Integers a and b are such that
[tex](a + 3 \sqrt{5} ) {}^{2} + a - b \sqrt{5} = 51 [/tex]
Find the possible values of a and the corresponding values of b.
(Answers are a= -3 , 2 and b= -18 , 12. But I don't know how to show the working so please help, thx)​

Answer:

B

Step-by-step explanation:

Answer:

The radius of the circle is 24.5 units

Step-by-step explanation:

The formula for the circumference of a circle is [tex]C=2\pi r[/tex]

Where [tex]C[/tex] is the circumference and [tex]r[/tex] is the radius.

Lets solve for [tex]r[/tex].

Divide both sides of the equation by [tex]2\pi[/tex].

[tex]\dfrac{C}{2\pi}=r[/tex]

Now we have an equation to evaluate the radius.

Numerical Evaluation

In this example we are given

[tex]C=153.86153\\\pi =3.14[/tex]

Substituting our values into our equation for the radius yields

[tex]r=\dfrac{153.86153}{2*3.14}[/tex]

[tex]r=\dfrac{153.86153}{6.28}[/tex]

[tex]r=\dfrac{153.86153}{6.28}[/tex]

[tex]r=24.5[/tex]

This is known as the Collatz Conjecture, and it remains an unsolved problem in mathematics. Despite extensive computational evidence suggesting that the conjecture is true, a proof or counterexample has yet to be found.

The conjecture states that no matter what positive integer you start with, applying the "3n+1" rule (multiply by 3 and add 1 if n is odd) or "n/2" rule (divide by 2 if n is even) repeatedly will eventually lead to the sequence 4, 2, 1, and then it will loop endlessly: 4, 2, 1, 4, 2, 1, and so on.

While the conjecture has been checked for all starting values up to at least 10^20, no one has been able to prove that it holds true for all positive integers. It is possible that there exists a starting value that does not eventually fall into the 4, 2, 1 loop, but no such value has been found.

Multiply equation 2 by 3. Then subtract the result from equation 1.

Explanation:

The 2nd equation given to us is -2x+y = 8

Triple both sides to get -6x+3y = 24

This means the original system

[tex]\begin{cases}2x-3y = 12\\-2x+y = 8\end{cases}[/tex]

is equivalent to

[tex]\begin{cases}2x-3y = 12\\-6x+3y = 24\end{cases}[/tex]

If we were to add the equations, then the y terms would cancel out because -3y+3y = 0y = 0.

Subtracting the equations will not cancel out the y terms.

-3y minus 3y = -3y -3y = -6y

Therefore the portion saying "Then subtract the result from equation 1." in answer choice B is not correct. It should be "add the equations" or "add the result to equation 1" after tripling both sides of the 2nd equation.

Answer choices A and C say the same thing more or less. Adding the original equations straight down will cancel out the x terms. Therefore, choices A and C are valid approaches to using elimination. We can cross choices A and C off the list.

The balance based on the.data given in the table is shown below.

Balance from day 1 to 10 = 11000

Balance from day 11 to 21= 8000

Balance from day 21 to 30- 5500

Balance on day 31 7500

Balance from day 1 to 10= 11000

Balance from day 11 to 21= 8000 Balance from day 21 to 30 5500

Balance on day 31- 7500

The average daily balance of the credit card for the month of december is:

Average daily balance The ratio of total balance from each day in cycle to the total number of days in cycle.

Total balance from each day in cycle=

(11000 x 10)+(8000 × 10)+(5500 x 10) + (7500 x 1) = 104000

Average daily balance=104000 / 31

= 8145

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Step-by-step explanation:

138 = 10 log R

13.8 = log R       anti-log both sides of the equation

10^13.8 = R       Expand L side

6.3 x 10^13  = R

Answer: -1/2x

Step-by-step explanation:

Rise over run method if you know it is really helpful

Answer:

20 Kg

Step-by-step explanation:

let b represent the weight of 1 bag of sand

after using [tex]\frac{1}{4}[/tex] of 1 bag there is [tex]\frac{3}{4}[/tex] bag left along with the unused bag , then

