Consider. F (x) = log2 x
Graph g(x) =log2(x-3) on the graph

The given triangles GEF and HKJ are congruent by the AAA test of congruency and the value of a = √2.

The two triangles are proved to be congruent and the value of a is to be determined.

Both triangles are right-angled triangles.

We know that similar triangles are those triangles that have similar properties,i.e. angles and proportionality of sides.

In triangle GEF,

since the sum of the angles in a triangle is 180

∠ G + ∠ F + ∠ E =180

90 + 45 + E = 180

∠ E = 45

similarly, ∠ J = 45

In Triangle GEF and Triangle HKJ,

∠ G = ∠ K (both are 90)

∠  F = ∠ H  (both are 45)

∠ E = ∠ J  (both are 45)

So, Triangle GEF and Triangle HKJ are congruent triangles by AAA congruency.

Now in triangle GEF,

cos 45 = 3a/6

1/√2 = a/2

a = √2

Thus, the given triangles are congruent by AAA . and the value of a = √2.

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The complete question is -

Given right triangles EFG and HJK. find the value of a that satisfies the congruency of the triangles. in two or more complete sentences, explain the relevance between the congruency of the triangles and the calculations made in finding the value of a.

The algebraic expression -

{ (4x² - 9y²)³ + (9y² - 16y²)³ + (16z² - 4x²)³ } / { (2x - 3y)³ + (3y - 4z)³ + (4z - 2x)³ } is simplified to give

The given algebraic expression is in fraction and we solve by breaking the fraction in to two: numerator and denominator and solve separately then bring them together.

The numerator

= (4x² - 9y²)³ + (9y² - 16y²)³ + (16z² - 4x²)³

bringing the powers together

= (4x² - 9y² + 9y² - 16y² + 16z² - 4x²)³

collecting like terms together

= (4x² - 4x² - 9y² + 9y² - 16y² + 16z² )³

= (0x² - 16y² + 16z² )³

= (16y² + 16z² )³

The denominator

= (2x - 3y)³ + (3y - 4z)³ + (4z - 2x)³

bringing the powers together

= (2x - 3y + 3y - 4z + 4z - 2x)³

collecting like terms

= (2x - 2x - 3y + 3y - 4z + 4z )³

= (0x - 0y - 0z )³

= 0

with a denominator of zero the solution is undefined, this may mean infinite many solutions

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Answer:

[tex]a=5.1[/tex]

Step-by-step explanation:

[tex]2.4x-1.2x-3.9y+ay=1.2x+y(a-3.9) \\ \\ \implies 1.2=a-3.9 \implies a=5.1[/tex]

The numerator is 3 and the denominator is 5. IN order to determine the reciprocal 4 becomes the denominator and 5 becomes the numerator. The reciprocal is 5/4.

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Answer:

d= Deondre's age

d + 3

c = the number of people in the class

1/2c

w = wildcat's score

a = away team's score

w = 2a + 7

c = cost of the sweater

s = the regular cost of the sweater

c = 1/2s

y = total in the bank

x = number of weeks

y = 75x + 300

y = altitude

x = number of minutes

y = 200x + 1000

Step-by-step explanation:

The number of blue tickets is 50 tickets. The result is obtained by using the concept of operation with fractions.

In a hat, we have:

Find the number of blue tickets!

Let's say all tickets is a. Then,

The remaining section (pink tickets) is

= 1 - (green + white + blue) tickets

= 1 - (1/10 + 1/2 + 1/4)

= 1 - (2/20 + 10/20 + 5/20)

= 20/20 - 17/20

= 3/20

All tickets are

3/20 = 30/a

a = (20 × 30)/3

a = 20 × 10

a = 200 tickets

The blue tickets are

= 1/4 a

= 1/4 (200)

= 50 tickets

Hence, the blue tickets in the hat are 50 tickets.

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The LCD (Least Common Denominator) of 10t2 is 10t2. To determine the equivalent expression is the sum of the fractions is 8t2.

A/B = C/D

Where A and B are the two fractions, and C and D are the equivalent fractions.

We will use this formula to determine the equivalent expression of the given fractions:

3t/20 = C/10t2

We can solve for C by multiplying both sides by 10t2, yielding:

30t2/20 = C/10t2

We can solve for C by multiplying both sides by 10t2, yielding:

30t2/20 = C/10t2

We can then simplify by cancelling out the 10t2 on the denominator, yielding:

C = 30t2/20

The equivalent expression for the given fractions is therefore 30t2/20.

