what are the foci of the ellipse given by the equation 225x^2 144y^2=32400

It is proved here that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9. This is known as divisibility test for 9.

How to test divisibility for 9?

To show that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9, we can use the concept of congruence.

Let's start by representing an integer as the sum of its decimal digits. Consider an integer n expressed in decimal notation as:

[tex]n = d_k * 10^k + d_(k-1) * 10^(k-1) + ... + d_2 * 10^2 + d_1 * 10^1 + d_0 * 10^0[/tex],

where [tex]d_i[/tex] represents the i-th decimal digit of n, and k is the number of digits in n (k >= 0).

We want to prove that n is divisible by 9 if and only if the sum of its decimal digits, [tex]d_k + d_(k-1) + ... + d_2 + d_1 + d_0[/tex], is divisible by 9.

1. If n is divisible by 9:

Assume n is divisible by 9, which means there exists an integer q such that n = 9q. We can express n as:

[tex]n = (d_k * 10^k + d_(k-1) * 10^(k-1) + ... + d_2 * 10^2 + d_1 * 10^1 + d_0 * 10^0) = 9q[/tex]

Since 10 is congruent to 1 modulo 9 (10 ≡ 1 (mod 9)), we can rewrite the above equation as:

[tex](d_k + d_{(k-1)} + ... + d_2 + d_1 + d_0)[/tex]  ≡ [tex]9q (mod\ 9)[/tex].

The left-hand side of the congruence represents the sum of the decimal digits, and the right-hand side is a multiple of 9. Therefore, the sum of the decimal digits is divisible by 9.

2. If the sum of the decimal digits is divisible by 9:

Assume the sum of the decimal digits is divisible by 9, which means there exists an integer p such that [tex](d_k + d_{(k-1)} + ... + d_2 + d_1 + d_0) = 9p.[/tex]

We can express n as:

[tex]n = (d_k * 10^k + d_{(k-1)} * 10^{(k-1)} + ... + d_2 * 10^2 + d_1 * 10^1 + d_0 * 10^0) = (9p + d_k * 10^k + d_{(k-1)} * 10^{(k-1)} + ... + d_2 * 10^2 + d_1 * 10^1 + d_0).[/tex]

Since 10 is congruent to 1 modulo 9 (10 ≡ 1 (mod 9)), we can rewrite the above equation as:

n ≡ [tex](9p + d_k + d_{(k-1)} + ... + d_2 + d_1 + d_0)[/tex] ≡ 0 (mod 9).

This shows that n is congruent to 0 modulo 9, or in other words, n is divisible by 9.

Therefore, we have shown that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9.

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Answer:

3x-10y=-41

Step-by-step explanation:

"standard form of the line" is ax+by=c, where a, b, and c are free coefficients

first, we need to find the slope (m) of the line

that is calculated with the formula (y2-y1)/(x2-x1)

we have the points (3,5) and (-7,2)

label the points:

x1=3

y1=5

x2=-7

y2=2

substitute into the equation

m=(2-5)/(-7-3)

m=-3/-10

m=3/10

the slope is 3/10

before we put a line into standard form, we need to put it into another form first-- like slope-intercept form (y=mx+b, where m is the slope and b is the y intercept)

we already know the slope

here's our line so far:

y=3/10x+b

we need to find b; since the line will pass through the points (3,5) and (-7,2) we can use either one of them to find b

let's use (3,5) as an example. Substitute into the equation

5=3/10(3)+b

5=9/10+b

41/10=b

b is 41/10

this is the equation:

y=3/10x+41/10

now we can find the equation in standard form. Subtract 3/10x from both sides

-3/10x+y=41/10

a (the number in front of x cannot be negative OR less than one. First, let's multiply both sides by -1)

3/10x-y=-41/10

multiply both sides by 10 to clear the fraction

3x-10y=-41

^^ is the equation

hope this helps!

Answer: 240 in 2

Step-by-step explanation:

just did it

Answer:

It’s either A OR B

Step-by-step explanation:

Step-by-step explanation:

Volume=base area * height

=πr^2h

22/7 * 10^2 *11

=3457.1cm3

Answer:

The volume of a cylinder with a diameter of 10 inches and height of 20 inches is 1,570.8 cubic inches.

Answer:

522

Step-by-step explanation:

A good strategy to avoid depending on wrong answers is to conduct rigorous testing and validation.

