What is the cost of 10. 0 gallons of gasoline priced at 1. 449 per gallon?

The final answer is: ∫(2x-1)÷[(x-3)(x+2)]dx = ln|x-3| + ln|x+2| + C

What is Integration ?

In calculus, integration is the inverse operation of differentiation. It is a mathematical technique used to find the integral of a function. The integral of a function f(x) is another function F(x), which gives the area under the curve of f(x) from a certain point to another.

To perform the integration of the given function:

∫(2x-1)÷([tex]x^{2}[/tex]-x-6)dx

First, we need to factor the denominator:

[tex]x^{2}[/tex]- x - 6 = (x-3)(x+2)

So we can rewrite the integral as:

∫(2x-1)÷[(x-3)(x+2)]dx

Next, we need to decompose the fraction into partial fractions:

(2x-1)÷[(x-3)(x+2)] = A÷(x-3) + B÷(x+2)

Multiplying both sides by (x-3)(x+2), we get:

2x-1 = A(x+2) + B(x-3)

Substituting x=3, we get:

5A = 5

A = 1

Substituting x=-2, we get:

-5B = -5

B = 1

So we have:

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

Substituting this back into the integral, we get:

∫(2x-1)÷[(x-3)(x+2)]dx = ∫[1÷(x-3) + 1÷(x+2)]dx

Using the first rule of integration, we get:

∫[1÷(x-3) + 1÷(x+2)]dx = ln|x-3| + ln|x+2| + C

where C is the constant of integration.

Therefore, the final answer is: ∫(2x-1)/[(x-3)(x+2)]dx = ln|x-3| + ln|x+2| + C

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[tex] \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - x - 6 } dx[/tex]

Answer:

[tex] \underline{\boxed{\rm = ln |x + 2| + ln |x - 3| + C}}[/tex]

Step-by-step explanation:

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - x - 6 } dx[/tex]

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ {x}^{2} - 3x + 2x - 6 } dx[/tex]

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ x(x - 3) + 2(x - 3) } dx[/tex]

[tex] = \displaystyle \int \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3)} dx[/tex]

[tex] \rm \: Let : \displaystyle \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3) } = \dfrac{A }{x + 2} + \dfrac{B}{x - 3} [/tex]

[tex]\rm\implies\displaystyle \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3) } = \dfrac{A(x - 3) + B(x + 2) }{(x + 2)(x - 3)} \\ [/tex]

[tex] \rm \implies\displaystyle \rm \: {2x - 1}{ } = {A(x - 3) + B(x + 2) } \\ [/tex]

Put x = 3 , we get

[tex] \rm \implies\displaystyle \rm \: {6 - 1}{ } = {A(3- 3) + B(3 + 2) } \\ [/tex]

[tex] \rm \implies\displaystyle \rm \: {5}{ } = 5 B \\ [/tex]

[tex] \implies \rm \: B = 1[/tex]

Again

put put x = -2

[tex] \rm \implies\displaystyle \rm \: { - 4- 1}{ } = {A( - 2- 3) + B( - 2 + 2) } \\ [/tex]

[tex] \rm \implies\displaystyle \rm \: { - 5}{ } = {A( - 5) } \\ [/tex]

[tex] \rm \implies\displaystyle \rm A = 1 \\ [/tex]

Thus ,

[tex] \displaystyle \int \rm \: \dfrac{2x - 1}{ (x + 2) (x - 3)} dx = \int\dfrac{1}{x + 2} dx + \int \dfrac{1}{x - 3} dx[/tex]

[tex] \rm = ln |x + 2| + ln |x - 3| + C[/tex]

[tex] \tt\int \dfrac{dx}{ {x}^{2} + {a}^{2} } = \frac{1}{a} { \tan}^{ - 1} \frac{x}{a} + c \\ [/tex]

[tex] \tt\int \dfrac{dx}{ {x}^{2} - {a}^{2} } = \frac{1}{2a} log \frac{x - a}{x + a} + c \\ [/tex]

