8.G.1.2 A mathematical puzzle uses four triangles with the dimensions show below. Which of the following triangles are congruent? A. P and Q B. Q and R C. R and S D. S and P

The values of angles x in the parallelogram maze are calculated below

The question is solved by taking the shapes from left to right on each row

So, we have

x + 65 = 180 --- adjacent angles

x = 115

x + 103 = 180 --- adjacent angles

x = 77

9x = 90 --- angles in a rectangle

x = 10

x + 85 = 180 --- adjacent angles

x = 95

x + 60 = 180 --- adjacent angles

x = 120

2x = 90 --- angles in a rectangle

x = 45

x = 98 -- opposite angles in a parallelogram

3x = 81 -- opposite angles in a parallelogram

x = 27

4x + 76 = 180 --- adjacent angles

x = 26

3x = 78 -- opposite angles in a parallelogram

x = 26

8x + 68 = 180 --- adjacent angles

x = 14

4x = 72 -- opposite angles in a parallelogram

x = 18

x + 126 = 180 --- adjacent angles

x = 54

x = 90 -- opposite angles in a parallelogram

6x = 90 -- opposite angles in a parallelogram

x = 15

2x + 62 = 180 --- adjacent angles

x = 59

6x = 106 --- opposite angles in a parallelogram

x = 17.6

5x = 90 --- angles in a rectangle

x = 18

5x + 80 = 180 --- adjacent angles

x = 20

10x = 90 --- angles in a rectangle

x = 9

2x = 100 --- opposite angles in a parallelogram

x = 50

6x = 96 --- opposite angles in a parallelogram

x = 16

x + 105 = 180 --- adjacent angles

x = 75

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Note that the perimeter of the parallelogram is 42

recall that opposite sides of a parallelogram are congruent always

We have to to find the distance between the  points Q(6, 6 ) and R(1, -6) using the distance formula which is

d = √[(x2 - x1) ² + (y2 - y 1)²]

where d is the distance between two points with paris   (x1 , y1)

and (x2, y2).

PS = QR = √(6-1)²  + (6+6)²

    = √5² + 12²
= √(25+144

= √(169)

= 13

PQ = SR = 8

Perimeter  = 13 + 13 + 8 + 8
Perimeter = 42.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

See attached image.

Answer:

the function f(x) is f(x) = 5x + 4.

Step-by-step explanation:

To find f(x), we need to use the formula:

(gof)(x) = g(f(x)) = 3f(x) - 2

We are given that (gof)(x) = 15x + 10, so we can substitute this expression into the formula to get:

3f(x) - 2 = 15x + 10

Simplifying this equation, we get:

3f(x) = 15x + 12

Dividing both sides by 3, we get:

f(x) = 5x + 4

Therefore, the function f(x) is f(x) = 5x + 4.

The range, median and mode of the data-set are given as follows:

Considering the stem-and-leaf plot, the data-set is given as follows:

54, 55, 60, 66, 69, 72, 73, 74, 75, 75, 81, 82, 88, 89, 95, 96.

The range of a data-set is calculated as the difference between the highest value and the lowest value in the data-set, thus:

96 - 54 = 52.

The data-set has an even cardinality of 16, hence the median is calculated as the mean of the two middle elements as follows:

Median = (74 + 75)/2

Median = 74.5.

The mode of a data-set is the observation that appears the most times in the data-set, hence it is of 75, which is the only observation that appears twice in the data-set.

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The accounting terms when matched with their relevant actions would be;

A company's free cash flow is an indicator of its ability to cover expenses and invest in future growth opportunities. A high level of free cash flow suggests that the business is doing well financially, while a low amount indicates that the opposite is true.

In order to generate income or save money over time, companies will make capital expenditures. However, if investors deem a company to be underperforming or unlikely to provide returns on investment, they may choose to sell their shares as a preventative measure against further financial losses.