In the business world, many analytical models are found to be incorrect, as studies have shown. To avoid relying on flawed answers, it is crucial to implement a strategy that emphasizes rigorous testing and validation. This involves thoroughly evaluating the model's performance by comparing its outputs with known or expected outcomes. By subjecting the model to various scenarios and testing its predictions against real-world data, discrepancies can be identified and corrected.

Regularly testing and validating analytical models helps to uncover potential flaws and inaccuracies. This iterative process allows for adjustments and improvements to be made, ensuring that the model provides reliable and accurate results. By implementing a robust testing and validation strategy, businesses can minimize the risks associated with using incorrect analytical models and make informed decisions based on reliable insights.

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The general solution of the differential equation y" - 4y' + 5y = 16 cos (1) using determinant coefficients is given by y =

yh + yp = c1 e^(2x) cos(x) + c2 e^(2x) sin(x) + 16/((5^2 + 1)√26) cos(1).

In order to find the solution using determinant coefficients, first, we solve the homogeneous equation y" - 4y' + 5y = 0. The characteristic equation is given by r^2 - 4r + 5 = 0, which has roots r = 2 ± i. Therefore, the general solution of the homogeneous equation is yh = c1 e^(2x) cos(x) + c2 e^(2x) sin(x).

Next, we find the particular solution of the non-homogeneous equation using the method of undetermined coefficients. Since the forcing function is cos(1), we assume the particular solution to be of the form yp = a cos(1). Substituting this into the differential equation, we get -a + 4a + 5a cos(1) = 0, which implies a = 16/(5^2 + 1). Hence, the particular solution is yp = 16/((5^2 + 1)√26) cos(1).

Therefore, the general solution of the given differential equation is y = yh + yp = c1 e^(2x) cos(x) + c2 e^(2x) sin(x) + 16/((5^2 + 1)√26) cos(1).

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Answer:

La razón entre el número de gallinas y el total de animales es: [tex]\frac{5}{7}[/tex]

Step-by-step explanation:

La razón es una comparación entre dos magnitudes comparables.

En otras palabras, la razón es el cociente entre dos números o dos cantidades comparables entre sí, expresado como fracción.

En este caso, la cantidad total de animales es la suma de la cantidad de conejos y la cantidad de gallinas:

2 conejos + 5 gallinas= 7 animales

Entonces la razón entre el número de gallinas y el total de animales es: [tex]\frac{5}{7}[/tex]

Answer:

7 m

Step-by-step explanation:

you divide the volume by the radius I'm pretty sure. that's what I did and I got 7

Answer:

-1.25, -1/4, -1/8, 0.125

Step-by-step explanation:

Answer:-1.25, -1/4, -1/8, 0.125

Step-by-step explanation:

Answer: 132ft

Step-by-step explanation:

9x4=36+36=72

30-9=21

21+21+9+9=60

60+72=132ft

a) The derivative of g(x) at x=2 is g'(2) = 2.

b) The second derivative of k(x) at x=1 is k"(1) = -30.

c) The derivative of n(x) at x=0 is n'(0) = -18.

a) To find g'(x), we need to take the derivative of g(x) with respect to x. Let's differentiate each term separately:

g(x) = 3x³ - 8x² - 2x + 35

The derivative of 3x³ is obtained by applying the power rule, which states that if we have a term of the form [tex]ax^n[/tex], the derivative is given by [tex]nax^{(n-1)[/tex]. In this case, the derivative of 3x³ is 3 * 3x², which simplifies to 9x².

The derivative of -8x² is obtained in a similar manner, resulting in -16x.

The derivative of -2x is -2.

Since 35 is a constant term, its derivative is zero.

Now, let's combine these derivatives to find g'(x):

g'(x) = 9x² - 16x - 2

To find g'(2), we substitute x = 2 into the derivative:

g'(2) = 9(2)² - 16(2) - 2

= 9(4) - 32 - 2

= 36 - 32 - 2

= 2

Therefore, g'(2) = 2.

b) To find k"(x), we need to take the second derivative of k(x) with respect to x. Let's differentiate each term:

k(x) = 1 / [tex]x^5[/tex]

The derivative of 1/[tex]x^5[/tex] can be found using the power rule and the chain rule. The power rule states that the derivative of [tex]x^n[/tex] is n[tex]x^{(n-1)[/tex], and the chain rule applies when we have a function within another function. In this case, the derivative of 1/[tex]x^5[/tex] is -5/[tex]x^6[/tex].