[tex]\frac{3}{4}[/tex] b + b = 35 ( multiply through by 4 to clear the fraction )

3b + 4b = 140

7b = 140 ( divide both sides by 7 )

b = 20

that is one full bag weighs 20 Kg

[tex]\huge\text{Hey there!}[/tex]

[tex]\mathtt{6\dfrac{2}{3} - 4\dfrac{1}{3}}\\\\\mathtt{\rightarrow \dfrac{6\times3 + 2}{3} - \dfrac{4\times3 + 1}{3}}\\\\\mathtt{\rightarrow \dfrac{18 + 2}{3} - \dfrac{12 + 1}{3}}\\\\\mathtt{\rightarrow \dfrac{20}{3} -\dfrac{13}{3}}\\\\\mathtt{\rightarrow\dfrac{7}{3}}\\\\\mathtt{\rightarrow2 \dfrac{1}{3}}[/tex]

[tex]\huge\text{Therefore your answer should be:}[/tex]

[tex]\huge\boxed{\mathtt{2\dfrac{1}{3}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

Answer:

Step-by-step explanation:

The hypotenuse length is not given here. So we have to find it.

We take hypotenuse length as X

by using the Pythagoras theorem

x² = 6²+10²

x= √136

x=2√34

the perimeter of the garden = 6+10+2√34

                                                = 16+2√34

                                                = 5.8+16

                                                =21.8m

Answer:

25

Step-by-step explanation:

To find the value of the expression, all you have to do is plug 9 in for w and plug 8 in for x.

When you plug the numbers in, your equation should look like 9(9)-7(8)

Then you just simplify the expression and get your answer of 25

[tex]9(9)-7(8)\\81-56\\25[/tex]

the result by 5 and adds 4 one last time to get the final output value. This can also be written in shorthand as:=(f o f o f) (X) = 5(5(5X + 4) + 4) + 4 = 125X + 124.

How to solve a  function?

To find (f o f o f) (X), we need to apply the function f three times to X. Let's start by finding (f o f) (X):

(f o f) (X) = f(f(X)) = f(5X + 4) = 5(5X + 4) + 4 = 25X + 24

Now, we need to apply f to (f o f) (X) to get the final result:

(f o f o f) (X) = f((f o f) (X)) = f(25X + 24) = 5(25X + 24) + 4 = 125X + 124

Therefore, (f o f o f) (X) = 125X + 124.

In words, the function (f o f o f) (X) takes any input value X, multiplies it by 5, adds 4, multiplies the result by 5 again, adds 4, and finally multiplies the result by 5 and adds 4 one last time to get the final output value. This can also be written in shorthand as:

(f o f o f) (X) = 5(5(5X + 4) + 4) + 4 = 125X + 124.

This process of applying a function multiple times to an input value is known as function composition, and it is a fundamental concept in mathematics and computer science.

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By adding the set of parentheses as shown above, we get an expression [tex]65 e^{0.68} \simeq 17.09[/tex]

In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bⁿ, where b is the base and n is the power; this is pronounced as "b to the n".

One possible way to create an expression using parentheses to get a value of 17 is:

((8 ÷ 2) + (9 × 9) - (10 × 2)) exponent 0.5

First, we evaluate the multiplication and division operations inside parentheses, since they have higher precedence than addition and subtraction.

[tex]((8 \div 2) + (9 \times 9) - (10 \times 2)) = (4 + 81 - 20) = 65[/tex]

Then, we raise the result to the power of 0.68 which is the same as taking the square root:

65 exponent 0.68 ≈ 17.09

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Drag the following expressions accordingly to complete the proof:

1/sinx· cos x)/ (sinx/cosx  + cosx/sinx) = cos²x

(cos x/sinx)/ ([sin²x/[sinx·cosx] + cos²x/[sinx·cosx]) = cos²x

(cos x/sinx)/ ([sin²x+cos²x] / sinx·cosx) = cos²x

(cos x/sinx)/ ( 1 / sinx·cosx) = cos²x

(cos x/sinx) · (sinx·cosx) = cos²x

Given: (csc x cos x)/ (tan x + cot x) = cos²x

We want to prove the trigonometric identity.