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The probability that event A or B takes place is P ( A ∪ B ) = 6/17

Given data ,

Let the probability that event A or B takes place is P ( A ∪ B )

Now , the probability of A is P ( A ) = 2/17

And , the probability of B is P ( B ) = 4/17

where P ( A ∩ B ) = 0

On simplifying the equation , we get

P ( A ∪ B ) = P ( A ) + P ( B ) - P ( A ∩ B )

So , P ( A ∪ B ) = 2/17 + 4/17

P ( A ∪ B ) = 6/17

Hence , the probability is 6/17

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The limit [tex]\lim _{x\to \infty \:}\frac{1}{3^x}+3^{\frac{1}{x}}[/tex] without using L'Hopital's Rules has a value of 1

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

[tex]\lim _{x\to \infty \:}\frac{1}{3^x}+3^{\frac{1}{x}}[/tex]

By substitution, we have

[tex]\lim _{x\to \infty \:}\frac{1}{3^x}+3^{\frac{1}{x}} = \frac{1}{3^{}\infty}+3^{\frac{1}{\infty}}[/tex]

So, we have

[tex]\lim _{x\to \infty \:}\frac{1}{3^x}+3^{\frac{1}{x}} = 0+3^0[/tex]

Evaluate the exponents

[tex]\lim _{x\to \infty \:}\frac{1}{3^x}+3^{\frac{1}{x}} = 0+1[/tex]

Evaluate the sum

[tex]\lim _{x\to \infty \:}\frac{1}{3^x}+3^{\frac{1}{x}} = 1[/tex]

Hence, the solution is 1

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Therefore, the volume of the composite solid is approximately 1357.28 cubic feet, rounded to the nearest hundredth.

Volume is a measure of the amount of space occupied by an object or a substance. In other words, it is the three-dimensional space that an object or substance occupies. The SI unit for volume is cubic meters (m³), although other units such as cubic centimeters (cm³) and liters (L) are also commonly used.

Here,

The composite solid is made up of a hemisphere and a cone with the same base radius and height. The volume of the hemisphere is given by (2/3)πr³, and the volume of the cone is given by (1/3)πr²h, where r is the radius and h is the height. Given that the radius is 6 feet and the height is 12 feet, we can find the volumes of the hemisphere and the cone as follows:

Volume of hemisphere = (2/3)π(6³)

= 288π cubic feet

Volume of cone = (1/3)π(6²)(12)

= 144π cubic feet

The total volume of the composite solid is the sum of the volumes of the hemisphere and the cone, which is:

Total volume = Volume of hemisphere + Volume of cone

= 288π + 144π

= 432π

To find the numerical value of this volume, we can use the approximation π ≈ 3.14 and round the result to the nearest hundredth:

Total volume ≈ 432(3.14)

= 1357.28 cubic feet

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The correct answers are:

They are inverses of one another

They are symmetric over the line y=x

When a number is too big or too little, exponential notation can express it as a single number and 10 increased to the power of the appropriate exponent.

The exponential form [tex]y=5^x[/tex]

The logarithmic form [tex]y = log_5\ x[/tex] are inverse functions of each other.

If we [tex]y = 5^x[/tex], then [tex]x = log_5\ y[/tex].

As long as x is a real number and y is positive, this is true for any value of x and y.

The graphs of [tex]y = 5^x[/tex] and [tex]y = log_5\ x[/tex] are symmetric over the line y = x.

We obtain the other graph by reflecting the first graph across this line.

Over the line y = x, the inverse functions are always symmetric.

If we swap the x and y coordinates of any point on one graph, we get a comparable position on the other graph.

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The function that has the greatest rate of change is y = x

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

y = -x ,

y = 3/4x +1,

y = 1/2x +2,

y = x

A linear function is represented as

y = mx + c

Where

rate of change = m

So, we have

y = -x , rate = -1

y = 3/4x +1, rate = 3/4

y = 1/2x +2, rate = 1/2

y = x, rate = 1

This means that the function with the highest rate is y = x

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The solution of the matrix is [tex]4G + 2F = \begin{bmatrix}32 &-20 &-32& -8 &-40 \\-24& -28 &4 &36 &8 \\16& 24 &12 &28 &20 \\-16&-12 &0 &-40 &-36\end{bmatrix}[/tex]

A matrix is a rectangular array of numbers arranged in rows and columns. The size of a matrix is given by its dimensions, which indicate the number of rows and columns in the matrix. In this question, both matrices G and F have 4 rows and 5 columns, so we say that they are 4x5 matrices.

Scalar multiplication is performed by multiplying each element of a matrix by a scalar, which is simply a number.