To find the sum of the fractions, we first need to convert all fractions to equivalent fractions with a common denominator. In this case, the LCD is 10t2.

The equivalent expressions for the fractions are therefore:

3t/20 + 5/10t2 = 30t2/20 + 50/10t2

The sum of the fractions is then the sum of the numerators divided by the denominator, or:

30t2/20 + 50/10t2 = 80t2/10t2

We can simplify this expression by cancelling out the 10t2 on the denominator, yielding:

80t2/10t2 = 8t2

Therefore, the sum of the fractions is 8t2.

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The sum of 14t/10t^2 +  (5t^2 + 40t)/10t^2 is (5t^2 + 54t)/10t^2

In this question we have been given that Elias is adding 7t/5t^2 and (t + 8)/2t

He finds the least common divisor (LCD) of 7t/5t^2 and (t + 8)/2t which 10t^2

He uses multiplication to find an equivalent expression.

7t/5t^2 = 7t/5t^2 * 2/2

            = 14t/10t^2

and (t + 8)/2t = ((t + 8) * 5t)/(2t * 5t)

                     = (5t^2 + 40t)/10t^2

We need to find the sum 14t/10t^2 +  (5t^2 + 40t)/10t^2

We know that if the base of two rational expressions a/b and c/b then the denominator of the result is same and we just add numerators i.e., a/b + c/b = (a + c)/b

So, 14t/10t^2 +  (5t^2 + 40t)/10t^2

= [14t + (5t^2 + 40t)]/10t^2

= (5t^2 + 54t)/10t^2

Hence, 14t/10t^2 +  (5t^2 + 40t)/10t^2 = (5t^2 + 54t)/10t^2

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The 3rd graph has a relationships which is a function as a vertical can be drawn through any point and it only intersects once.

A function is a particular kind of relationship in which inputs are mapped to just one output. Consider a function as a machine for useful insight. The domain of the function, which describes the types of input that the machine will accept, states that it will produce one and only one output for each input.

A function's fundamental notation is of the form f ( x), where x is the input variable and f is the function's name. "F of X" should be read as this. Any time you see this notation, f(x) generally refers to a function's output, though these letters are subject to change.

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Loosen monetary policy causes interest rates to decrease and consumption and investment to increase.

When a central bank, such as the Federal Reserve in the United States, loosens monetary policy, it is typically trying to stimulate economic growth by making borrowing cheaper for individuals and businesses. One of the ways it can do this is by lowering interest rates.

When interest rates decrease, it becomes less expensive for individuals and businesses to borrow money, which can encourage them to spend more on things like homes, cars, and business expansion. This increased spending can lead to a boost in economic growth, as consumer and business spending are two major drivers of economic activity.

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Answer:

Step-by-step explanation:

Write k for the divisor.

Write the coefficients of the dividend.

Bring the leading coefficient down.

Multiply the leading coefficient by k.

Add the terms of the second column.

Multiply the result by k.

Repeat steps 5 and 6 for the remaining columns.

For each part of the circle in the table, provide a description of where you see that part in the pizza is given below.

A circle is a shape that has no sides or corners and is made up of only one curved line, which forms a perfect loop. It is the most basic and fundamental shape in geometry, and is used to create many other shapes, such as squares, rectangles, and triangles. It is also an important symbol in many cultures and religions, representing the cycle of life, eternity, and unity.

1. Arc: The arc is the curved edge of the pizza, starting at one end and ending at the other. This part can be seen along the outer circumference of the pizza.

2. Diameter: The diameter is the length of a straight line that passes through the center of the pizza and connects two opposite points on the circumference. It is the longest line that can be drawn on the pizza.

3. Radius: The radius is the distance from the center of the pizza to any point on the circumference. It is half the length of the diameter.

4. Chord: A chord is a straight line that connects two points on the circumference of the pizza.

5. Sector: A sector is a section of the pizza that is created when two radii are drawn from the center of the pizza to two points on the circumference.

6. Segment: A segment is a section of the pizza created when a chord is drawn between two points on the circumference.

7. Central Angle: A central angle is an angle formed by two radii drawn from the center of the pizza to two points on the circumference.

