Question 1
Fill in the blank question.
The sum of two numbers is 20.5. Their difference is 6.5. Find the two numbers.



Part A



Write a system of equations that represents the situation.



x+y=_____



x−y=_____



Part B



Solve the system of equations. Express the coordinates as decimals if necessary.



( _____, _____)



Part C



Interpret the solution.



The greater number is
____________________
and the lesser number is
____________________

The solution to the given differential equation is:

\(\[y(x)\ =\ a_0 + a_1x + \frac{1}{4}a_1x^4 + \sum_{n=2}^{\infty} \frac{2a_{n-2}}{(n-2)(n-1)}x^{n+3}\]\)

To solve the given differential equation

\(x^3y'''\ -\ 6y\ =\ 0,\)

we can use the method of power series. Let's assume a power series solution of the form

\(y(x)\ =\ \sum_{n=0}^{\infty} a_nx^n.\)

Differentiating y(x) with respect to x gives:

\(\[y'(x)\ =\ \sum_{n=0}^{\infty} n a_n x^{n-1}\ =\ \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n\]\)

Differentiating again gives:

\(\[y''(x)\ =\ \sum_{n=0}^{\infty} (n+1)na_{n+1}x^{n-1}\ =\ \sum_{n=0}^{\infty} (n+2)(n+1)a_{n+2}x^n\]\)

Differentiating one more time gives:

\(\[y'''(x)\ =\ \sum_{n=0}^{\infty} (n+2)(n+1)na_{n+2}x^{n-1}\ =\ \sum_{n=0}^{\infty} (n+3)(n+2)(n+1)a_{n+3}x^n\]\)

Substituting these expressions into the differential equation, we have:

\(\[x^3 \sum_{n=0}^{\infty} (n+3)(n+2)(n+1)a_{n+3}x^n - 6 \sum_{n=0}^{\infty} a_n x^n\ =\ 0\]\)

Rearranging the terms and combining like powers of x, we get:

\(\[\sum_{n=0}^{\infty} (n+3)(n+2)(n+1)a_{n+3}x^{n+3} - 6 \sum_{n=0}^{\infty} a_n x^n\ =\ 0\]\)

Now, let's equate the coefficients of like powers of x to zero:

For n=0:

\(\[(3)(2)(1)a_3 - 6a_0 = 0 \implies 6a_3 - 6a_0 = 0 \implies a_3 = a_0\]\)

For n=1:

\(\[(4)(3)(2)a_4 - 6a_1 = 0 \implies 24a_4 - 6a_1 = 0 \implies a_4 = \frac{1}{4}a_1\]\)

\(For \: n\geq 2:

\[(n+3)(n+2)(n+1)a_{n+3} - 6a_n = 0 \implies a_{n+3} = \frac{6a_n}{(n+3)(n+2)(n+1)}\]

\)

Now we can write the solution as:

\(\[y(x)\ =\ a_0 + a_1x + \frac{1}{4}a_1x^4 + \sum_{n=2}^{\infty} \frac{6a_{n-2}}{n(n-1)(n-2)}x^{n+3}\]

\)

Simplifying the series, we get:

\(\[y(x)\ =\ a_0 + a_1x + \frac{1}{4}a_

1x^4 + \sum_{n=2}^{\infty} \frac{2a_{n-2}}{(n-2)(n-1)}x^{n+3}\]

\)

Therefore, the solution to the given differential equation is:

\(\[y(x)\ =\ a_0 + a_1x + \frac{1}{4}a_1x^4 + \sum_{n=2}^{\infty} \frac{2a_{n-2}}{(n-2)(n-1)}x^{n+3}\]\)

where a₀ and a₁ are arbitrary constants to be determined based on the initial conditions or boundary conditions given in the problem.

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The distance between point B and point B prime is 13 units.

Distance Formula: Let (x₁, y₁) and (x₂, y₂) be two points in the coordinate plane.

Then the distance between these points is given by the formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

So, let's begin by finding the coordinates of point B prime.

Since point B is translated 5 units up and 12 units to the right to create B prime, the coordinates of B prime are:

(x,y)(x,y)=(−6+12,−6+5)=(6,-1)

Using the distance formula to find the distance between the points, we have:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]d

= √[(6 − (-6))² + ((-1) − (-6))²]d

= √[(6 + 6)² + (5)²]d

= √[12² + 5²]d

= √(144 + 25)d

= √169d

= 13

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Answer:

41.57 in²

Step-by-step explanation:

Answer:

41.57 in² round it up and it's 42

Step-by-step explanation:

I just did it

Brainliest pls

The chance that a hypothesis test will incorrectly reject the null hypothesis when the p-value cutoff is 1%. The correct option is A.

A hypothesis test is used to test the validity of a hypothesis by calculating the probability that a sample statistic occurred by chance.

The null hypothesis is the hypothesis that is being tested, and it is usually assumed to be true unless there is evidence to the contrary.

The p-value is the probability of obtaining a sample statistic at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level, which is the p-value cutoff chosen by the researcher, then the null hypothesis is rejected.

If the p-value is greater than or equal to the significance level, then the null hypothesis is not rejected. The p-value cutoff is the significance level chosen by the researcher, and it represents the maximum probability of rejecting the null hypothesis when it is actually true.

In this case, the p-value cutoff is 1%, which means that the maximum probability of rejecting the null hypothesis when it is actually true is 1%. Therefore, the chance that a hypothesis test will incorrectly reject the null hypothesis when the p-value cutoff is 1%.

Therefore, the correct option is A.

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Coordinates of point B are (5x₂+6x₁)/11 and (5y₂+6y₁)/11

What is the section formula in coordinate geometry?

When the line segment is divided internally in the ratio m:n, we use this formula.  That is when point C lies somewhere between points A and B. p= (mx₂+nx₁)/(m+n) & q= (my₂+ny₁)/(m+n)

Given here,  point B on AC such that the ratio of AB to AC is 5:6

Let the point B(p,q) internally divide the line joining points A(x₁, y₁) and C(x₂, y₂) in the ratio m : n , then,

p= (mx₂+nx₁)/(m+n)

q= (my₂+ny₁)/(m+n)

Here. m=5 and n=6

p=(5x₂+6x₁)/11 , q=(5y₂+6y₁)/11

Hence, the Coordinates of point B are (5x₂+6x₁)/11 and (5y₂+6y₁)/11

I hope this helps <3

To find a value of c such that P(z ≥ c) = 0.55, we need to use a standard normal distribution table or calculator.From the standard normal distribution table, we find that the z-score corresponding to a right-tailed area of 0.55 is approximately 0.126.

Therefore, we have:

P(z ≥ c) = 0.55

P(z ≤ c) = 1 - P(z ≥ c) = 1 - 0.55 = 0.45

Using the standard normal distribution table, we find that the z-score corresponding to a left-tailed area of 0.45 is approximately -0.126.

So, c = -0.126.

Therefore, the value of c that satisfies P(z ≥ c) = 0.55 is approximately -0.126.

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Answer: 150.95

Step-by-step explanation:144 -\(\sqrt{243}\) -\(\sqrt{5}\) -5 +16 /64 -\(\sqrt{3}\)

144-5+16/64=139+0.25

                   =139.25 or 139,1,4

\(\sqrt{243} -\sqrt{5} -\sqrt{3}\) =15.6-2.2-1.7

                           =11.7

added together

11.7+139.25=150.95

Answer:

Step-by-step explanation:

slope=-2

y-intercept=(0,2)

Answer:

75

Step-by-step explanation:

(0.45)(30) = (0.18)x

x = (0.45)(30) / (0.18)

x = 13.5 / 0.18 = 75

Check the answer:

(0.45)(30) = (0.18)(75)

13.5 = 13.5   answer is correct!

Answer:

Step-by-step explanation:

45% of 30 = 18% of some number

In this kind of questions, we usually assume the missing number as as something

Let the number be x

So assuming the number is x, the equation will be as follows:

45% of 30 is equal to 18% of x

45/ 100 * 30 = 18/100 * x

This means that 0.45 *30= 0.18 * x

Therefore, 13.5 = 0.18 x

Thus X = 13.5 / 0.18

Which is equal to 75

Thus 45% of 30 is equal to 18% of 75

Proving the above situation as follows:

45% OF 30 = 13.5

And 18% of 75 = 13.5

Thus both equations are equal.

Solution =75

When two players A and B alternately flip a fair coin, the probability  E(T) is 4. (A tosses the coin first, then B tosses the coin, then A, then B and so on).

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.

Two players, A and B, turn a fair coin in succession (A tosses the coin first, then B tosses the coin, then A, then B and so on). The order of the heads and tails is noted. If there is a head followed by a tail, the game is done, and the person who throws the tail wins (HT sub-sequence).

We must ascertain the game's estimated duration. Allow T coin tosses for the game to proceed. Find E (T).

\(E(T) = first instance of HT in seriesE(T) =0.5×(1+2)+0.5×(1+E(T))\\0.5×E(T)=0.5×4\\E(T) =4\)

As a result, when two players A and B alternately flip a fair coin, The E(T) is 4. (A tosses the coin first, then B tosses the coin, then A, then B and so on).

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The condition imposed such that a₁X₁ + a₂X₂ +.....+ aₙXₙ is an unbiased estimator of μ must be that ∑a = 1

We are given that X₁, X₂, ... Xₙ constitute a random sample from the population with the mean μ.