Now, to find the value of 4G + 2F, we need to perform scalar multiplication on each matrix and then add the results. We get:

[tex]4G = 4 * \begin{bmatrix}8 &-5 &-8& -2 &-10 \\-6& -7 &1 &9 &2 \\4& 6 &3 &7 &5 \\-4&-3 &0 &-10 & -9\end{bmatrix}\\= \begin{bmatrix}32 &-20 &-32& -8 &-40 \\-24& -28 &4 &36 &8 \\16& 24 &12 &28 &20 \\-16&-12 &0 &-40 & -36\end{bmatrix}[/tex]

[tex]2F = 2 * \begin{bmatrix}1 &8 &-2& -5 &9 \\-9& 10 &6 &-3 &0 \\4& 5 &-4 &3 &7 \\2&-10 &-6 &-1 & -8\end{bmatrix}= \begin{bmatrix}2 &16 &-4& -10 &18 \\-18& 20 &12 &-6 &0\\8& 10 &-8 &6 &14 \\4&-20 &-12 &-2 & -16\end{bmatrix}[/tex]

Now, we can perform matrix addition on 4G and 2F to get:

[tex]4G + 2F = \begin{bmatrix}32 &-20 &-32& -8 &-40 \\-24& -28 &4 &36 &8 \\16& 24 &12 &28 &20 \\-16&-12 &0 &-40 &-36\end{bmatrix}[/tex]

Therefore, the value of 4G + 2F is the 4x5 matrix given above.

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Answer:

True proportion of homeowners in the population lies between 0.333 and 0.847

Step-by-step explanation:

If 9 out of 22 randomly picked people said they rented their homes, then the remaining 13 must own their homes. Therefore, the point estimate of the proportion of homeowners is:

Point estimate = Number of homeowners / Total number of people surveyed

Point estimate = 13 / 22

Point estimate = 0.59 (rounded to two decimal places)

To calculate the 95% confidence interval for the true proportion of homeowners, we can use the following formula:

Confidence interval = Point estimate ± (Z-value x Standard error)

where the Z-value corresponds to the desired level of confidence and the standard error is given by:

Standard error = sqrt [ (Point estimate x (1 - Point estimate)) / Sample size]

For a 95% confidence interval, the Z-value is 1.96 (from the standard normal distribution). Using the point estimate obtained earlier, the sample size is 22, and we can calculate the standard error as:

Standard error = sqrt [ (0.59 x 0.41) / 22 ]

Standard error = 0.131 (rounded to three decimal places)

Substituting these values into the formula, we get:

Confidence interval = 0.59 ± (1.96 x 0.131)

Confidence interval = 0.59 ± 0.257

Confidence interval = (0.333, 0.847)

Therefore, with 95% confidence, we can say that the true proportion of homeowners in the population lies between 0.333 and 0.847.

Answer: 134.

Step-by-step explanation:

First revert 67% into it's decimal form. You get 0.67. Then, multiply 0.67 by 200 because there were 200 problems.

0.67 x 200

67 x 2

134

So, christian got 134 out of 200 problems correct.

Answer:

[tex]f(2\pi) = 32\pi^3 - 2[/tex]

Step-by-step explanation:

Main steps:

Step 1: Use integration to find a general equation for f

Step 2: Find the value of the constant of integration

Step 3: Find the value of f for the given input

Step 1: Use integration to find a general equation for f

If [tex]f'(x) = g(x)[/tex], then [tex]f(x) = \int g(x) ~dx[/tex]

[tex]f(x) = \int [12x^2 - sin(x)] ~dx[/tex]

Integration of a difference is the difference of the integrals

[tex]f(x) = \int 12x^2 ~dx - \int sin(x) ~dx[/tex]

Scalar rule

[tex]f(x) = 12\int x^2 ~dx - \int sin(x) ~dx[/tex]

Apply the Power rule & integral relationship between sine and cosine:

[tex]f(x) = 12*(\frac{1}{3}x^3+C_1) - (-cos(x) + C_2)[/tex]

Simplifying

[tex]f(x) = 12*\frac{1}{3}x^3+12*C_1 +cos(x) + -C_2[/tex]

[tex]f(x) = 4x^3+cos(x) +(12C_1 -C_2)[/tex]

Ultimately, all of the constant of integration terms at the end can combine into one single unknown constant of integration:

[tex]f(x) = 4x^3 + cos(x) + C[/tex]

Step 2: Find the value of the constant of integration

Now, according to the problem, [tex]f(0) = -2[/tex], so we can substitute those x,y values into the equation and solve for the value of the constant of integration:

[tex]-2 = 4(0)^3 + cos(0) + C[/tex]

[tex]-2 = 0 + 1 + C[/tex]

