is y = (c₁x + c₂)e¹⁰ˣ.

10) 2y" + 2y' + y = 0 or (2D²+ 2D + 1)y =0

Auxiliary equation can be written as, 2m² + 2m + 1 = 0 , a quadratic equation solving it by using quadratic formula, \(m =\frac{ - (2) ± \sqrt {4 - 4×1×2}}{2}\)

=> m = (- 2 ± 4i)/2 ( since, √-1 = i)

=> m = -1 + 2i , -1 - 2i

The roots of equation are imaginary and unequal. So, general solution is y = e⁻ˣ (c₁cos(2x) + c₂sin(2x)). Hence, required solution of differential equation is y = e⁻ˣ (c₁cos(2x) + c₂sin(2x)).

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Answer:Week 1:  $35.40

              Week 2: $40.90

              Week 3: $46.40

               Week 4: $51.90

Step-by-step explanation:

29.90 + 5.50(x) = y

replace x with 1, 2, 3, or 4 for each week.

Answer:

8

Step-by-step explanation:

2+6

The slopes of GH, HI, IJ, and JG include the following:

In Mathematics and Geometry, the slope of any straight line can be determined by using the following mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = rise/run

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

By substituting the given data points into the formula for the slope of a line, we have the following;

Slope GH = (-3 + 9)/(-4 + 7)

Slope GH = 6/3

Slope GH = 2.

Slope HI = (5 + 3)/(-6 + 4)

Slope HI = -8/2

Slope HI = -4.

Slope IJ = (-1 - 5)/(-9 + 6)

Slope IJ = -6/-3

Slope IJ = 2.

Slope JG = (-9 + 1)/(-7 + 9)

Slope JG = -8/2

Slope JG = -4.

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The equation of the given the roots x=4 x=2-i x=2+i is y = x³ - 8x² + 21x - 12

What is a root?

The roots of a quadratic equation are the values of the variables that fulfill the equation. In other words, if f(α) = 0, then x = α is a root of the quadratic equation f(x). The x-coordinates of the sites where the curve y = f(x) intersects the x-axis are the real roots of equation f(x) = 0.

Here, we have

x=4 x=2-i x=2+i

Roots are the points where the graph intercepts with the x-axis y = 0 at the roots.

The root at x = 4 was found by solving for x when x−(4) = y and y = 0

The factor is x−4

The root at x = 2-i was found by solving for x when x−(2-i) = y and y = 0.

The factor is x−2+i

The root at x = 2+i was found by solving for x when x−(2+i) = y and y = 0

The factor is x-2-i

Combine all the factors into a single equation.

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

y = x³ - 8x² + 21x - 12

Hence, the equation of the given the roots x=4 x=2-i x=2+i is y = x³ - 8x² + 21x - 12

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.

The perimeter of the triangle is 170 inches.

To solve for the perimeter, we first need to find the length of the perpendicular height. We can do this using the sine function:

sin(35°) = 46/x

x = 46/sin(35°) = 65 inches

Now that we know the length of the perpendicular height, we can find the length of the base of the triangle using the cosine function:

cos(35°) = 65/x

x = 65/cos(35°) = 75 inches

The perimeter of the triangle is the sum of the lengths of the three sides, so the perimeter is:

P = 65 + 75 + 30

= 170 inches

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Answer:

y=1/2x-3

Step-by-step explanation:

Answer:

que

Step-by-step explanation:

no Tengo carnitas yo quero sopes

Answer:

16.  25/45 ; 18/45

17  4/7; 4/7

Step-by-step explanation:

5/9 and 2/5

The common denominator is 45

5/9 * 5/5 = 25/45

2/5 *9/9 = 18/45

4/7  and 12/21

Simplify 12/21 by dividing by 3 in the numerator and denominator

12/21 = 4/7

4/7; 4/7

The common denominator is 7

16:25/45;18/45

17: 4/7;4/7

Answer:

16:5/9;2/5

Common denominator is 9*5=45

now

make a equal denominator

5/9=5/9*5/5=25/45

2/5=2/5*9/9=18/45

and

17: 4/7;12/21

common denominator =7

4/7

12/21=4/7

If the first bus is boarded at a rate of 15 people per minute, the domain of the function is; (15 ≤ x ≤ 45)

We are told that;

Total number of Juniors at Smithers High School that are taking a class trip = 130

Number of buses taking each student to their destination = 3 buses

Capacity of each bus = 45 people

Now, if the function  f(x) describes the total number of people on the first

bus, where x is the number of minutes spent loading the bus.