Taking the derivative of -5/[tex]x^6[/tex], we apply the power rule again, resulting in 30/[tex]x^7[/tex].

Now, let's find k"(x) by differentiating -5/[tex]x^6[/tex] again:

k"(x) = -30/[tex]x^7[/tex]

To find k"(1), we substitute x = 1 into the second derivative:

k"(1) = -30/([tex]1^7[/tex])

= -30/1

= -30

Therefore, k"(1) = -30.

c) To find n'(x), we need to take the derivative of n(x) with respect to x. We can apply the product rule to differentiate the two factors of n(x):

n(x) = (-4x + 2)(3x² - 5x + 2)

Using the product rule, the derivative of n(x) is given by:

n'(x) = (-4x + 2)(d/dx)(3x² - 5x + 2) + (3x² - 5x + 2)(d/dx)(-4x + 2)

To differentiate each term, we use the power rule:

(d/dx)(3x² - 5x + 2) = 6x - 5

(d/dx)(-4x + 2) = -4

Substituting these derivatives back into n'(x), we get:

n'(x) = (-4x + 2)(6x - 5) + (3x² - 5x + 2)(-4)

Now, let's find n'(0) by substituting x = 0 into the derivative:

n'(0) = (-4(0) + 2)(6(0) - 5) + (3(0)² - 5(0) + 2)(-4)

= (2)(0 - 5) + (0 - 0 + 2)(-4)

= (2)(-5) + (2)(-4)

= -10 - 8

= -18

Therefore, n'(0) = -18.

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Answer:

1 7/10

Step-by-step explanation:

Given that:

Weight of bag 1 = 2/5 pounds

Weight of bag 2 = 7/10 pounds

Weight of bag 3 = 3/5 pounds

Total weight of the three bags :

2/5 + 7/10 + 3/5

Take the lcm of 5 and 10

Lcm of 5 and 10 = 10

(4 + 7 + 6) / 10

17 /10

= 1 7/10

The value of det(3A^-2B^T) + 1, given det(A) = 1 and B is a singular matrix, is :

1

To find the determinant of the given expression, let's break it down step by step.

Matrix A is 4x4 with det(A) = 1.

Matrix B is a singular matrix.

Find the inverse of matrix A.

Since A is given to be a 4x4 matrix with det(A) = 1, we know that A is invertible. Therefore, A^-1 exists.

Find the determinant of the expression 3A^-2B^T.

Let's calculate the determinant of 3A^-2B^T:

det(3A^-2B^T) = det(3) * det(A^-2) * det(B^T)

We know that det(A^-2) = (det(A))^(-2) = 1^(-2) = 1.

Also, det(B^T) = det(B) because the determinant of a transpose is the same as the determinant of the original matrix.

So, det(3A^-2B^T) = 3 * 1 * det(B) = 3 * det(B)

Determine the value of det(3A^-2B^T) + 1.

Since B is given to be a singular matrix, its determinant is 0.

Therefore, det(3A^-2B^T) + 1 = 3 * det(B) + 1 = 3 * 0 + 1 = 1.

So, the value of det(3A^-2B^T) + 1 is 1.

Therefore, the correct answer is 1.

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Answer:

x >12

Step-by-step explanation:

2 + 5(x-3) > 3x + 11

Distribute

2 + 5x - 15 > 3x+11

Combine like terms

5x -13 > 3x+11

Subtract 3x from each side

5x-13-3x> 3x+11-3x

2x-13 > 11

Add 13 to each side

3x-13+13> 11+13

2x> 24

Divide by 2

2x/2 > 24/2

x >12

Answer:

Positive association is correct

Step-by-step explanation: The dots form are going up, forming a positive line. If they were to go down they would be negative. So in this case positive association is correct

A)The probability that the yield strength is greater than 60 is approximately 0.0003.

B)The yield strength value that separates the strongest 75% from the others is approximately 45.3725 ksi.

What is probability?

Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood or chance of an event occurring. It provides a numerical measure of uncertainty or the relative frequency with which an event is expected to happen. In simpler terms, probability is a way of expressing how likely it is for a particular outcome or event to take place.

(a) The probability that yield strength is at most 39:

Using the standard normal distribution, we can calculate the z-score as follows:

[tex]\[ z = \frac{{39 - 42}}{{5.0}} = -0.6 \][/tex]

The cumulative probability associated with a z-score of -0.6 represents the probability of obtaining a value less than or equal to 39. Using a standard normal distribution table or a calculator, we find that this cumulative probability is approximately 0.2743.