[tex]-2 = 1 + C[/tex]

[tex]-3 = C[/tex]

Knowing the constant of integration, we now know the full equation for the function f:

[tex]f(x) = 4x^3 + cos(x) -3[/tex]

Step 3: Find the value of f for the given input

So, to find [tex]f(2\pi)[/tex], use 2 pi as the input, and simplify:

[tex]f(2\pi) = 4(2\pi)^3 + cos(2\pi) -3[/tex]

[tex]f(2\pi) = 4*8\pi^3 + 1 -3[/tex]

[tex]f(2\pi) = 32\pi^3 - 2[/tex]

Answer:

[tex]f(2 \pi)=32\pi^3-2[/tex]

Step-by-step explanation:

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given:

If g(x) is the first derivative of f(x), we can find f(x) by integrating g(x) and using f(0) = -2 to find the constant of integration.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]     [tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $\sin x$}\\\\$\displaystyle \int \sin x\:\text{d}x=-\cos x+\text{C}$\end{minipage}}[/tex]

[tex]\begin{aligned} \displaystyle f(x)&=\int f'(x)\; \text{d}x\\\\&=\int g(x)\;\text{d}x\\\\&=\int (12x^2-\sin x)\;\text{d}x\\\\&=\int 12x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\int x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\cdot \dfrac{x^{(2+1)}}{2+1}-(-\cos x)+\text{C}\\\\&=4x^{3}+\cos x+\text{C}\end{aligned}[/tex]

To find the constant of integration, substitute f(0) = -2 and solve for C:

[tex]\begin{aligned}f(0)=4(0)^3+\cos (0) + \text{C}&=-2\\0+1+\text{C}&=-2\\\text{C}&=-3\end{aligned}[/tex]

Therefore, the equation of function f(x) is:

[tex]\boxed{f(x)=4x^3+ \cos x - 3}[/tex]

To find the value of f(2π), substitute x = 2π into function f(x):

[tex]\begin{aligned}f(2 \pi)&=4(2 \pi)^3+ \cos (2 \pi) - 3\\&=4\cdot 2^3 \cdot \pi^3+1 - 3\\&=32\pi^3-2\\\end{aligned}[/tex]

Therefore, the value of f(2π) is 32π³ - 2.

________________________________

________________________________

Answer:

7

Step-by-step explanation:

We can use the fact that the corresponding sides of similar triangles are in proportion to each other.

Let's compare the corresponding sides of the two triangles:

Corresponding sides and ratio

Ratio

Since the triangles are similar, we know that these ratios are equal. That is,

6 / (x-3) = 12 / (x+1)

We can simplify this equation by cross-multiplying:

6(x+1) = 12(x-3)

Expanding and simplifying:

6x + 6 = 12x - 36

42 = 6x

x = 7

Therefore, x has a value of 7.

Answer:

a= Acute

b=Obtuse

c= Acute

d= Right

Answer:

Step-by-step explanation:

-9+6-1-2-3-4+5-6

-3-3-7-1

-6-8

-14

Answer:

The answer is -14.

Step-by-step explanation:

-(9 - 6) + (-1 -2) - (3 + 4) + (5 - 6)

Subtract 6 from 9 to get 3.

-3 -1 -2 - (3 + 4) + 5 - 6

Subtract 1 from -3 to get -4.

-4 - 2 - (3 + 4) + 5 - 6

Subtract 2 from -4 to get -6.

-6 - (3 + 4) + 5 - 6

Add 3 and 4 to get 7.

-6 - 7 + 5 - 6

Subtract 7 from -6 to get -13.

-13 + 5 - 6

Add -13 and 5 to get -8.

-8 - 6

Subtract 6 from -8 to get -14.

-14.

The volume of the fully - inflated balloon is 4186.67 cubic inches. The solution has been obtained by using the formula for sphere.

What is a sphere?

The geometric equivalent of a circle in two dimensions in three dimensions is a sphere. A collection of three-dimensional points with the same r separation between them are referred to as a sphere.

We are given diameter as 20 inches.

So, the radius is half of the diameter which comes out to be 10 inches.

From this, we get the volume as

⇒ Volume = [tex]\frac{4}{3}[/tex] π [tex]r^{3}[/tex]

⇒ Volume = [tex]\frac{4}{3}[/tex] π [tex]10^{3}[/tex]

⇒ Volume = [tex]\frac{4000}{3}[/tex] π

⇒ Volume = 4186.67 cubic inches

Hence, the volume of the fully - inflated balloon is 4186.67 cubic inches.

