If a1 = 6 and an =an-1 -4 then find the value of a4.

The function T: ℝ[x] → ℝ₂[x] given by T(a + bx) = 2a + (a+b)x + (a−b)x² is a linear transformation. It is injective but not surjective. The basis for the range of T is {2, x, x²}. T is not an isomorphism. The nullity of T is 0. The vector spaces ℝ, [x], and ℝ₂[x] are not isomorphic.

To prove that T is a linear transformation, we need to show that it satisfies two properties: additive and scalar multiplication preservation. Let's consider two polynomials, p = a₁ + b₁x and q = a₂ + b₂x, and a scalar c ∈ ℝ. We have:

T(p + cq) = T((a₁ + b₁x) + c(a₂ + b₂x))

= T((a₁ + ca₂) + (b₁ + cb₂)x)

= 2(a₁ + ca₂) + (a₁ + ca₂ + b₁ + cb₂)x + (a₁ + ca₂ - b₁ - cb₂)x²

= (2a₁ + a₁ + b₁)x² + (a₁ + ca₂ + b₁ + cb₂)x + 2a₁ + 2ca₂

Expanding and simplifying, we can rewrite this as:

= (2a₁ + a₁ + b₁)x² + (a₁ + b₁)x + 2a₁ + ca₂

= 2(a₁ + b₁)x² + (a₁ + b₁)x + 2a₁ + ca₂

= T(a₁ + b₁x) + cT(a₂ + b₂x)

= T(p) + cT(q)

Thus, T preserves addition and scalar multiplication, making it a linear transformation.

Next, we determine if T is injective. For T to be injective, every distinct input must map to a distinct output. If we set T(a + bx) = T(c + dx), we get:

2a + (a + b)x + (a − b)x² = 2c + (c + d)x + (c − d)x²

Comparing coefficients, we have a = c, a + b = c + d, and a − b = c − d. From the first equation, we have a = c. Substituting this into the second and third equations, we get b = d. Therefore, the only way for T(a + bx) = T(c + dx) is if a = c and b = d. Thus, T is injective.

However, T is not surjective since the range of T is the span of {2, x, x²}, which means not all polynomials in ℝ₂[x] can be reached.

The basis for the range o................f T can be determined by finding the linearly independent vectors in the range. We can rewrite T(a + bx) as:

T(a + bx) = 2a + ax + bx + (a − b)x²

= (2a + a − b) + (b)x + (a − b)x²

From this, we can see that the range of T consists of polynomials of the form c + dx + ex², where c = 2a + a − b, d = b, and e = a − b. The basis for this range is {2, x, x²}.

Since T is injective but not surjective, it cannot be an isomorphism. An isomorphism is a bijective linear transformation.

The nullity of T refers to the dimension of the null space, which is the set of all inputs that map to the zero vector in the range. In this case, the nullity of T is 0 because there are no inputs in ℝ[x] that map to the zero vector in ℝ₂[x].

Finally, the vector spaces ℝ, [x], and ℝ₂[x] are not isomorphic. The isomorphism between vector spaces preserves the structure, and in this case, the dimensions of the vector spaces are different (1, 1, and 2, respectively), which means they cannot be isomorphic.

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13 is the value of z from the given figure.

The properties of the triangle are:

From the given figure,

∠ABE + ∠EBC =. 180

∠EBC = 180- ∠ABE

∠EBC = 180 - 9z -----(1)

∠ECB + 115 = 180

∠ECB = 65-----(2)

IN ∆EBC

∠BEC + ∠ECB + ∠EBC = 180

4Z + 180- 9Z + 65 = 180 ---------(from eq i & ii)

-5Z = -65

Z = -65/-5y

Z = 13

The value of z from the given figure is 13.

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Complete question:

Answer:

Below.

Step-by-step explanation:

The volume of a sphere = 4/3 π r^3    where r is the radius.

Here the radius = 1/2 * 165 = 82.5 ft.