[tex] \tt\int \dfrac{dx}{ {a}^{2} - {x}^{2} } = \frac{1}{2a} log \frac{a + x}{a - x} + c \\ [/tex]

[tex] \tt\int \: \dfrac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log|x + \sqrt{ {a}^{2} + {x}^{2} } | + c \\ [/tex]

[tex] \tt\int \: \dfrac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log|x + \sqrt{ {x}^{2} - {a}^{2} } | + c \\ [/tex]

[tex] \tt \int \: \dfrac{dx}{ {a}^{2} - {x}^{2} } = { \sin }^{ - 1} \bigg(\dfrac{x}{a} \bigg) + c \\ [/tex]

[tex] \tt \int \: \sqrt{ {x}^{2} + {a}^{2} } dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\= \tt \dfrac{x}{2} \sqrt{ {a}^{2} + {x}^{2} } + \dfrac{ {a}^{2} }{2} log |x + \sqrt{ {x}^{2} + {a}^{2} }| + c[/tex]

Answer:

102

Step-by-step explanation:

(b•h)/2

(12•17)/2

204/2

The coordinates of point Y are (10 - x1, 4y1 - 10), where (x1, y1) are the coordinates of point T.

If W is the midpoint of segment TY, then the coordinates of W are the average of the coordinates of T and Y. Using the midpoint formula, we can find the coordinates of Y:

Let the coordinates of T be (x1, y1) and the coordinates of Y be (x2, y2).

x-coordinate of W = (x-coordinate of T + x-coordinate of Y) / 2

y-coordinate of W = (y-coordinate of T + y-coordinate of Y) / 2

Putting in the coordinates of W and T, we get:  5 = (x1 + x2) / 2

y-coordinate of Y = 2y1 - y-coordinate of W

y-coordinate of Y = 2y1 - (y1 + y2) / 2

Simplifying these equations, we get:

x1 + x2 = 10

y2 = 4y1 - 10

From the first equation, we can solve for x2:  x2 = 10 - x1

Putting this into the second equation, we get:  y2 = 4y1 - 10

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Answer:

3.8

Step-by-step explanation:

using sinθ = opp/hypo

sin(25) = x/9

0.4226 = x/9

x = 9(0.4226) = 3.8

The constraints of inequalities are 3x + 4y ≤ 640 and 75x + 60y ≤ 12900

Represent the types of cellphones with x and y

Using the problem statements, we have the following table of values

                         x             y       Available

Labor (hours)   3            4         640

Materials ($)     75         60       12900

From the above, we have the following constraints of inequalities:

3x + 4y ≤ 640              

75x + 60y ≤ 12900

The graph of the inequalities and the shaded region are added as an attachment

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Answer:

We can start by using the formula:

distance = speed x time

We know that Oliver traveled 66 miles in 4 hours, so we can use this information to find his speed:

speed = distance / time

speed = 66 miles / 4 hours

speed = 16.5 miles per hour

Now that we know Oliver's speed, we can use the same formula to find how many miles he could travel in 14 hours:

distance = speed x time

distance = 16.5 miles per hour x 14 hours

distance = 231 miles

Therefore, Oliver could travel 231 miles in 14 hours at this speed.

√111: Irrational number

0: Whole number, Rational number

Π (Pi): Irrational number

71: Natural number, Whole number, Rational number

√-81: Non-real number

√111: Irrational number - The square root of 111 is an irrational number because it cannot be expressed as a fraction or a terminating or repeating decimal.

0: Whole number, Rational number - Zero is a whole number because it is a non-negative integer. It is also a rational number because it can be expressed as the ratio 0/1.

Π (Pi): Irrational number - Pi is an irrational number because it is a non-repeating, non-terminating decimal. It cannot be expressed as a fraction.

71: Natural number, Whole number, Rational number - 71 is a natural number because it is a positive integer. It is also a whole number and a rational number because it can be expressed as the ratio 71/1.

√-81: Non-real number - The square root of -81 is a non-real number because it involves the square root of a negative number. It cannot be expressed as a real number.