Then since the maximum number of people on the first bus is 45, then it means that the domain of the function is;

(15 ≤ x ≤ 45)

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the C-CAPM with power utility and lognormal consumption growth predicts that the representative agent in country A will consume more in the current period and that the price of an identical financial asset will be lower compared to country B.

The C-CAPM is a financial model that relates the consumption patterns and asset prices in an economy. In this scenario, the difference in subjective discount factors implies that the representative agent in country A values future consumption relatively less compared to country B. As a result, the representative agent in country A tends to consume more in the current period, prioritizing immediate consumption over saving for the future.

Furthermore, the C-CAPM suggests that the price of an identical financial asset, such as a stock or bond, will be lower in country A. This is because the higher subjective discount factor in country A implies a higher expected return requirement for investors. As a result, investors in country A will demand a higher risk premium, leading to a lower price for the financial asset.

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The correct answer for (C) the standard distribution for Jack is \(2.5\)

Diana is \(0.79\)

For part (C).

Given:

For jack:

Sample size \(n = 4\)

Standard deviation \(\sigma = 5\)

For Diana:

Sample size \(n = 40\)

The standard deviation for Jack's distribution:

\(\sigma_x = \dfrac{\sigma}{\sqrt{x} } \\= \dfrac{5}{\sqrt{4} }\)

\(= 2.5\)

The standard deviation for Diana's distribution:

\(\sigma_x = \dfrac{\sigma}{\sqrt{n} } \\= \dfrac{4}{\sqrt{40} } \\\\= 0.79\)

The standard deviation for (C) Jack is  \(2.5\) and Diana's is \(0.79\).

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Answer:

\(x =23^o\)

Step-by-step explanation:

Given

See attachment

Required

Find x

To solve for x, we make use of:

\(5x + 65= 180\) --- angle on a straight line

\(5x = 180-65\)

\(5x = 115\)

Solve for x

\(x = 115/5\)

\(x =23^o\)

Answer:

The answer is 6.

Step-by-step explanation:

Simply by calculation.

Power series solutions have a minimum radius of convergence of R of 10.0498 around the normal point x = 0 and 10 units around the normal point x=1.

A differential equation in mathematics is an equation that connects the derivatives of one or more unknown functions.

Applications often involve functions that reflect physical quantities, derivatives that depict the rates at which those values change, and a differential equation that establishes a connection between the three.

The given equation: \(\left(x^2-2 x+26\right) y^{\prime \prime}+x y^{\prime}-4 y=0\)

It is necessary to determine the power series solutions' minimal radius of convergence R around the typical points x = 0 and x = 1.

The separation between the ordinary point and the differential equation's singularity is now the minimal radius of convergence.

The polynomial's root, which is connected to the second derivative, is the singularity point.

The singularity points will be determined as follows:

\(\begin{aligned}& x^2-2 x+26=0 \\& x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\& x=\frac{2 \pm \sqrt{(-2)^2-4 \times 1 \times 26}}{2} \\& x=1 \pm \sqrt{-100} \\& x=1 \pm 10 i\end{aligned}\)

In this case, x1 = 1+10i and x2 = 1-10i are the singularity sites.

The ordinary points at this time are z1 = 0+01 and z2 = 1+0i.

One can compute the minimum radius of convergence using the formula:

\(\begin{aligned}& r_1=\left|z_1-x_1\right| \\& =|0+0 i-1-10 i| \\& =\sqrt{101} \\& =10.0498 \\& r_2=\left|z_2-x_1\right| \\& =\sqrt{100} \\& =10\end{aligned}\)

Therefore, power series solutions have a minimum radius of convergence of R of 10.0498 around the normal point x = 0 and 10 units around the normal point x = 1.

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Answer:

The roots are {-4, 2}.

Step-by-step explanation:

It seems strange to have x terms on both sides of your equation.  If this is indeed the correct equation, then

x^2-x-8=-3x becomes x^2 +3x - x - 8= 0, and this latter result simplifies to

x^2 + 2x - 8 = 0.  

One set of factors of -8 is {-2, 4} and another is {1, -8).  Hence we could try evaluating this quadratic at x = 4:

4^2 + 2(4) - 8 = 16 (since this is not zero, 4 is not a root and (x - 4) is not a factor.  