8. Circumference: The circumference is the outermost edge of the pizza. It is the line that forms the boundary between the inside and outside of the pizza.

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If and only if m is a divisor of 76, then there exists a full m-ary tree with 76 leaves and height 3. 1, 2, 4, 19, 38, and 76 are the divisors of the number 76.

One node at the root, 76 child nodes at the second level, and [tex]76^{2}[/tex] = 5776 child nodes at the third level make up the 5776 + 76 = 5852 nodes of a full 1-ary tree with 76 leaves and height 3. As a result, there is no whole 1-ary tree with 76 leaves and a height of 3.

 There would be one node at the root, 38 child nodes at the second level, and [tex]38^{2}[/tex] = 1444 child nodes at the third level, making a total of 1444 + 38 = 1482 nodes in a full 2-ary tree with 76 leaves and height 3. As a result, there is a full 2-ary tree that is 3 in height and has 76 leaves.

 There would be one node at the root, 19 child nodes at the second level, and [tex]19^{2}[/tex] = 361 child nodes at the third level, for a total of 361 + 19 = 380 nodes in a full 4-ary tree with 76 leaves and height 3. As a result, there is a full 4-ary tree that is 3 in height and has 76 leaves.

A full 19-ary tree with 76 leaves and height 3 would have a single node at the root, four children at the second level, and four and a half children, or 16 at the third level, for a total of 16 + four and a half and twenty nodes. As a result, there is a full 19-ary tree that is 3 in height and has 76 leaves.

 A full 38-ary tree would contain a single node at the root, two child nodes at the second level, and two and a half child nodes at the third level, for a total of four + two + six nodes. Consequently, there is a complete 38-ary tree that is 3 in height and has 76 leaves.

 There would be one node at the root, one child node at the second level, and [tex]1^{2}[/tex]= one child node at the third level, for a total of 1 + 1 = 2 nodes in a full 76-ary tree with 76 leaves and height 3. Consequently, there is a full 76-ary tree with 76 leaves and a height of 3.

  As a result, for m = 2, 4, 19, 38, and 76, a full m-ary tree with 76 leaves and height 3 exists.

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If functions f(x) = 2x² + 4x - 5, g(x) = - 6x³+2x² - 3 then (f + g)(x)=6x³+ 4x² +4x-8

A relation is a function if it has only One y-value for each x-value.

The given two functions are f(x) = 2x² + 4x - 5

and g(x) = - 6x³+2x² - 3

We need to find  (f + g)(x).

(f + g)(x)=f(x)+g(x)

=2x² + 4x - 5 - 6x³+ 2x² -3

Now add the like terms

= 6x³+ 4x² +4x-8

(f + g)(x)=6x³+ 4x² +4x-8

Hence, if functions f(x) = 2x² + 4x - 5, g(x) = - 6x³+2x² - 3 then (f + g)(x)=6x³+ 4x² +4x-8

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A) A 3 × 4 matrix can have independent columns but not independent rows.

B) A 3 × 4 matrix can have independent columns but not independent rows.

C) If A is an mxn matrix and rank A = m, it follows that m ≤ n.

D) A nonsquare matrix can have its rows independent.

E) The null space of a 3 × 6 matrix cannot have dimension 2.

F) If A is 5x4 and null(A) = x for some column x ≠ 0, dim(im A) cannot be 2.

A) A 3 × 4 matrix can have independent columns but not independent rows. This is because a column vector can be linearly independent, meaning no linear combination of other columns in the matrix can create it, but a row vector can't be linearly independent because it can always be written as a linear combination of the columns.

B) If A is 4 × 3 and rank A = 2, it cannot have independent columns but it can have independent rows. This is because if rank A = 2, the number of linearly independent columns is 2, which is less than the number of columns, meaning that not all columns can be linearly independent. However, since rank A = 2, the number of linearly independent rows is also 2, which is less than the number of rows, meaning that some rows can be linearly independent.

C) If A is an mxn matrix and rank A = m, it follows that m ≤ n. This is because the rank of a matrix is the maximum number of linearly independent columns, which cannot be greater than the number of columns.

D) A nonsquare matrix can have its rows independent and its columns independent, as long as the number of rows is not equal to the number of columns.

E) The null space of a 3 × 6 matrix cannot have dimension 2. This is because the dimension of the null space is always less than or equal to the number of columns of the matrix, which is 6 in this case.