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The difference in what you have as a balance in your check register versus your online statment by 11/18 is $34.78.

The combined total of the 2 transactions they have in common is $206.99.

Use the amount in 11/1 as your balance, then compute by deducting the checks (chk) and debit purchases and adding the deposits. Then get the difference.

Check Register:

Total = 397.45 - 50.32 - 16.32 + 190.67

Total = 521.48

Online Statement:

Total = 486.44 - 84.80 - 19.73 - 16.32 + 190.67

Total = 556.26

Difference = online - check register

Difference = 34.78

On 11/10 and 11/18, check and online have the same transactions (same date, same amount) which are 16.32 and 190.67. Add them together to get 206.99

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The distance of the point from the line is d =

Given data ,

Let the point be P ( 1 , 1 )

Let the equation of line be represented as e

where A ( -5 , 0 ) and B ( 1 , -2 ) lies on the line

So , slope m = ( 0 + 2 ) / ( -5 - 1 )

m = 2/-6

m = -1/3

So , the equation of line is

y - 0 = ( -1/3 ) ( x + 5 )

y = ( -1/3 ) ( x + 5 )

( 1/3 )x + y + 5/3 = 0

Now , the Distance of a point to line D = | Ax₀ + By₀ + C | / √ ( A² + B² )

On simplifying , we get

D = | ( 1/3 )( 1 ) + ( 1 ) ( 1 ) + ( 5/3 ) | / √[ ( 1/3 )² + ( 1 )² ]

D = | ( 6/3 ) + 1 | / √ [ ( 1/9 ) + 1 ]

D = 3 / ( 10 ) / 9

D = 27/10

D = 2.7 units

Hence , the distance is 2.7 units

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All the numbers are rational numbers.

We have,

Rational numbers are those numbers that are terminating and recurring non-terminating.

Irrational numbers are those numbers that are non-terminating and non-recurring.

The numbers are:

1)

0.020202

This is non terminating recurring number.

It is a rational number.

2)

0.8

This is a terminating number.

It is a rational number.

3)

0.020202

This is non terminating recurring number.

It is a rational number.

Thus,

All the numbers are rational numbers.

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Step-by-step explanation:

T = 5200 = .2 ( E - 10600)

       5200 / .2 = E - 10600

            E = 5200/.2   + 10 600 = 36600 pounds

Answer:

40 degrees and 140 degrees

Step-by-step explanation:

To solve this problem you can add x+3x+20 and set that equal to 180. (We can do this because angle x and angle 3x + 20 make a linear pair. Knowing this we can estimate that both angles added together will equal 180)

Let us add x + 3x + 20 = 180 to find x. We can then substitute that into the equation.

[tex]x+3x+20=180 :a\\\\4x+20=180 :b\\\\4x + 20 -20=180-20:c\\\\4x=160:d\\\\\frac{4x}{4} = \frac{160}{4}:e\\\\x=40:f[/tex]

a: So in this part, we have rewritten the equation to make it easier to solve

b:  In this step, you combine the like terms x+3x to get 4x

c: In step c, you are subtracting 20 from both sides to keep constants on one side and variables on the right

d: In this last step the equation has been simplified to make it easier to solve.

e: To isolate x you have to divide both sides by 4, we do this because the coefficient of x is 4 so you divide the equation by 4 to cancel it out.

f: You rewrite and simplify the equation.

Now to find the measure of both angles you substitute x into the equation.

The first angle's value is 40 degrees and the second is 140 degrees.

These are our answers.

Answer: We can use Bayes' theorem to find the probability that a randomly chosen applicant has a graduate degree, given that they are male. Bayes' theorem states:

P(A|B) = P(B|A) * P(A) / P(B)

where A and B are events, P(A|B) is the conditional probability of event A given that event B has occurred, P(B|A) is the conditional probability of event B given that event A has occurred, P(A) is the prior probability of event A, and P(B) is the prior probability of event B.