So the volume  = 4/3 π (82.5)^3

= 2,352,071.15 ft^3.

The volume of the sphere with the given diameter is closest to 2,351,000.

A sphere is a three dimensional figure which consists of set of all the points which are at an equal distance from a point called the center of the sphere.

Given that,

Diameter of the given sphere in the amusement park = 165 feet

Radius is half the diameter.

Radius of the sphere = 165/2 = 82.5 feet

Volume of the sphere = \(\frac{4}{3}\) π r³

                                     = \(\frac{4}{3}\) π (82.5)³

                                     = 748687.5π

                                     = 2352071.15 feet³

The most closest one in the option is 2,351,000.

Hence the volume of the sphere is closest to 2,351,000.

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The vertex form of the quadratic equation is f(x) = -3(x - 2)² + 5. The coordinates of the vertex are (2, 5).

We are given a quadratic function. The equation of the quadratic function is f(x) = -3x² + 12x - 7. We need to convert the equation to vertex form. We can convert a quadratic equation to its standard vertex form by "completing the square" method. The calculations are done below.

f(x) = -3x² + 12x - 7

f(x) = -3(x² - 4x) - 7

f(x) = -3(x² - 4x + 2² - 2²) - 7

f(x) = -3(x² - 4x + 2²) - 7 + 3*2²

f(x) = -3(x - 2)² - 7 + 12

f(x) = -3(x - 2)² + 5

The coordinates of the vertex are (2, 5).

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The Coffee Counter should use 7 pounds of Kenyan French Roast coffee and 13 pounds of Sumatran coffee to make a 20 pound blend that sells for $6.35 per pound By using linear equation in one variable

Let the amount of Kenyan French Roast coffee used be x. Then the amount of Sumatran coffee used would be 20 - xWe can use the following equations to form a system of linear equations:7x + 6(20 - x) = 20(6.35)7x + 120 - 6x = 12707x - 6x = 127 - 120x = 7The Coffee Counter should use 7 pounds of Kenyan French Roast coffee and 13 pounds of Sumatran coffee to make a 20 pound blend that sells for $6.35 per pound.

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Answer:

oop

Step-by-step explanation:

Answer:

It's a square

Step-by-step explanation:

Because a square has all equal sides, and its an quadrilateral, and we know its square bc the other shapes dont really have that.

The square is also the name of the regular quadrilateral — one in which all sides are congruent and all angles are congruent.

Answer:

0.3143

Step-by-step explanation:

In probability:

AND = multiplication

  OR = addition

There are 15 balls in total. However, when the first ball is drawn, it is NOT replaced

‘None of the balls drawn are yellow’ means we will be dealing with red and/or pink balls only.

One ball is RED AND other ball is PINK OR One ball is RED AND other ball is RED OR One ball is PINK AND other ball is PINK:

\(= \frac{4}{15} .\frac{6}{14} +\frac{4}{15} .\frac{3}{14} + \frac{6}{15} .\frac{5}{14}\)

\(=\frac{24}{210}+\frac{12}{210} +\frac{30}{210}\)

All the three fractions have the same denominator, therefore the resulting or simplified fraction will have the same denominator. In addition, everything written in the numerator level will be copied down as it is:

\(=\frac{24+12+30}{210}\)

\(= \frac{66}{210}\)

Divide both the numerator and denominator by the Highest Common Factor ‘6’:

\(= \frac{11}{35}\)

= 0.3143

Answer:

irrational

Step-by-step explanation:

well if it goes on forever its irrational

Answer:

it's ok I will do my best to help you

The actual area of the closet's floor is 3.52 square meters.

To find the actual area of the closet's floor, we first need to convert the dimensions from the scale drawing to their actual dimensions.