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1. 55%

2. 60%

3. 35%

4. 95%

5. 40%

1. The probability that a randomly selected student is male is 0.55 (55%).

2. The probability that a randomly selected female student is registered in the 2-year diploma program is 0.6 (60%).

3. The probability that a randomly selected male student is registered in the 1-year certificate program is 0.35 (35%).

4. The probability that a randomly selected student is female or is registered in a 2-year diploma program is 0.95 (95%).

5. The events “1-year program” and “male” are not independent as the probability of one event affects the probability of the other event. For example, the probability of a randomly selected male student being registered in the 1-year certificate program is 0.35 (35%), which is lower than the overall probability of a randomly selected student being registered in the 1-year certificate program (0.4 or 40%).

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False: It is the same to reflect the graph of y = cos x across the y-axis as it is across the x-axis.

A 180° rotation about the origin is a transformation that would have the same outcome as a reflection over the x-axis followed by a reflection over the y-axis. The x-coordinate of each point must be negated while reflecting across the Y axis, but the -value must remain unchanged.

By graphing y=-f(x), we may reflect the graph of any function f about the x-axis, and by graphing y=f, we can reflect the graph about the y-axis (-x). By graphing y=-f, we can even reflect it about both axes (-x).

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The function that models the amount of money in Yasmin's savings account for a given week can be written as: f(x) = 3x + 5

where x represents the number of weeks since Yasmin opened the account.

The constant term of 5 represents the initial amount Yasmin deposited into the account when she opened it, and the coefficient of 3 represents the amount she adds at the end of each week.

For example, after 1 week, the amount of money in the account would be:

f(1) = 3(1) + 5 = 8

After 2 weeks:

f(2) = 3(2) + 5 = 11

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Answer:

x  = -26/7

Step-by-step explanation:

Cancel terms that are in both the numerator and denominator

Multiply the numbers

Distribute

Subtract the numbers

Rearrange terms

Subtract the numbers

Multiply the numbers

Answer:

To solve this equation, we need to use the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

First, we need to simplify the expression inside the parentheses:

6 - 3/3(7x + 2) = 6(8 - 3)

6 - 1(7x + 2) = 6(5)

6 - 7x - 2 = 30

4 - 7x = 30

Next, we need to isolate the variable (x) on one side of the equation. We can do this by subtracting 4 from both sides:

4 - 7x - 4 = 30 - 4

-7x = 26

Finally, we can solve for x by dividing both sides by -7:

x = -26/7

Therefore, the solution to the equation is x = -26/7.

Answer:

1

Step-by-step explanation:

i think

Answer: 20%

Step-by-step explanation:

150,000 - 125,000 = 25,000

20 percent of 125,000 = 25k

a. Covariance = 10.11
b. Correlation coefficient = 0.376

Considering the following sample data:

x 12 18 20 22 25

y 15 20 25 22 27

a. Calculation of covariance

Covariance can be calculated by the formula:

Cov (X, Y) = [Σ(X - μx) (Y - μy)] / n

where, Σ denotes the sum of, X and Y are the variables, μx and μy are the means of X and Y respectively, and n is the sample size.

x  y  x-μx  y-μy  (x-μx)(y-μy) (-)^2  (-)^2

12  15  -6.6  -5.6  37.12  43.56  31.36

18  20  -0.6  -0.6  0.36  0.36  0.36

20  25  1.4  4.4  6.16           1.96  19.36

22  22  3.4  -2.6  -8.84  11.56  6.76

25  27  6.4  2.4  15.36  41.16  5.76

Σx = 97, Σy = 109, Σ(x-μx) = 5.00, Σ(y-μy) = -2.00, Σ(x-μx)(y-μy) = 50.56

Covariance is: Cov (X, Y) = [Σ(X - μx) (Y - μy)] / n= 50.56/5= 10.11

Thus, the covariance between the variables is 10.11.

b-1. Calculation of correlation coefficient.

Correlation coefficient is a statistical measure that measures the degree to which two random variables are associated. It can be calculated by the formula:

= Cov (X, Y) /  where, Cov (X, Y) is the covariance between X and Y, σX and σY are the standard deviations of X and Y respectively.