Try x = -4 instead:

(-4)^2 + 2(-4) - 8 = 0 simplifies to 16 - 8 - 8 = 0, which is true.

Then x + 4 is a factor and x - 2 is another.  Multiplying these two together as a check results in the given x^2- x - 8 = -3x.

The roots are {-4, 2}.

The capacity of fish tank in quarts is 246.

Therefore the answer is 246.

A pint is a unit of volume in the US customary system of measurement and is equal to 16 fluid ounces. A quart is also a unit of volume in the US customary system of measurement and is equal to 32 fluid ounces or 2 pints.

There are 2 pints in 1 quart, so to convert from pints to quarts, we need to divide by 2.

The capacity of the fish tank in pints is 492.

Therefore, the capacity of the fish tank in quarts is

= 492 pints / 2 pints per quart

= 246 quarts

So, the capacity of the tank in quarts is 246 quarts.

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The surface area of a hexagonal prism willl be 251.13844 cm².

The total land area of all the faces of a three-dimensional object is its surface area. When we wish to wrap something in real life, we employ the notion of surface areas of distinct things.

A=6ah+3√3a²

A=6×4×7+3√3×4²

A≈251.13844

Hence the surface area of a hexagonal prism willl be 251.13844 cm².

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1/2 pounds = 56 ounces (since 1 pound = 16 ounces)

Now we can add the weight of both packages in ounces:

56 ounces + 20 ounces = 76 ounces

Therefore, the total weight of both packages is 76 ounces.

The total weight of both packages is, 76 ounces.

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

Given that;

Jazmin is mailing two packages. One package weighs 3 1/2 pounds, and the other package weighs 20 ounces.

Since, We know that;

⇒ 1 pound = 16 ounces

Hence, We get;

3 1/2 pounds = 7/2 x 16 pounds

                    = 56 ounces

Now, we can add the weight of both packages in ounces:

56 ounces + 20 ounces = 76 ounces

Therefore, the total weight of both packages is 76 ounces.

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Answer: ( 1.26 x 10 ^ 5 ) / ( 2.8 x 10 ^ 4 )

= 4.5 times bigger

Answer:

4.5

Step-by-step explanation:

always remember the ^. This is scientific notation.

( 1.26 x 10 ^ 5 ) / ( 2.8 x 10 ^ 4 )

Answer:

30 days

Step-by-step explanation:

2 pints : 6 days = 10 pints : x days

2/6 = 10/x

1/3 = 10/x

1 × x = 3 × 10

x = 30

Answer: 30 days

Answer:

(A) 560 students

(B) P = 0.41

Step-by-step explanation:

(Please check the attached picture for more details)

As given, a survey of 1000 students found that:

41% of the surveyed students were freshmen

=> 1000 x 41/100 = 410 surveyed students were freshmen

31% were juniors

=> 1000 x 31/100 = 310 surveyed students were junior

67% of surveyed students lived on the college campus

=> 1000 x 67/100 = 670 surveyed students lived on the campus

(A) 260 of the surveyed students were freshmen living on the campus

=> A = 410 - 260 = 150 surveyed students were freshmen NOT living on the campus

=> B = 670 - 260 = 410 surveyed students were not freshmen living on the campus

=> The number of surveyed students who were either freshmen or living on the campus:

N = A + B = 150 + 410 = 560

(B)  From a previous survey you know that of interviewed juniors, only 23% live on campus (we don't really need this)

=>The probability a surveyed student is a junior or someone that lives on campus: P = (670 - 260)/1000 = 0.41

I hope this helps!

The expression that is equivalent to 81/16 is 22163164. To understand why, we need to simplify 81/16 first. To simplify, we can factor 81 and 16 into their prime factors:

81 = 3 x 3 x 3 x 3
16 = 2 x 2 x 2 x 2

Then, we can cancel out any common factors between the numerator and denominator:

81/16 = (3 x 3 x 3 x 3)/(2 x 2 x 2 x 2)
       = (3/2)^4

Now we have 3/2 raised to the fourth power, which can be rewritten as (3/2) x (3/2) x (3/2) x (3/2) = 81/16.

Therefore, we can see that 81/16 is equivalent to (3/2)^4. To write this in a simplified form without exponents, we can rewrite it as:

(3/2)^4 = (3/2) x (3/2) x (3/2) x (3/2)
        = 9/4 x 9/4
        = 81/16

This means that the expression equivalent to 81/16 is 22163164, which can be obtained by adding the digits of 81 and 16 together in the order they appear: 8+1+1+6+2+0+1+6 = 22, and then repeating this process until we end up with a single digit: 2+2 = 4. Then we add a, b, and c to the number 4, based on their corresponding coefficients in the original expression, to get 4a4b6c8.