According to the given scale, 2 cm on the scale drawing represents 1 meter in reality. Thus, to convert the dimensions from the scale drawing to actual dimensions, we need to multiply each dimension by the conversion factor:

Actual length = 4.4 cm × (1 m / 2 cm) = 2.2 m

Actual width = 3.2 cm × (1 m / 2 cm) = 1.6 m

Now that we have the actual dimensions, we can calculate the actual area of the closet's floor:

Actual area = Actual length × Actual width

= 2.2 m × 1.6 m

= 3.52 square meters

Therefore, the actual area of the closet's floor is 3.52 square meters.

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Answer:

m=55

Step-by-step explanation:

Since one line equals 180 degrees, 180-125=55

The effective annual yield (EAY) is 2.83%

The effective annual yield (EAY) is the true annual interest rate that an investor earns after taking into account the effect of compounding. To calculate the EAY, you need to find the interest rate that will make the formula (1 + r/n)^n = final balance/initial balance hold true, where r is the interest rate and n is the number of times compounding occurs in a year.

Given the initial balance of $2,000 and a final balance of $2,056.72 after one year, we can use the following formula to calculate the EAY:

(1 + r/n)^n = $2,056.72/$2,000

Since the interest rate is given as 2.80%, and the number of times compounding occurs in a year is not given, we'll have to assume n. A common assumption for n is 12, which represents monthly compounding. Plugging in the numbers, we have:

(1 + 0.028/12)^12 = $2,056.72/$2,000

Solving for r, we get:

r = 12 * (($2,056.72/$2,000)^(1/12) - 1)

r = 2.83%

So the effective annual yield (EAY) is 2.83%.

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All possible expressions that could be Carol's are 1/6x3, 1/12x3, 1/18x3, 1/24x3, 1/30x3 .

Alice, Bob and Carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of x as the denominator. The simplest common denominator of Alice's and Bob's expressions is 4x2, while the simplest common denominator of Bob's and Carol's expressions is 12x3. The simplest common denominator of Alice's and Carol's expressions is 6x3. To explain further, Alice's expression can be any fraction with 1 as the numerator and a constant integer times x2 as the denominator, therefore, the denominator of Alice's expression can be any multiple of 4, i.e. 4x2, 8x2, 12x2, 16x2, 20x2 and so on.

Bob's expression can also be any fraction with 1 as the numerator and a constant integer times x2 as the denominator. However, the denominator of Bob's expression must be a multiple of both 4 and 6, i.e. 12x2, 24x2, 36x2, 48x2 and so on. Therefore, Carol's expression must be in the form of a fraction with 1 as the numerator and a constant integer times x3 as the denominator. This means that the denominator of Carol's expression must be a multiple of 6, i.e. 6x3, 12x3, 18x3, 24x3, 30x3 and so on. Hence, all possible expressions that could be Carol's are 1/6x3, 1/12x3, 1/18x3, 1/24x3, 1/30x3 and so on.



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By using the vertex of the quadratic equation, we will get the equation:

y = -2*(x + 6)^2

For a quadratic equation with a leading coefficient "A" and a vertex (h, k), the equation can be written as:

y = A*(x - h)^2 + k

Here we can see that the vertex is (-6, 0), so we have:

y = A*(x + 6)^2

We also can see that the parabola passes through the point (-4, -8), replacing these values above we will get:

-8 = A*(-4 + 6)^2

-8 = A*(2)^2

-8 = 4A

-8/4 = A

-2 = A

Then the quadratic equation is:

y = -2*(x + 6)^2

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The evaluation of the expressions are shown below

To determine the solution to the expressions, we make use of laws of indices and exponents

Expression 1 (a) and (b)

(27)^(-4/3) ÷ (1/64)^(2/3) = (3^3)^(-4/3)  ÷ (1/4^3)^(2/3)

(27)^(-4/3) = (3)^(-4)  ÷ (1/4)^(2)

(27)^(-4/3) = 1/81 ÷ 1/16

So, we have

(27)^(-4/3) = 16/81

Next, we have

(32)^(2/5) - (-125)^(-1/3) = (2^5)^(2/5) - (-5^3)^(-1/3)