We have:

(csc x· cos x)/ (tan x + cot x) = cos²x

Recall the following trig. expressions and substitute for them:

csc x = 1/sinx

tan x = sinx/cosx

cot x = cosx/sinx

(1/sinx· cos x)/ (sinx/cosx  + cosx/sinx) = cos²x

(cos x/sinx)/ ([sin²x/[sinx·cosx] + cos²x/[sinx·cosx]) = cos²x

(cos x/sinx)/ ([sin²x+cos²x] / sinx·cosx) = cos²x

Recall sin²x+cos²x = 1. Subsitiute for it to get:

(cos x/sinx)/ ( 1 / sinx·cosx) = cos²x

(cos x/sinx) · (sinx·cosx) = cos²x

cosx  · cos x = cos²x

cos²x = cos²x

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Answer:

9

Step-by-step explanation:

the distance between the two points is just the difference in x-values: -7-2 = 9.

Answer:

The distance between two points  (-7, 5) and (2, 5) is [tex]\sqrt{35}[/tex]

Step-by-step explanation:

Given: Points (-7, 5) and (2, 5)            

We have to find the distance between the given points (2, 5) and (5, 7)

Consider the given points (2, 5) and (5, 7)

Distance between two points is calculated using formula

[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1^2}[/tex]

[tex](x_1,y_1)=(-7,5), \ (x_2,y_2)=(2,5)[/tex]

Thus, [tex]D=\sqrt{(5- (-7))^2+(5-2)^2}=\sqrt{35}[/tex]

Thus, The distance between two points  (-7, 5) and (2, 5) is [tex]\sqrt{35}[/tex]

Evaluating the function f(x) = [x - 1] at different values of x, f(-1.30)=-3, f(-3)=-4, f(2.3)=1, f(4)=3, and f(7.1)=6

Given the function f(x) = [[x - 1]], we are asked to evaluate f at several values of x.

For x=-1.30, we have:

[tex]$f(-1.30) = [[-1.30 - 1]] = [[-2.30]] = -3$[/tex]

For x=-3, we have:

[tex]f(-3) = [[-3 - 1]] = [[-4]] = -4[/tex]

For x=2.3, we have:

[tex]$f(2.3) = [[2.3 - 1]] = [[1.3]] = 1$[/tex]

For x=4, we have:

[tex]$f(4) = [[4 - 1]] = [[3]] = 3$[/tex]

For x=7.1, we have:

[tex]$f(7.1) = [[7.1 - 1]] = [[6.1]] = 6$[/tex]

Therefore, f(-1.30)=-3, f(-3)=-4, f(2.3)=1, f(4)=3, and f(7.1)=6

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Answer:

(-5,3)

Step-by-step explanation:

The surface area of the given geometric shape(cube) is 24[tex]cm^2[/tex]. To find the surface area of a cube, you need to determine the total area of all the six faces of the cube.

Each face of a cube is a square with all sides equal in length to the edge length of the cube. The formula to find the surface area of a cube is:

Surface Area of Cube = 6 × (edge length[tex])^2[/tex]

Therefore surface are of the given cube = 6 x [tex](2cm[/tex][tex])^2[/tex]

= 6 x 4[tex]cm^2[/tex]

= 24[tex]cm^2[/tex]

Therefore, the surface area of the cube is 24 square centimeters.

To understand this conceptually, we have a cube with 2cm sides. It has six faces, and each face is a square with a side length of 2cm. Therefore, the area of each face is (2cm x 2cm) = 4[tex]cm^2[/tex]. The total surface area of the cube is obtained by adding up the area of all six faces, which gives us:

Total surface area = 6 x 4[tex]cm^2[/tex] = 24[tex]cm^2[/tex]

Therefore, the surface area of a cube with side length 2cm is 24 square centimeters.

The complete question is:

What is the surface area of the given geometric shape in the image, in square centimetres?

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The area of the following shape is 76 units.

What is a hexagon?