This implies, E(X) = μ

Now we have another set of data here, a₁X₁ + a₂X₂ +.....+ aₙXₙ

For a₁X₁ + a₂X₂ +.....+ aₙXₙ to be an unbiased estimator of μ, we should have

E(∑aX) = μ

or, ∑a E(X) = μ

Now we already know that

E(X) = μ

Hence we get

∑a = 1

Hence condition imposed must be that ∑a = 1

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Correct Question

If X₁, X₂, ... Xₙ constitute a random sample from the population with the mean μ, what condition must be imposed on the constants a₁, a₂, ... aₙ so that a₁X₁ + a₂X₂ +.....+ aₙXₙ is an unbiased estimator of μ?

Answer:

See Below

Step-by-step explanation:

If DF = 42, What is DE?

D__________E_______F

(7x + 1)         +    ( 4x-3)         = 42

7x+1+ 4x -3 = 42        Solve for x

11x -2 = 42

   +2    +2

11x = 44

11x / 11  = 44/ 11

x = 4

Now plug x in for the equation of DE

7x+1

7(4) + 1

28 + 1 = 29

DE = 19

Answer:

This is called the product.

Step-by-step explanation:

Answer:

The result of a multiplication operation is called a product. The multiplication of whole numbers may be thought of as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier.

Step-by-step explanation:

Answer:

The value of x is 10.

3x+15=5x-5

3x=5x-5-15

3x=5x-20

3x-5x=(-20)

(-2x)=(-20)

x=10

Answer: In the figure AB is about 8.4 inches and AC is about 13.05 inches.

Step-by-step explanation: We can use cosine to find the hypotenuse. \(cos(40)=\frac{10}{x} \\cos(40) (x)=\frac{10}{x}(x)\\cos(40) (x) =10\\\frac{cos(40) (x)}{cos (40)} =\frac{10}{cos (40)} \\x=\frac{10}{cos(40)}\)

Using a calculator x is about 13.05

Using tangent we can find the length opposite of <C

\(tan(40)=\frac{x}{10} \\tan(40) (10)=\frac{x}{10}(10)\\tan(40) (10) = x\)

Using a calculator x would be about 8.4

Answer:

Step-by-step explanation:

Answer:

(\(\frac{6}{v}\)) - 7

Step-by-step explanation:

7 minus the quotient of 6 and v

 \/

(\(\frac{6}{v}\)), 7 minus

\/

(\(\frac{6}{v}\)) - 7

Have a nice day!

     If this is not what you are looking for - comment! I will edit and update my answer accordingly. (ノ^∇^)

- Heather

Answer:

9

Each value is 3 times the number on the ratio.

10 x 3 = 30.

So, the ratio for oranges would be 9 because:

3 x 3 = 9.

So, the answer is 9.

The value of BA.BC vector is 18 cm² as side lengths are 6 cm.

A triangle is a closed two-dimensional geometric figure with three sides, three angles, and three vertices. It is one of the basic shapes in geometry and is formed by connecting three non-collinear points with straight line segments. The sum of the interior angles of a triangle is always 180 degrees, and there are various types of triangles, such as equilateral, isosceles, scalene, right-angled, acute-angled, and obtuse-angled triangles. Triangles are used in a wide range of applications, from architecture and engineering to physics and computer graphics.

Here,

To find the area of the AABC, we can use Heron's formula:

s = (AB + BC + CA)/2 = (6 + 6 + 6)/2 = 9

Area = √(s(s-AB)(s-BC)(s-CA))

= √(9(9-6)(9-6)(9-6))

= 27/4 cm²

To find the height of the AABC corresponding to side AB, we can use the formula:

Area = 1/2 * AB * height

Rearranging, we get:

height = 2 * Area / AB = 2 * 27/4 / 6 = 9/4 cm

Therefore, the area of AABC is 27/4 cm², and the height corresponding to side AB is 9/4 cm.

For the second part of the question, we can use the Law of Cosines to find the angle between sides BA and BC:

cos(A) = (BA² + BC² - CA²) / (2 * BA * BC)

= (6² + 6² - 6²) / (2 * 6 * 6)

= 1/2

A = arccos(1/2) = 60 degrees

Using this angle, we can find the length of side BA . BC:

BA . BC = 2 * AB * BC * cos(A)

= 2 * 6 * 6 * cos(60)

= 18 cm²

Therefore, BA . BC = 18 cm²

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The point (-5, -9) was reflected over the y-axis to become the point (5, -9). The x-coordinate changed sign the y-coordinate remained the same, confirming that the reflection took place over the y-axis.

The point (-5, -9) was reflected over the y-axis to become the point (5, -9).

To understand which axis the reflection occurred over, we need to analyze the changes in the coordinates.

In a reflection over the x-axis, the y-coordinate of a point changes sign while the x-coordinate remains the same.

Similarly, in a reflection over the y-axis, the x-coordinate changes sign while the y-coordinate remains the same.