σx2 = [Σ(x-μx)2] / (n-1)σy2 = [Σ(y-μy)2] / (n-1)σx = √[Σ(x-μx)2] / (n-1)σy = √[Σ(y-μy)2] / (n-1)

x  y  (x-μx)  (y-μy)  (x-μx)2  (y-μy)2  (-)(-)

12  15  -6.6          -5.6         43.56  31.36               1  

18  -0.6  -0.6          0.36         0.36  0.32                5

20  25    1.4          4.4          1.96          19.36               22  

22  3.4  -2.6          11.56  6.76          -8.84                25  

27  6.4  2.4          41.16  5.76          15.36

Σx = 97, Σy = 109, Σ(x-μx) = 5.00, Σ(y-μy) = -2.00, Σ(x-μx)(y-μy) = 50.56

σx2 = 30.70

σy2 = 25.70

σx = √30.70 = 5.54

σy = √25.70 = 5.07

Correlation coefficient is:

= Cov (X, Y) /  = 10.11 / (5.54*5.07)= 0.376

Thus, the correlation coefficient is 0.376.

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From the given data, the effectiveness of epoetin alfa, which is a medication used to treat anemia, can be determined through a blood test called a complete blood count (CBC).

The CBC measures the number of red blood cells, white blood cells, and platelets in the blood, as well as the levels of hemoglobin and hematocrit.

In patients receiving epoetin alfa, the goal is to increase the hemoglobin level and hematocrit to a target range that is appropriate for their condition. Therefore, monitoring these levels through regular CBCs can help determine whether the medication is effective and whether the dosage needs to be adjusted.

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Answer:

Farrah borrowed $155 from her brother and has paid back $15 so far. She plans to pay back $35 each month until the debt is paid off.

To determine the number of months it will take to pay off the debt, we need to solve the equation:

x * 35 + 15 = 155

where x is the number of months it will take to pay off the debt.

Simplifying the equation, we get:

x * 35 = 155 - 15

x * 35 = 140

x = 4

Therefore, it will take 4 months to pay off the debt.

Options that describe the number of months it will take to pay off the debt are:

Step-by-step explanation:

Answer:

Step-by-step explanation:

[tex]y=x^2+10x+8=(x+5)^{2}-17[/tex]

Answer:

y = (x + 5)^2 - 17

Step-by-step explanation:

To write the quadratic function y = x^2 + 10x + 8 in vertex form, we need to complete the square. We start by adding and subtracting the square of half of the coefficient of x, which is (10/2)^2 = 25:

y = x^2 + 10x + 8

= (x^2 + 10x + 25) - 25 + 8

= (x + 5)^2 - 17

Therefore, the quadratic function in vertex form is:

y = (x + 5)^2 - 17

The vertex of this parabola is at the point (-5, -17), and the axis of symmetry is the vertical line x = -5. The term (-17) represents the minimum value of the function.

Answer: -6(p - 6) can be simplified using the distributive property of multiplication:

-6(p - 6) = -6 * p - (-6 * 6)

= -6p - (-36)

= -6p + 36

Therefore, -6(p - 6) is equivalent to -6p + 36.

Step-by-step explanation:

The factored form of 196x²- y² is (14x + y)(14x - y).

A factored form is a parenthesized algebraic expression. In effect a factored form is a product of sums of products, or a sum of products of sums. Any logic function can be represented by a factored form, and any factored form is a representation of some logic function.

The slope intercept form of a straight line is one of the most common forms used to represent the equation of a line. The slope intercept formula can be used to find the equation of a line when given the slope of the straight line and the y-intercept. The formula is y=mx+b.

The expression 196x² - y² can be factored using the difference of squares formula, which states that:

a²- b² = (a + b)(a - b)

In this case, we have a = 14x and b = y, so we can write:

196x² - y² = (14x + y)(14x - y)

Therefore, the factored form of 196x²- y² is (14x + y)(14x - y).

The expression (14x + y)(14x - y) is the factored form of a quadratic expression and does not represent a linear equation that can be written in slope-intercept form.