Therefore, the probability that the yield strength is at most 39 is approximately 0.2743.

The probability that yield strength is greater than 60:

Converting 60 to a z-score:

[tex]\[ z = \frac{{60 - 42}}{{5.0}} = 3.6 \][/tex]

The cumulative probability associated with a z-score of 3.6 represents the probability of obtaining a value greater than 60. Using a standard normal distribution table or a calculator, we find that this cumulative probability is approximately 0.9997.

Since we want the probability of a value greater than 60, we subtract this cumulative probability from 1:

[tex]\[ P(\text{{yield strength}} > 60) = 1 - 0.9997 = 0.0003 \][/tex]

Therefore, the probability that the yield strength is greater than 60 is approximately 0.0003.

(b) The yield strength value that separates the strongest 75% from the others:

To find the yield strength value that separates the strongest 75% from the others, we need to find the z-score corresponding to the cumulative probability of 0.75. Using a standard normal distribution table or a calculator, we find that the z-score associated with a cumulative probability of 0.75 is approximately 0.6745.

Next, we can use the z-score formula to find the yield strength value:

[tex]\[ z = \frac{{x - \mu}}{{\sigma}} \][/tex]

Rearranging the formula to solve for x:

[tex]\[ x = \mu + (z \times \sigma) \][/tex]

Substituting the values into the formula:

[tex]\[ x = 42 + (0.6745 \times 5.0) = 45.3725 \][/tex]

Therefore, the yield strength value that separates the strongest 75% from the others is approximately 45.3725 ksi.

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Answer:

B. 11

Step-by-step explanation:

The 50th percentile represents the halfway point of a data set and therefore, it is simply another name for the median.

We can use the following steps to find the median:

Step 1:  Arrange the numbers in ascending numerical order:

10, 10, 11, 12, 13.

Step 2:  Find the middle of the numbers:

Since there are 5 numbers, the median will have two numbers to the left and right of it.  11 satisfies this requirement so it is the median and thus the 50th percentile of the numbers.

Answer:

b

Step-by-step explanation:

Answer:

What is it love ?

Step-by-step explanation:

Answer:

3, 1/2

Step-by-step explanation:

x + 2y = 4

2x - 2y = 5

_______________ +

2x + x + 2y + (-2y) = 4 + 5

3x + 0 = 9

3x = 9

x = 9/3

x = 3

if you want to find the value of y, you just have to choose one of the equation. I will choose x + 2y = 4, even if you choose 2x - 2y = 5 the result remains same

x + 2y = 4

3 + 2y = 4

2y = 4 - 3

y = 1/2

x, y = 3, 1/2

i'm sorry, i'm not good at english ^^

a) Formal explanation of risk aversion An agent with utility function u is more risk averse than an agent with utility function ˜u if the former has a higher marginal utility of consumption and a diminishing marginal utility of consumption.

The marginal utility of consumption is defined as the amount of utility gained from an additional unit of consumption.

b) Show that an agent with utility function u(x) = log x is more risk averse than an agent with utility function ˜u(x) = √ x. An agent with utility function u(x) = log x is more risk averse than an agent with utility function ˜u(x) = √ x. To show this, we need to find the Arrow-Pratt coefficient of risk aversion, also known as the coefficient of relative risk aversion. The Arrow-Pratt coefficient of risk aversion is given by :-u''(x)/u'(x)Where u'(x) is the first derivative of u with respect to x and u''(x) is the second derivative of u with respect to x.

The Arrow-Pratt coefficient of risk aversion measures the curvature of the utility function. A higher value of the Arrow-Pratt coefficient of risk aversion indicates greater risk aversion. Let us calculate the Arrow-Pratt coefficient of risk aversion for both functions:-For u(x) = log x, u'(x) = 1/x, and u''(x) = -1/x². Therefore, the Arrow-Pratt coefficient of risk aversion for u(x) is given by:-u''(x)/u'(x) = -1/x² ÷ (1/x) = -x For ˜u(x) = √ x, ˜u'(x) = 1/2√ x, and ˜u''(x) = -1/4x^(3/2). Therefore, the Arrow-Pratt coefficient of risk aversion for ˜u(x) is given by:-˜u''(x)/˜u'(x) = -1/4x^(3/2) ÷ (1/2√ x) = -1/2√ x

Therefore, since -x < -1/2√ x, the agent with the utility function u(x) = log x is more risk averse than the agent with the utility function ˜u(x) = √ x.