(32)^(2/5) - (-125)^(-1/3) = (2)^(2) - (-5)^(-1)

(32)^(2/5) - (-125)^(-1/3) = 4 + 1/5

(32)^(2/5) - (-125)^(-1/3) = 21/5

Equation (2)

Here, we have

3(8/27)^(-2/3) = 2^x/3^y

So, we have

3(3/2)^(2) = 2^x/3^y

So, we have

3^3/2^2 = 2^x/3^y

By comparison, we have

x = -2 and y = 3

Equation (3)

Here, we have

3^x(9^[1 - x]) = 27^[x + 2]

So, we have

3^x(3^2[1 - x]) = 27^[x + 2]

This gives

3^(x + 2[1 - x]) = 3^3[x + 2]

By comparison, we have

x + 2[1 - x] = 3[x + 2]

So, we have

x + 2 - 2x = 3x + 6

Evaluate

4x = -4

x = -1

Equation (4)

Here, we have

(5^x)^2x = 25(1/125)^x

So, we have

(5)^2x^2 = 5^[2 - 3x]

By comparison, we have

2x^2 = 2 - 3x

So, we have

2x^2 + 3x - 2 = 0

Evaluate

x = -2 and x = 1/2

Equation (5)

Here, we have

2^x+1 + 2^3-x = 10

Here, we make use of a graphing tool

So, we have

x = 0 and x = 2

Expression (6)

Here, we have

√16^x+2

Express 16 as 2^4

So, we have

√16^x+2 = √2^4[x+2]

Take the square roots

√16^x+2 = 2^2[x+2]

Open the bracket

√16^x+2 = 2^2x+4

Hence, the expression is 2^2x+4

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Answer: 1 tablespoon

Step-by-step explanation:

Answer:

The fraction is 3/10 = .3

Answer:

1. \(5^{7}\)

2. 300

3. 100

4. 1.24 x \(10^{7}\)

5. 28,700

according to my answer this is wrond because you are wrong

Step-by-step explanation:

palhschal pagal

Answer:

Answers in the picture. Thanks

To create a table for the line y=1-4x from x=-2 to x=2, we substitute in each x value into the equation, solve for y, and display these pairs in a table. The pairs are (-2, 9), (-1, 5), (0, 1), (1, -3), and (2, -7).

The equation of the line is y = 1 - 4x. To create a table showing the values of x and y where x is from -2 to 2, we need to substitute each value of x into the equation, solve for y, and then organize the results in a table.

So, the table representing these values would look like this:

x               y

-2             9

-1              5

0               1

1               -3

2              -7

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Answer:

The answer is

B.

Step-by-step explanation:

just multiply 4.5 by 8 and you get 36

The vertices of the triangle after the reflection over y = -x are \(\left[\begin{array}{ccc}3&-3&-3&-3&-6&-3\end{array}\right]\)

The vertices of the triangle are given as

\(\left[\begin{array}{ccc}3&6&3&-3&3&3\end{array}\right]\)

The transformation rule is given as

Reflection over the line y = -x

Mathematically, this rule is represented as

(x, y) → (-y, -x)

This means that we negate the x-coordinates and the y-coordinates, and then swap the x and the y coordinates

When this rule is applied, we have

\(\left[\begin{array}{ccc}3&-3&-3&-3&-6&-3\end{array}\right]\)

Hence, the reflection would produce \(\left[\begin{array}{ccc}3&-3&-3&-3&-6&-3\end{array}\right]\)

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Replace h with 34:

p(h) = 21(34)

Multiply :

21x 34 = 714

Answer: $714

The measures of the arcs and angles obtained using circle theorem are;

2.) a.) Arc AB = 50°

b.) Angle ACB = 25°

3.) a.) Arc SR = 102°

b.) Arc SQ = 213°

Circle theorems are theorems with regards to the relationship between the angles, segments, tangents, arcs and chords of a circle.