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6.2y - 3.7 = 6.2y + n n = -3.7 is the value we use to put this equal to and then solve for n. Hence, -3.7 is the value of p that equalises the two equations.

We need to simplify both equations and set them equal to one another in order to get the value of p that makes the expressions comparable.

Putting the left half of the equation first: Group like words to get (-y + 5.3) + (7.2y - 9) as -y + 7.2y - 9 + 5.3.

We will now put this equal to 6.2y + n and get n: 6.2y - 3.7 = 6.2y + n \sn = -3.7

Hence, -3.7 is the value of p that renders the equations equal.

A statement proving the equivalence of two mathematical expressions, sometimes incorporating one or more unknown variables, is known as an equation. Usually, an equal sign is used to denote it.

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The values of the variables can be proved by solving the the equations to get;

8. y = 3

9. k = -2

10. w = 14

11. x = -9

An equation is a statement that indicates that two expressions are equivalent by joining them with the '=' sign.

The method used to prove the value of the variable is by solving the equations as follows;

8. (5·y - 1)/2 = 7

Therefore;

2 × 7 = 5·y - 1

14 = 5·y - 1

5·y = 14 + 1 = 15

y = 15/5 = 3

y = 3

9. 10·k - 4 = 2·k - 20

Therefore;

10·k - 2·k = 8·k = 4 - 20 = -16

8·k = -16

k = -16/8 = -2

Therefore;

k = -2

10. -8·(w + 1) = -5·(w + 10)

-8·w - 8 = -5·w - 50

Therefore;

-5·w + 8·w = 50 - 8 = 42

3·w = 42

w = 42/3 = 14

Therefore;

w = 14

11. 14 - 2·(x + 8) = 5·x - (3·x - 34)

Therefore;

14 - 2·x - 16 = 5·x - 3·x + 34

14 - 2·x - 16 = -2·x - 2

5·x - 3·x + 34 = 2·x + 34

Therefore;

-2·x - 2 = 2·x + 34

2·x + 2·x = -2 - 34 = -36

4·x = -36

x = -36/4 = -9

x = -9

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the number that is not part of the solution set is A) 0.

How to find and what does variable mean?

To solve the inequality x + 2 ≥ 3, we need to isolate the variable x.

x + 2 ≥ 3

Subtract 2 from both sides:

x ≥ 1

This means that any value of x that is greater than or equal to 1 is part of the solution set.

To check which of the given numbers is not part of the solution set, we need to substitute each of them in the inequality and see if it is true or false.

A) 0 + 2 ≥ 3 --> 2 ≥ 3 (False)

B) 2 + 2 ≥ 3 --> 4 ≥ 3 (True)

C) 3 + 2 ≥ 3 --> 5 ≥ 3 (True)

D) 6 + 2 ≥ 3 --> 8 ≥ 3 (True)

Therefore, the number that is not part of the solution set is A) 0.

In mathematics, a variable is a symbol or letter that represents a value or a quantity that can vary or change. It is often used to represent unknown or undefined values or quantities, and is commonly denoted by letters such as x, y, z, a, b, and c.

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The distance from first base to dead center field in a certain baseball stadium is 338 feet.

Explanation:

The distance from first base to dead center field in a certain baseball stadium is 338 feet. Given,The distance from home plate to dead center field in a certain baseball stadium is 407 feet.A baseball diamond is a square with a distance from home plate to first base of 90 feet.

To find,How far is it from first base to dead center field?

Solution:Given that the distance from home plate to dead center field is 407 feet.The baseball diamond is a square with a distance from home plate to first base of 90 feet.Now we have to find the distance from first base to dead center field.We can find the distance by using the Pythagorean theorem which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.Let us consider a right triangle ABC where AB represents the distance from home plate to first base, AC represents the distance from home plate to dead center field, and BC represents the distance from first base to dead center field.