A regular hexagon is a closed shape polygon which has six equal sides and six equal angles. In case of any regular polygon, all its sides and angles are equal.

The structure in the figure is hexagon in which two parallel lines are present.

These parallel lines are dividing this hexagon into three parts:

1. Rectangular part with length = 8 units and width = 7 units

2. Triangle of height 2 units.

3. Triangle of height 3 units.

We know,

1. Area of rectangle = L x B

  Area of rectangle = 8 x 7

  Area of rectangle = 56 units

2. Area of triangle 1 = 1/2 x base x height

   Area of triangle 1 = 1/2 x 2 x 8

   Area of triangle 1 = 8 units

3. Area of triangle 2 = 1/2 x base x height

   Area of triangle 2 = 1/2 x 3 x 8

   Area of triangle 2 = 12 units

So, area of hexagon = area of rectangle + Area of triangle 1 + area of triangle 2

Area of hexagon = 56 + 8 + 12

Area of hexagon = 76 units

Therefore, the area of the shape is 76 units.

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To solve this problem, we need to use vector addition to find the resultant velocity of Julia.

Let's assume that the east direction is the positive x-axis and the south direction is the negative y-axis.

The velocity of Julia in still water is 8 km/hr in the positive x-axis direction.

The velocity of the river is 3 km/hr in the negative y-axis direction.

To find the magnitude and direction of Julia's velocity, we need to find the resultant velocity vector, which is the vector sum of her velocity in still water and the velocity of the river.

Using the Pythagorean theorem, the magnitude of the resultant velocity can be calculated as:

|V| = √(Vx² + Vy²)

where Vx is the x-component of the resultant velocity and is equal to Julia's velocity in still water, and Vy is the y-component of the resultant velocity and is equal to the velocity of the river.

Vx = 8 km/hr

Vy = -3 km/hr

|V| = √(8² + (-3)²) = √(64 + 9) = √73 km/hr

The direction of the resultant velocity can be calculated as:

θ = tan⁻¹(Vy / Vx)

θ = tan⁻¹(-3 / 8) = -20.56°

The negative sign indicates that the resultant velocity vector makes an angle of 20.56° below the positive x-axis (east direction).

Therefore, the magnitude of Julia's velocity is approximately 8.54 km/hr, and the direction of her velocity is 20.56° below the positive x-axis (east direction).

Using simple mathematical operations we know that 19.03 min will be required to make 88 pretzels by 5 makers.

In mathematics, an operation is a function that transforms zero or more input values ​​into well-defined output values.

The number of operands is the arity of the operation.

The four basic operations in mathematics for all real numbers are:

Addition (sum; '+') Subtraction (difference formation; '-') Multiplication (product formation; '×') Division (quotient formation; '÷')

So, insert values as follows: t is time

2*6/44 = 5*t/88

t = 8*88/444*5 = 0.3171 hours

0.3171*60 = 19.026 min

Therefore, using simple mathematical operations we know that 19.03 min will be required to make 88 pretzels by 5 makers.

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Complete question:

If 2 pretzel makers can make 444 pretzels in 6 hours, how long does it take 5 pretzel makers to make 88 pretzels?

The solution of the given system of equation is x = 6 and y = -6.

The places where the lines representing the intersection of two linear equations intersect are referred to as the solution of a linear equation. In other words, the set of all feasible values for the variables that satisfy the specified linear equation is the solution set of the system of linear equations.

The two equations are x + y = 0 and 5x + 4y = 6.

The first equation can be written as:

x = - y

Substituting the value of x in equation 2 we have:

5(-y) + 4y = 6

-5y + 4y = 6

-y = 6

y = -6

Substitute the value of y in equation 1:

x - 6 = 0

x = 6

Hence, the solution of the given system of equation is x = 6 and y = -6.

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Answer:

d = 523.8 miles

Step-by-step explanation:

law of cosines

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Step-by-step explanation:

The drawing attached represents the question. We create a triangle with the distances from each pair of cities, and we call the distance from B to C by 'd'.