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Answer:

hi

Step-by-step explanation:

hope it helps

have a nice day

Answer:

3.14, 5.8, 0.3, negative 2

Step-by-step explanation:

3.14 is found bn 2 and 3 but very close to 3 u can just take 3.1 and for 5.8 it's bn 5 and 6 very close to 6 but not six , and we'll negative 2 is right on negative 2

Answer:

3,2,4, then 1

Answer:

yes

Step-by-step explanation:

they both equal 5x=1 where x = 1/5

Answer:

3(x+4)=3x+11

3x + 12 = 3x + 11

3x + 1 = 3x

No solution

-2(x+3)=-2x-6

-2x - 6 = -2x - 6

x = x

Infinite solutions

4(x-1) = 1/2(x-8)

4x - 4 = 1/2x - 4

8x - 8 = x - 8

7x = 0

x = 0

One solution

3x-7=4+6 +4x

3x - 7 = 10 + 4x

x = -17

One solution

The probability is equal to 0 for the option "a) Less than 5.0."Explanation:Given, the mean of the normal distribution, μ = 7.5 and the standard deviation of the normal distribution, σ = 2.5 The formula to find the Z-score, z is given by;z = (x - μ)/σwhere x is the value of the random variable under consideration.

a) To find the probability of the random variable being less than 5, we find the Z-score;z = (5 - 7.5)/2.5 = -1 Therefore, P(X < 5) = P(Z < -1)Using the standard normal table, the probability corresponding to the Z-score -1 is 0.1587.Therefore, the probability of the random variable being less than 5 is 0.1587.

b) To find the probability of the random variable being between 4.0 and 10.0, we find the Z-score corresponding to each value and calculate the difference in their probability. The probability required will be the absolute value of this difference. z 1 = (4 - 7.5)/2.5 = -1.4 z 2 = (10 - 7.5)/2.5 = 1 Therefore, P(4 < X < 10) = P(-1.4 < Z < 1) = P(Z < 1) - P(Z < -1.4) = 0.8413 - 0.0808 = 0.7605

c ).To find the probability of the random variable being greater than 9.0, we find the Z-score;z = (9 - 7.5)/2.5 = 0.6 Therefore, P(X > 9) = P(Z > 0.6)Using the standard normal table, the probability corresponding to the Z-score 0.6 is 0.2743.Therefore, the probability of the random variable being greater than 9.0 is 0.2743

d) To find the probability of the random variable being between 6.0 and 8.0, we find the Z-score corresponding to each value and calculate the difference in their probability. The probability required will be the absolute value of this difference. z 1 = (6 - 7.5)/2.5 = -0.6z2 = (8 - 7.5)/2.5 = 0.2Therefore, P(6 < X < 8) = P(-0.6 < Z < 0.2) = P(Z < 0.2) - P(Z < -0.6) = 0.5793 - 0.2743 = 0.305Therefore, the probability is equal to 0 for the option "a) Less than 5.0."

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A random variable has a normal distribution with mean 7.5 and standard deviation 2.5. The following is the probability equal to 0.

Option a) Less than 5.0 is correct.

Note that the normal distribution is symmetrical. So, the probabilities of events occurring are equal on either side of the mean.

The probability is zero when the range of events is beyond the limits of the standard normal distribution, which is from -3 to +3. Now let's standardize the values below:

a. less than 5.0: The formula to standardize is

[tex]z = (x - \mu) / \sigma[/tex]

z = (5 - 7.5) / 2.5

z = -1

Thus, the area of the left side of the standard normal distribution is zero, indicating that the probability of less than 5 is zero. Therefore, option a) is correct.

Other options are: b. Between 4.0 and 10.0: The probability that the values fall between 4 and 10 is 0.974 c. Greater than 9.0: The probability that the values are greater than 9 is 0.080 d. Between 6.0 and 8.0: The probability that the values fall between 6 and 8 is 0.329.

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Answer:

Question 7 = $2

Question 8 = $16

Answer:

its 9 signs

Step-by-step explanation:

in order to find number of  signs you gonna divide total distance by distance of a small sign

3/4  ÷ 1/12  

= 3/4 × 12/1 = 3 × 3

therefore, the answer is 9 sign