F) If A is 5x4 and null(A) = x for some column x ≠ 0, dim(im A) cannot be 2. This is because the dimension of the image of A is equal to the rank of A, which cannot be less than the dimension of the null space. Therefore, dim(im A) must be greater than or equal to the dimension of the null space, which is at least 1.

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Answer:

Below in bold.

Step-by-step explanation:

I have assumed that it's + 24.

a. This is a cubic expression (highest power is 3).

By the theorem there are 3 roots with 2 possible complex roots.

b. f(x) = 3x^3-7x^2-14x+ 24

Using Descartes Rule of signs, the signs from left to right are:

+ - - +

Here there are 2 sign changes (+ - and - +)

So, the maximum number of positive real roots is 2.

f(-x)= 3(-x)^3 - 7(-x)^2 - 14(-x) + 24

Sign changes are:

- - + +

There is one change of sign so maximum number of possible negative real roots is 1.

c. Using the Rational roots theorem, the list of all possible rational roots is:

+/- (factors of 24 / factors of 3)

= +/- [1, 2, 3, 4, 6, 8, 12, 24) / (1, 3) ]

= +/- 1, +/-2, +/-3, +/- 4, +/-6, +/-12, +/- 8, +/-24, +/- 1/3, +/-2/3, +/-4/3, +/-8/3

d. Possible roots, using the Remainder theorem:

x = 1:

f(1) = 3 - 7 - 14 + 24 = 8 so 1 is not a root

f(-1) = 28 - so x = -1 not a root

f(2) = -8 so 2 not a root

f(-2) = 0 So, x = -2 is a root

So, x + 2 is a factor of f(x) so we divide

x + 2)3x^3 - 7x^2 - 14x + 24(3x^2 - 13x + 12     <--- Quotient

        3x^3 + 6x^2

                 - 13x^2 - 14x

                 - 13x^2 - 26x

                                 12x + 24

                                 12x + 24

                                  ..............

So, we solve the following equation to find the other 2 roots:

3x^2 - 13x + 12 = 0

3x^2 - 9x - 4x + 12 = 0

3x(x - 3) - 4(x - 3) = 0

3x - 4 = 0 or x - 3 = 0

x = 4/3, 3.

The 3 roots are -2, 4/3, 3.

The equation giving the boiling point b of the​ liquid, in degrees​ Fahrenheit, in terms of altitude​ a, in feet is b = -0.0016a + 210

The boiling point of the liquid at 2300 ​ft is 206.32°F

From the question, we have the following parameters that can be used in our computation:

These parameters mean that

(a, b) = (8100, 197.04) and (4500, 202.8)

The relationship is a linear equation

So, we have

slope = (b2 - b1)/(a2 - a1)

Substitute the known values in the above equation, so, we have the following representation

slope = (202.8 - 197.04)/(4500 - 8100)

Evaluate

slope = -0.0016

The equation is then represented as

b = slope * a + c

Using one of the points, we have

197.04 = -0.0016 * 8100 + c

So, we have

197.04 = -12.96 + c

Evaluate

c = 210

This gives

b = -0.0016a + 210

This means that

a = 2300

So, we have

b = -0.0016 *  2300 + 210

Evaluate

b = 206.32

Hence, the boiling point is 206.32°F

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Answer:38

Step-by-step explanation: it is 180 degrease because it is spushed so (180-94)-50=38

Answer:

36⁰

Step-by-step explanation:

Angles in a triangle add upto 180⁰

94⁰+ 50⁰+b= 180⁰

144⁰+b=180⁰

b=180⁰-144⁰

b=36⁰

The factorization of the quadratic equation:

x^2 + 3x - 54

Is (x + 9)*(x - 6).

To factorize the quadratic equation we first need to find the roots:

x^2 + 3x - 54 = 0

Using the quadratic formula we will get:

[tex]x = \frac{-3 \pm \sqrt{3^2 - 4*1*(-54)} }{2} \\\\x = \frac{-3 \pm15 }{2}[/tex]

So the two roots are:

Then the factorized form of the quadratic equation is:

(x - (-9))*(x - 6)

(x + 9)*(x - 6)

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An equivalent expression is an expression that is equal in value to a given expression. For example, consider the following expression: 4x + 2y

An equivalent expression is an expression that represents the same value as a given expression, regardless of the specific values of any variables that appear in the expression. Equivalent expressions can be obtained by applying the rules of algebra, such as the associative, commutative, and distributive properties.