The y-coordinate of the point (-5, -9) remains the same as -9 in both the original and reflected points.

The x-coordinate changes from -5 to 5.

This means that the reflection involved a change in the x-coordinate, indicating that it occurred over the y-axis.

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Answer:

The perimeter of the figure will be 62.8 or B

Step-by-step explanation:

1) The radius is 10

2) Perimeter basically means circumference

3) The formula for circumference is 2πr

4) 2π*10

5)2*pi=6.28*10=62.8

His shadow changing at the instant when he is 24 feet from the pole Is \(\frac{95}{13}\)

What do triangles that are similar mean?

The related angle measurements and proportional side lengths of similar triangles are the same.

In this diagram, x represents the distance between the man and the pole, while y represents the distance between the shadow's tip and the pole. My assumption is that the man and the pole are both standing straight up, making the two triangles comparable.

by comparison, \(\frac{y-x}{y} = \frac{6}{19}\)

\(19(y-x)=6(y)\\19y-19x=6y\\19y-6y=19x\\13y=19x\\y=\frac{19}{13} y\)

differentiate both sides with respect to t or time.

\(\frac{dy}{dt}=\frac{19}{13}\frac{dx}{dt}\)

You know \(\frac{dx}{dt}\) = 5 ft/s because the man is edging away from the pole at that pace. Find out \(\frac{dy}{dt}\) how quickly the shadow's tip is moving.

That Means, \(\frac{dy}{dt}= \frac{19}{13} * 5 = \frac{95}{13}\)ft/s
Actually, it doesn't matter how far the man is from the pole because only his speed impacts how quickly his shadow moves.

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Answer:

(a) hundreds = 817

(b) hundredths = 817.09

(c) 3 d.p = 817.086

(d) 4 s.f. = 817.1

(e) whole number = 817

Serena leans a 24-foot ladder against a 22.8-foot-tall building to climb on the roof is 71.8 degree.

Measurement of Angle:

The measurement of angles is done using basic geometric tools such as protractors and compasses. This tool helps to find precise measurements of angles. A protractor helps provide accurate measurements of angles, while compasses help construct angles. Angles are measured in three ways: degrees, radians, and revolutions.

According to the Question:

Suppose the measure of angle is θ, then we have

   Sinθ = 22.8/24

⇒ Sinθ = 0.95

Therefore,

θ = Sin⁻¹ (0.95)

  = 71.8 degree.

Therefore, Serena leans a 24-foot ladder against a 22.8-foot-tall building to climb on the roof is 71.8 degree.

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The perimeter of the polygon is  22 units.

What is the area of the polygon?

The area of a polygon is defined as the area enclosed by the polygon's boundary. In other words, the region occupied by any polygon determines its area. In this lesson, we will learn how to calculate polygon area and the difference between polygon perimeter and area in detail.

we know that

the perimeter of a polygon is the sum of the length sides

in this problem, we have five vertices

so the polygon has five sides

Let,

A(-1, 3)

B(−1, 6)

C(2, 10)

D(5, 6)

E(5, 3).

the perimeter is equal to

P = AB + BC + CD + DE + AE

The formula to calculate the distance between two points is equal to

d = √(y₂ - y₁)² + (x₂ - x₁)²

The distance of AB:

A(-1, 3)

B(−1, 6)

d = √(6 - 3)² + (-1 - (-1))²

d = √(3)² + (0)²

d = 3

The distance of BC:

B(−1, 6)

C(2, 10)

d = √(10 - 6)² + (2 - (-1))²

d = √(4)² + (3)²

d = 5

The distance of CD:

C(2, 10)

D(5, 6)

d = √(6 - 10)² + (5 - 2)²

d = 5

The distance of DE:

D(5, 6)

E(5, 3).

d = √(3 - 6)² + (5 - 5)²

d = 3

The distance of AE:

A(-1, 3)

E(5, 3)

d = √(3 - 3)² + (5 - (-1))²

d = 6

Find the perimeter

the perimeter is equal to

P = AB + BC + CD + DE + AE

P = 3+5+5+3+6

P = 22 units.

Hence, the perimeter of the polygon is  22 units.

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this is a number is can not devide

Find the derivate of the expression using the general procedure to find the derivative of a polynomial function. To do it, multiply the term times the exponent of the variable and raise the variable to the original exponent minus 1:

Now that we have the expression for the derivative of the function, evaluate it at the value of x that is in the question statement, which is 1:

The derivative of the function is -2 at x=1.

Answer:Total Commission

$12

Commission for Each Agent

$6.00

Total Amount Seller Receives

$188

Step-by-step explanation:

Answer: 56.118

Step-by-step explanation:

300/4 ÷ 4/4 [Divide the numerator and denominator by 4] = 75%. Answer: 75% (ii) Express 12/5 as a percent. Solution: 12/5 × 100 = (12 × 100)/5 = 1200/5