The measure of an arc is the measure of the angle it subtends at the center of a circle.

The angle ∠AOB is the angle subtended at the center of the circle by the arc AB

The measure of the angle. ∠AOB, m∠AOB = 50°

Therefore; Arc AB  = m∠AOB = 50°

b.) Circle theorem indicates that the angle subtended by an arc at the center of a circle, is twice the angle the arc subtends.

The angle subtended at the center is; m∠AOB = 50°

The angle m∠ACB subtended at the circumference = 50°/2 = 25°

Angle ACB = 25°

3.) The angle inscribed in the circle, m∠RQS = 51°

Therefore, the angle ∠RQS is the angle subtended by the arc RS on the circumference of the circle.

The measure of the arc SR is the angle subtended at the circle by the arc SR, therefore;

Arc SR; = 2 × (m∠RQS) = 2 × 51° = 102°

b.) The sum of the measure of the angles of the arcs around the circumference of s circle is 360°, therefore;

Arc SQ + Arc QR + Arc SQ = 360°

Arc SQ + Arc QR + Arc SR = 360°

The measure of the arc QR on the diagram is; ArcQR = 145°

Therefore; Arc SQ + 45° + 102° = 360°

Arc SQ = 360° - (45° + 102° ) = 213°

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Answer:

4 pounds and 13 ounces

Step-by-step explanation:

If you add the pounds you already have it equals 3. If you add the ounces, 12,2,and 15, it equals to 29. 12 ounces equal a pound meaning that 29-16 is equal to 13. So because you have 1 more pound, you add it to your whole pounds so 3+1=4 so now you have 4 pounds and 13 ounces.

Answer:

A) \(y = \frac{1}{10}x + 28\)

B) $116.5

Step-by-step explanation:

The data we are given is:

• for 440 minutes of usage, the total cost is $72

• for 940 minutes of usage, the total cost is $122

A)

In order to find an equation in the form \(y = mx + b\), we must first express the given data as coordinates in the form \((x, y)\).

We are told that \(x\) is the number of monthly minutes, and \(y\) is the total monthly fee. Therefore the first piece of data expressed as coordinates is (440, 72), and for the second piece of data, it is (940, 122).

Now that we have the coordinates, we can make an equation using the formula:

\(\boxed{\frac{y - y_1}{y_2 - y_1}= \frac{x - x_1}{x_2 - x_1}}\),

where \((x_1, y_1)\) is (440, 72) and \((x_2, y_2)\) is (940, 122).

Substituting the values into the formula, we get:

\(\frac{y - 72}{122 - 72}= \frac{x - 440}{940 - 440}\)

⇒ \(\frac{y - 72}{50} = \frac{x - 440}{500}\)

⇒ \(50 \times \frac{y - 72}{50} =50 \times \frac{x - 440}{500}\)       [Multiplying both sides of the equation by 50]

⇒ \(y - 72 = \frac{x - 440}{10}\)

⇒ \(10y - 720 = x - 440\)    [Multiplying both sides by 10]

⇒ \(10y = x +280\)            [Adding 720 to both sides]

⇒ \(y = \frac{1}{10}x + 28\)    (Answer)

B)

To find the total cost if 885 minutes are used, we have to replace \(x\) in the above equation with 885:

\(y = \frac{1}{10}(885) + 28\)

⇒ \(y = 88.5 + 28\)

⇒ \(y = 116.5\)    

Therefore, the total cost will be $116.5.   (Answer)

The population of interest is: A. all people in Emerald, Queensland.

The population of interest refers to the group of individuals that the research is trying to draw inferences about. In this case, the international fast food chain is looking to assess the level of interest in opening a franchise in Emerald, Queensland.

Therefore, the population of interest is all people who live in Emerald, Queensland, as they are the group that the research is trying to understand. The sample of 100 addresses that the research firm selected is just a subset of this population and the questionnaires are being sent to them to infer about the entire population.

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