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a) The 90% confidence interval for the population proportion is approximately (0.7777, 0.8423).

b) The 95% confidence interval for the population proportion is approximately (0.7737, 0.8463).

To calculate the confidence intervals for the population proportion, we can use the formula:

Confidence Interval = sample proportion ± margin of error

The margin of error can be calculated using the formula:

Margin of Error = critical value * standard error

where the critical value is determined based on the desired confidence level and the standard error is calculated as:

Standard Error = \(\sqrt{((p * (1 - p)) / n)}\)

Given that the sample proportion (p) is 0.81 and the sample size (n) is 500, we can calculate the confidence intervals.

a. 90% Confidence Interval:

To find the critical value for a 90% confidence interval, we need to determine the z-score associated with the desired confidence level. The z-score can be found using a standard normal distribution table or calculator. For a 90% confidence level, the critical value is approximately 1.645.

Margin of Error = \(1.645 * \sqrt{(0.81 * (1 - 0.81)) / 500)}\)

≈ 0.0323

Confidence Interval = 0.81 ± 0.0323

≈ (0.7777, 0.8423)

Therefore, the 90% confidence interval for the population proportion is approximately (0.7777, 0.8423).

b. 95% Confidence Interval:

For a 95% confidence level, the critical value is approximately 1.96.

Margin of Error = \(1.96 * \sqrt{(0.81 * (1 - 0.81)) / 500)}\)

≈ 0.0363

Confidence Interval = 0.81 ± 0.0363

≈ (0.7737, 0.8463)

Thus, the 95% confidence interval for the population proportion is approximately (0.7737, 0.8463).

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The computer equipment was acquired at a cost of $73,700 with an estimated residual value of $34,600 and a useful life of 5 years. The depreciable cost of the equipment is $39,100. The straight-line rate is 20%, and therefore, the annual straight-line depreciation for the computer equipment is $7,820.

a. To determine the depreciable cost, we subtract the estimated residual value from the initial cost: $73,700 - $34,600 = $39,100.

b. The straight-line rate is calculated by dividing 100% by the estimated useful life of the equipment. In this case, the straight-line rate is 100% / 5 = 20% per year.

c. The annual straight-line depreciation is found by multiplying the depreciable cost by the straight-line rate. Thus, the annual depreciation is $39,100 * 20% = $7,820 per year.

By following these calculations, we can determine that the depreciable cost of the computer equipment is $39,100, the straight-line rate is 20%, and the annual straight-line depreciation amounts to $7,820.

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Explanation:

The triangles are similar, allowing us to form a proportion

The vertical sides pair up on one side, the horizontal sides pair up on the other side. Divide in the same order.

QV/RT = VS/TS

5/3 = x/12

12*5 = 3x ... cross multiply

60 = 3x

3x = 60

x = 60/3

x = 20

VS = 20

The minimum number of shirt they will sell to make the given profit is 192 shirts

Inequalities are equations not separated by an equal sign.

If a cheerleading team plans to sell t-shirts as a fundraiser and the teams goal is to make profit of at least $1248 with each t-shirt sold at $6.50, then the equation required is;

6.50t ≥ 1246

Divide both sides by 6.50

6.50t/6.50 ≥1246/6.5

t ≥ 192

Hence the minimum number of shirt they will sell to make the given profit is 192 shirts

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The normal distribution have an importance in statistics of whether or not out appears in nature, because it fits several natural phenomena.

For exa. measurement error, heights, IQ scores, and physical activity measures all follow the normal distribution.

In this question we need to describe why does the normal distribution have an importance in statistics of whether or not out appears in nature.

We know that if we plot the probability distribution and it forms a bell-shaped curve. If the mean, mode, and median of the sample are equal then the variable has normal distribution.

A normal distribution is dependent on two parameters of the data set: mean and the standard deviation of the sample.

This characteristic of the normal distribution makes it extremely simple.

Also, the normal distribution is important because of the Central limit theorem.

It makes math easy - like calculating moments, correlations between variables, and other calculations that are domain specific.

So, the normal distribution have an importance in statistics of whether or not out appears in nature.

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