As per the Pythagorean theorem, we have

AC² = AB² + BC²

Putting the values, we have

AC² = (90)² + BC²AC² = 8100 + BC²AC² - BC² = 8100

Taking the square root on both sides, we getAC = √(8100 + BC²)

Now we have AC = 407 ft,AB = 90 ftAC² = AB² + BC²407² = 90² + BC²BC² = 407² - 90²BC² = 165649BC = √165649BC = 407 ft - 90 ft

BC = 338 ft

Therefore, the distance from first base to dead center field in a certain baseball stadium is 338 feet.

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Answer:

The answer is B

Step-by-step explanation:

it looks right

The basketball will reach its maximum height 1.735 seconds after it has been thrown vertically into the air.

A function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. A function can be represented as an equation, a graph, a table of values, or a verbal description. For example, the function f(x) = 2x + 1 represents a relationship between the input x and the output 2x + 1.

To find the maximum height of the basketball, we need to find the vertex of the parabola represented by the function f(x). The vertex of x-coordinate is:

x = -b/2a

The coefficients of the quadratic equation a[tex]x^2[/tex] + b[tex]x[/tex] + c are a, b, and c. In this case, a = -4.9 and b = 17, so:

x = -17/(2*(-4.9)) = 1.735 (rounded to 3 decimal places)

Therefore, the basketball will reach its maximum height 1.735 seconds after it has been thrown vertically into the air.

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Answer:

To show how x² + 6x + 5 is equivalent to (x + 5)(x + 1), we can use the FOIL method, which stands for First, Outer, Inner, and Last.

First, we multiply the first term of each factor: x and x, which gives x².

Next, we multiply the outer terms of each factor: x and 1, which gives x.

Then, we multiply the inner terms of each factor: 5 and x, which gives 5x.

Finally, we multiply the last term of each factor: 5 and 1, which gives 5.

Adding up these terms, we get:

x² + x + 5x + 5

Simplifying by combining like terms, we get:

x² + 6x + 5

This is the same as the original expression. Therefore, we have shown that:

x² + 6x + 5 = (x + 5)(x + 1)

Step-by-step explanation:

The probability of observing an 84°F temperature or higher in LA during a randomly chosen day in June is 0.1151 and the temperature of the coldest 10% of the days (days with lowest average high temperature) during June in LA is approximately 71.6°F.

Let X be the random variable that represents the average daily high temperature in LA in June. Then X ~ N(μ = 78, σ = 5). The probability of observing an 84°F temperature or higher in LA during a randomly chosen day in June is given by: P(X > 84) = P(Z > (84 - 78) / 5) = P(Z > 1.2) = 0.1151 (rounded to 4 decimal places)

To find the temperature of the coldest 10% of the days (days with lowest average high temperature) during June in LA, we need to find the 10th percentile of the distribution. Using a z-score table, we can find the z-score corresponding to the 10th percentile: z = -1.28. Thus, the temperature of the coldest 10% of the days during June in LA is given by: x = μ + zσ= 78 + (-1.28)(5)≈ 71.6°F (rounded to 1 decimal place)

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Answer:

vro I guess this is locating root 5

Step-by-step explanation:

mark me BRAINLIST

The correct option is C. Green image of triangle ABC.

How to find the rotated shape or coordinates of image about origin?

As we know that triangle ABC, is rotated 270° counterclockwise about the origin. So, the algebraic rule for a figure is,

90° Clockwise or 270° counter clockwise, (x,y) = (y,-x)

As, C(-3,3) which is equal to (3,3) from the algebraic rule.

And (3,3) belongs to 1st quadrant i.e. green image is the correct answer.

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Triangle below represents is  option  C. Green image of triangle ABC.

A triangle is a geometric shape with three sides and three angles. It is one of the most fundamental shapes in geometry and is used extensively in mathematics, physics, engineering, and many other fields. Triangles are often classified based on their angles and sides.

As we know that triangle ABC, is rotated 270° counterclockwise about the origin. So, the algebraic rule for a figure is,

90° Clockwise or 270° counter clockwise, (x,y) = (y,-x)

As, C(-3,3) which is equal to (3,3) from the algebraic rule.

And (3,3) belongs to 1st quadrant i.e. green image is the correct answer.

To know more about Triangle, visit:

brainly.com/question/2773823

#SPJ1