This expression is equivalent to the following expressions:

2(2x + y)

2y + 4x

(4 + 2)x + 2y

All of these expressions are equivalent because they all represent the same value, regardless of the specific values of x and y. They can all be simplified to the same expression by applying the rules of the algebra.

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Answer:

C = 2πr, where C is the circumference, π is the mathematical constant pi (approx. 3.14), and r is the radius (9 inches). Therefore, the circumference of the basketball hoop is 2π(9) = 18π inches.

Step-by-step explanation:

In other words use C=2*pi*radius to find the circumference, here you are given the radius is 9 so when you plug that in it gives you a circumfrence of 18pi inches. Hope this helps!

Answer:

9

Step-by-step explanation:

the square of 3 is 9

3 + 3 + 3 = 9

Step-by-step explanation:

fill the data in the expression

(14)-7+5(4)

27

1.
In the triangle that contains x (call the other 2 angles as a and b, where a is the angle that is supplemetary to the center angle.)

a = 180 - 110 = 70 (supplemetary angles add up to 180)

b = 30 (b and the angle measures 30 are in the same segment.)

-> x = 180 - 70 - 30 = 80.

2. In triangle PRT, v = 180 - 40 - 70 = 70.

Since P,Q,S,T all lies on the circumference, PQST is a cyclic quadrilateral.

-> w = 180 - PQS = 180 - (180 - 70) = 70.

3.
i) A 1:2 scale means that all sides of P must be shortened by twice the lengths.

Since D is the only shape with the sides shortened from P, it's the correct option.

ii) A 1:1/2 scale means that all sides of P must be enlarged by twice the lengths.

Therefore, the measures of the sides become sqrt(20),6,4,6 (clockwise from diagonal), which is B.

4. Using the Pythagoras' theorem, we can find the hypotenuse PR : [tex]\sqrt{12^{2} + 5^{2} } = 13[/tex]

sin(x) = QP/PR = 5/13 = 0.385 (rounded to 3 d.p)
tan(y) = QP/QR = 5/12 = 0.417 (rounded to 3 d.p)

The remaining two sides of the triangular piece are 360 cm each.

Evan cut a triangular piece of cloth to use in a quilt. The perimeter of the cloth is 934 cm. The base of the triangular cloth is 214cm. The remaining two sides are the same length.

Therefore, the triangular piece is an isosceles triangle. This means the legs are equal.

Hence,

2x + 214 = 934

where

Therefore,

2x + 214 = 934

subtract 214 from both sides of the equation

2x + 214 - 214 = 934 - 214

2x = 720

divide both side by 2

x = 720 / 2

x = 360

Hence, the sides of the triangle are 360 cm each.

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Answer:

Step-by-step explanation:

Answer:

go right 2 up 1 starting at (0,0)

Step-by-step explanation:

Answer:

so here you'll get the formula

Slope (m) = y2 - y1 / x2 -x1

answer will be 2

correct if I'm wrong

The vertex of the translated quadratic equation is:

(6, -4)

We know that we have a quadratic function f(x), such that the vertex is at (4, -4).

Here we have the translated function:

g(x) = f(x - 2)

Notice that this is a translation of 2 units to the right

Now, remember that the vertex of f(x) is at f(4), then the vertex of g(x) is at the value of x such that:

x - 2 = 4

x = 4 + 2

x = 6

So the vertex of g(x) is (6, -4)

(the y-value of the vertex does not change).

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Answer: The best I could come up with is √65, −√65 or, 1, you pick!

(Decimal: 8.06225774…, −8.06225774…)

Step-by-step explanation:

Rewrite the equation as

x^2 = 8^2 + 1^2.

One to any power is one.

x^2 = 8^2 + 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

x = ± √8^2 + 1

x = ± √64 + 1

Add 64 and 1.

x = ± √65

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the ± to find the first solution.

x = √65

Next, use the negative value of the ± to find the second solution.

x = −√65

The complete solution is the result of both the positive and negative portions of the solution.

x = √65, −√65

The result can be shown in multiple forms.

Exact Form:

x = √65, −√65

Decimal Form:

x = 8.06225774..., −8.06225774…

Answer:

----------------------------

Given a right triangle with the missing hypotenuse.

Use Pythagorean to find the missing side: