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The values of k that satisfy the equation 4y'' = -9y for the function y = cos kt are k = ±3/2.

This is a question in differential equations, where we are given a cosine function y = cos kt and asked to find values of k that satisfy the given equation 4y'' = −9y.

Recall, the second derivative of cos kt is -k²cos kt which is equivalent to -k²y.

Substituting that into the differential equation, we get 4(-k²y) = -9y. If we simplify this, we get k² = 9/4. Therefore, the values of k which satisfy the given equation are k = ±3/2.

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The values of k that satisfy the differential equation 4y'' = −9y are k = ±3/2.

We can find the values of k that satisfy the differential equation 4y'' = −9y by solving for k.

First, we can rewrite the differential equation in its general form:

ay'' + by' + cy = 0

where a = 4, b = 0, and c = -9.

Next, we can find the characteristic equation:

ar^2 + br + c = 0

Plugging in the values of a, b, and c, we get:

4r^2 + 0r - 9 = 0

Solving for r, we get:

r = ±3/2

Therefore, the general solution to the differential equation is:

y = Acos(3t/2) + Bsin(3t/2)

where A and B are arbitrary constants.

Finally, we can substitute y = cos kt into the original differential equation and solve for k:

4k^2cos kt + 9cos kt = 0

Dividing both sides by cos kt, we get:

4k^2 + 9 = 0

Solving for k, we get:

k = ±3/2

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The length of the other side is : 16cm

A right-angled triangle is a triangle with one of its angle 90 degrees and has two adjacent sides and a hypotenuse which is the longest side.

Analysis:

Since it is a right-angled triangle, we use Pythagoras theorem

[tex]hypotenuse^{2}[/tex] = [tex]opposite^{2}[/tex] + [tex]adjacent ^{2}[/tex]

hypotenuse = 20cm, opposite = 12cm, adjacent = xcm

[tex]20^{2}[/tex] = [tex]x^{2}[/tex] + [tex]12^{2}[/tex]

[tex]x^{2}[/tex] = 400 - 144

[tex]x^{2}[/tex] = 256

x = [tex]\sqrt{256}[/tex] = 16cm

In conclusion, the length of the other side is 16cm

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Answer:

Yes, y = 2x

Step-by-step explanation:

I took it on edge and got it right!!!

There will be definitely a direct relation and the relation will be y = 2x.

The ratio is the division of the two numbers.

For example, a/b, where a will be the numerator and b, will be the denominator.

Proportion is the relation of a variable with another. It could be direct or inverse.

Proportion is very useful to set relations between variables.

Given that the prepaid cell phone charges a preset number of minutes to use text messaging. The graph represents y, the number of minutes used for x, and the number of text messages sent and received

so it is clear that cell phone charge upon the number of minutes to use text messaging hence y = 2x will be the correct answer.

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The expression 3x + 2 - 3x + 7 +5x simplifies to 5x

when the terms containing variable 'x' are combined. Therefore, the coefficient in the simplified expression is 5.

The given expression is 3x + 2 – 3x + 7 +5x.

To simplify this expression, we need to combine like terms, which in this case are terms with x.

The like terms are 3x, -3x, and 5x. When added together, they simplify to: 3x - 3x + 5x = 5x.

The number part of the term, excluding x, is referred to as the coefficient. So the coefficient of x in this simplified expression is 5.

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x = children

y = adults

1.50x +4.00y =5050

y = 1000

1.50x + 4.00(1000) = 5050

1.50x +4000.00 = 5050

1.50x= 1050

x = 1050 / 1.50 = 700 children atttended

According to the given rectangle, approximately 112.27 feet of fence are required to enclose the rose garden.

The perimeter of a rectangle is the sum of all its sides. In this case, the rectangle has two sides of length 24 feet and two sides of length 18 feet.

Therefore, the perimeter of the rectangle can be calculated as follows:

Perimeter of rectangle = 2 * (Length + Width)

= 2 * (24 ft + 18 ft) = 2 * 42 ft = 84 ft.

The circumference of a circle can be calculated using the formula:

Circumference = π * Diameter.

Diameter of the Semicircle: The diameter of the semicircle is equal to the width of the rectangle, which is 18 feet.

Circumference of the Semicircle: Using the formula mentioned above:

Circumference = π * 18 ft

≈ 56.5487 ft (rounded to four decimal places).

Perimeter of the Semicircle: As it is a semicircle, we only need half of the circumference:

Perimeter of semicircle = Circumference / 2

≈ 56.5487 ft / 2 ≈ 28.2744 ft (rounded to four decimal places).

Total Fence Length Required: Now, we add the perimeter of the rectangle and the semicircle to find the total length of fence required to enclose the rose garden:

Total fence length = Perimeter of rectangle + Perimeter of semicircle

= 84 ft + 28.2744 ft ≈ 112.2744 ft (rounded to four decimal places).

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The expression √(4x^2/3y) simplifies to (2x)/√(3y), by taking the square root of the numerator and the denominator separately.

To simplify the expression √(4x^2/3y), we will start by simplifying the square root of a fraction, which can be done by taking the square root of the numerator and the denominator separately. We have √(4x^2) = 2x because the square root of 4 is 2 and the square root of x^2 is x. For the denominator, we have √(3y), which cannot be simplified further without additional information about y. Therefore, the simplified form of the expression is (2x)/√(3y).

Remember, if we encounter an expression with a higher root like a fourth root, we would raise the number to the 0.25 power, which is equivalent to taking the square root twice. While simplifying expressions, we may also need to solve for variables in an equation or work with fractional powers, which represent roots, such as x^2 = √x.

To solve the problem:

Let f = distance one way distance on level ground

Let h = distance rode on the hill

Write a time equation; where Time = distance/speed:

level time + uphill time + downhill time + level time = 1hr

f/9 + h/6 + h/18 + f/9 = 1 hr

Multiply equation by 18 to get rid of the denominators

2f + 3h + h + 2f = 18

4f + 4h = 18

Simplify by dividing this by 4

f + h = 4.5 miles is took her to get to Jon’s house.

The graph of the complex number 4 + 2i corresponds to the point (4, 2) on the complex plane, with 4 as the real part and 2 as the imaginary part.

To identify the graph of the complex number 4 + 2i, you can think of it as a point on the complex plane. In the complex plane, the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. So, for the complex number 4 + 2i, the real part is 4, and the imaginary part is 2.

Now, let's locate this point on the complex plane:

The real part (4) corresponds to the horizontal axis.

The imaginary part (2) corresponds to the vertical axis.

So, the point (4, 2) on the complex plane represents the complex number 4 + 2i. Therefore, the correct answer is (4, 2).

In summary, the graph of the complex number 4 + 2i corresponds to the point (4, 2) on the complex plane, where 4 is the real part, and 2 is the imaginary part.

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The total number of factors for a number 'n' with prime factorization [tex]n = p^r * q^s * ... * z^w[/tex] (where 'p', 'q', ..., 'z' are distinct primes and 'r', 's', ..., 'w' are their respective powers) is equal to (r + 1) * (s + 1) * ... * (w + 1).

When we have the prime factorization of a number, determining its total number of factors is a straightforward process. The number of factors of 'n' can be found by considering the powers of its prime factors.

Let's take an example to illustrate this. Consider the number 'n' with prime factorization:

[tex]n = p^r * q^s * ... * z^w,[/tex]

where 'p', 'q', ..., 'z' are distinct prime numbers, and 'r', 's', ..., 'w' are their respective powers.

To find the total number of factors for 'n', we need to consider all possible combinations of its prime factors. For any factor of 'n', we can choose to include 'p' raised to a power ranging from 0 to 'r', 'q' raised to a power ranging from 0 to 's', and so on, for all the distinct primes.

The total number of factors can be calculated using the formula:

(r + 1) * (s + 1) * ... * (w + 1)

This formula works because for each prime factor 'p', we have 'r + 1' options for its power, and similarly for other prime factors. By multiplying all these options, we get the total number of factors.

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LMF = 17x

so 17x + 3x =180

20x = 180

x = 180/20 = 9

17 *9 = 153

angle LMF = 153 degrees