In this case, we want to find P(grad degree | male), which is the probability that an applicant has a graduate degree given that they are male. Using Bayes' theorem, we have:

P(grad degree | male) = P(male | grad degree) * P(grad degree) / P(male)

We are given that 134 applicants are male, and 40 of those males have a graduate degree. Therefore:

P(male | grad degree) = 40 / total number of applicants with a graduate degree

We are not given the total number of applicants with a graduate degree, but we can find it using the fact that 40 applicants are male and have a graduate degree, and that there are 134 male applicants in total. Therefore:

total number of applicants with a graduate degree = 40 / (40/134) = 118.33 (rounded to the nearest whole number)

Next, we need to find the prior probability of having a graduate degree, P(grad degree). We can use the total number of applicants and the number of applicants with a graduate degree to calculate this probability:

P(grad degree) = number of applicants with a graduate degree / total number of applicants = 118 / 300 = 0.3933 (rounded to four decimal places)

Finally, we need to find the prior probability of being male, P(male). This probability can be calculated similarly:

P(male) = number of male applicants / total number of applicants = 134 / 300 = 0.4467 (rounded to four decimal places)

Now, we can substitute these values into Bayes' theorem to find the probability that a randomly chosen applicant has a graduate degree, given that they are male:

P(grad degree | male) = (40 / 118) * 0.3933 / 0.4467 = 0.332 (rounded to three decimal places)

Therefore, the probability that a randomly chosen applicant has a graduate degree, given that they are male, is approximately 0.332 or 33.2%.

Step-by-step explanation:

The calculated value of the maximum possible value of the objective function is 46

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

Objective function, F = 5x + 6y

The coordinates of the feasible region are

(0, 6), (1, 6.5) and (2, 6)

Substitute (0, 6), (1, 6.5) and (2, 6) in the above equation, so, we have the following representation

F(0, 6) = 5(0) + 6(6) = 36

F(1, 6.5) = 5(1) + 6(6.5) = 44

F(2, 6) = 5(2) + 6(6) = 46

The maximum value above is 46 at (2, 6)

Hence, the maximum possible value of the objective function is 46

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Answer:

Step-by-step explanation:

vertex form = y=a(x-h)^2+k

standard form = y=ax^2+bx+c

the most straightforward way to solve it is to expand the vertex form

we get :

-(x+1)(x+1)-1

= -(x^2+x+x+1) - 1

note: remember to leave the negative sign out of the parentheses and distribute it after - otherwise you may mix up signs

= (-x^2-x-x-1) - 1

= -x^2-2x-1-1

= -x^2 - 2x - 2

So, in standard form, it is y=-x^2-2x-2

Hope this helps!

If an angle measure 125° then the angle measures a fraction of 25/72 of a full circle.

A circle is a two-dimensional geometric shape that is defined as the set of all points that are equidistant from a single point, called the center. The distance from the center to any point on the circle is called the radius, and the distance across the circle, passing through the center, is called the diameter. The circumference of a circle is the distance around the edge or perimeter of the circle. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle, and π (pi) is a mathematical constant that represents the ratio of the circumference to the diameter of a circle, approximately equal to 3.14159.

A circle has 360 degrees. To find what fraction of a circle an angle measures, we can divide the angle by 360. In this case, the angle measures 125 degrees, so the fraction of a circle it turns can be calculated as:

125/360 = 0.347222222...

To simplify this fraction, we can multiply both the numerator and denominator by 2:

125/360 = (1252)/(3602) = 250/720

We can further simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 10:

250/720 = (2510)/(7210) = 25/72

Therefore, the angle measures a fraction of 25/72 of a full circle.

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To use the change of base formula, we first identify the base of the original logarithm and the desired base of the new logarithm.

The change of base formula is a useful tool in logarithmic functions that allows us to rewrite a logarithm with one base as an equivalent logarithm with a different base. The formula is as follows:

logᵦ a = logᵦ(x) / logₐ(x)

where a is the value being logged, β is the base of the original logarithm, and x is any positive number other than 1.

We then apply the formula by taking the logarithm of the value being logged with the desired base, and dividing it by the logarithm of the same value with the original base.

For example, let's say we want to rewrite the logarithm log₂ 8 in base 10. Using the change of base formula, we have:

log₂ 8 = log₁₀ 8 / log₁₀ 2

The value of log₁₀ 8 is simply 0.9031, and the value of log₁₀ 2 is 0.3010, so we have:

log₂ 8 = 0.9031 / 0.3010 ≈ 3

Therefore, log₂ 8 is equivalent to log₁₀ 8 ≈ 3. By using the change of base formula, we can simplify logarithmic expressions and evaluate them using common logarithms or natural logarithms.

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