First, we need to find the angle BAC:

55° + BAC + 7° = 90°

BAC = 28°

Then, we can use the law of cosines to find the value of d:

d^2 = 470^2 + 890^2 - 2*470*890*cos(BAC)

d^2 = 470^2 + 890^2 - 2*470*890*0.8829

d^2 = 274365.86

d = 523.8 miles

Player A has a smaller IQR and range, indicating less variability in their scores, therefore the correct option is: Player A is the most consistent, with an IQR of 1.5.

To determine the best measure of variability for the data, we need to consider the nature of the data and what we want to measure. Since the data is quantitative and consists of individual values, measures like range, interquartile range (IQR), and standard deviation (SD) are commonly used.

The range is the difference between the highest and lowest values in the data. The IQR is the difference between the 75th and 25th percentiles of the data. The SD measures the average distance of the values from the mean.

For Player A:

Range = 8 - 1 = 7

IQR = Q3 - Q1 = 3 - 1.5 = 1.5

SD = 1.96 (approximate)

For Player B:

Range = 6 - 1 = 5

IQR = Q3 - Q1 = 4 - 1.5 = 2.5

SD = 1.61 (approximate)

Based on these measures, we can see that Player A has a smaller IQR and range, indicating less variability in their scores, while Player B has a larger IQR and range, indicating more variability. Therefore, Player A is the more consistent player.

So the correct option is: Player A is the most consistent, with an IQR of 1.5.

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Using matrix method solve + 2 = 3 5 − = 7. the solution to the system of equations is x = 2 and y = 1.

To solve the system of equations:

x + 2y = 3

5x - 2y = 7

using matrix methods, we can represent the system as a matrix equation:

AX = B

where

A = [1 2; 5 -2]

X = [x; y]

B = [3; 7]

To solve for X, we need to find the inverse of A:

A^-1 = 1 / det(A) * adj(A)

where det(A) is the determinant of A and adj(A) is the adjugate of A.

det(A) = (1 * -2) - (2 * 5) = -12

adj(A) = [-2 -2; -5 1]

So, A^-1 = -1/12 * [-2 -2; -5 1] = [1/6 1/6; 5/6 -1/6]

Now, we can solve for X:

X = A^-1 * B = [1/6 1/6; 5/6 -1/6] * [3; 7] = [2; 1]

Therefore, the solution to the system of equations is x = 2 and y = 1.

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The portion of the Earth that you can see from the hot air balloon is approximately 68.1 miles.

What is horizon ?

The horizon is the apparent boundary where the Earth's surface seems to meet the sky. It's the line that separates the visible portion of the Earth's surface from the portion that is not visible due to the curvature of the Earth. The horizon appears to be a straight line, but it is actually a curved line due to the spherical shape of the Earth. The distance to the horizon depends on the observer's height above the ground, as well as the radius of the Earth. The higher the observer, the farther away the horizon appears.

The distance from the balloon to the horizon is the same as the radius of the Earth plus the height of the balloon. So in this case, it's 1.5 + 4000 = 4001.5 miles.

Now we need to find the distance AC, which is the distance from the center of the Earth to point C. To do this, we can use the Pythagorean theorem:

AC² + BC² = (radius of Earth)²

AC² + (4001.5)² = (4000)²

AC² = (4000)² - (4001.5)²

AC ≈ 68.2 miles

Now we can find the distance BD using similar triangles. In the triangle ABD, we know that AB is the radius of the Earth, which is 4000 miles. We also know that AC is 68.2 miles, and we want to find BD. So we can set up the following proportion:

BD / AB = AC / AD

Solving for BD, we get:

BD = AB * (AC / AD)

To find AD, we can use the Pythagorean theorem again:

AD² = AB²+ BD²

AD² = (4000)²+ (BD)²

AD ≈ 4000.3 miles

Now we can plug in the values we have:

BD = AB * (AC / AD)

BD = 4000 * (68.2 / 4000.3)

BD ≈ 68.1 miles

So the portion of the Earth that you can see from the hot air balloon is approximately 68.1 miles. Rounded to the nearest tenth, the answer is 68.1 miles.

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Answer:

Step-by-step explanation:

this is a good aswer