Find x: 2x=1

X=1/2

X=1

X=2

X=-1

The largest area of a rectangle that can be inscribed in a semicircle of radius 6 is 36.

What is the semicircle?

A semicircle is a one-dimensional locus of points that makes up one-half of a circle in mathematics. A semicircle's whole arc is always 180 degrees long. It just has one symmetry line.

The equation of the semicircle is

x² + y² = r²         .............(1)

The area of the rectangle is

A  = 2xy  ..................(2)

From equation (1), we get

y² = r² - x²

y = √r² - x²

Plugging this value in equation (2)

A  = 2x√r² - x²

Differentiating wrt. x using the product rule,

[tex]\frac{dA}{dx} = 2\sqrt{r^2 - x^2} - 2\frac{x^2}{\sqrt{r^2 - x^2} }[/tex]

[tex]\frac{2r^2 - 4x^2 }{\sqrt{r^2 - x^2} }[/tex]

The critical points are when

[tex]\frac{dA}{dx} = 0[/tex]

that is

[tex]\frac{2r^2 - 4x^2 }{\sqrt{r^2 - x^2} } = 0[/tex]

r² = 2x²

x = r/√2

Then,

y = √r² - x²

= √r² - (r/√2)²

= r/√2

The maximum area is

A = 2 .  r/√2 .  r/√2  = r²

A = r² = (6)² = 36

Hence, the largest area of a rectangle that can be inscribed in a semicircle of radius 6 is 36.

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The equation of the line that passes through the points (-5,1) and (1,3) is x - 3y = -8

From  the question, we are to determine the equation of the line that passes through the given points.

The given points are (-5, 1) and (1, 3)

Using the formula,

(y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁)

Then,

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

Simplify

(y - 1) / (x + 5) = 2 / (1 + 5)

(y - 1) / (x + 5) = 2 / 6

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

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

3y - 3 = x + 5

x + 5 = 3y -3

x - 3y = -3 - 5

x - 3y = -8

Hence, the equation is x - 3y = -8

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Local minimum at: (1, -7) , (3, -7).

Local maximum at : DNE

What to find: Local extrema.

To solve the given problem, we will use the steps below:

Local extrema:

A local extremum of a function is the point at which a maximum or minimum value of the function in some open interval containing the point is obtained.

                                                   OR

Local extrema on a function are points on the graph where the -coordinate is larger (or smaller) than all other -coordinates on the graph at points ''close to'' . A function has a local maximum at , if for every near .

Locate the local extremum from the graph.

From the graph given, we have two local minimum.

At x = 1, f(1) =-7

Hence, the local minimum is (1, -7)

At x=3, f(3) =-7

Hence, the local minimum is ( 3, - 7)

Therefore, the local minimum are (1, -7) , (3, -7).

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Answer:5/8

Step-by-step explanation:

[tex]\begin{gathered} \boxed{ \sf \frac{3}{4} - \frac{1}{8}} = \\ \\ \sf \frac{8 \div 4 \times 3 - 8 \div 8 \times 1}{8} = \\ \\ \sf\frac{6 - 1}{8} = \\ \\ \sf \boxed{ \sf\frac{5}{8}} \end{gathered}[/tex]

Therefore, the result of this subtraction of fractions is five eighths.

How to subtract unlike fractions:

Heterogeneous fractions are fractions that have different denominators.

To subtract heterogeneous fractions we follow these steps:

Plants C and D have the same ratios of green bottles to total bottles recycled.

The ratio is defined as a relationship between two quantities, it is expressed as one divided by the other.

The table is given in the question,

As per the given question, the required solution would be as:

For plant A,

The ratio of green bottles to total bottles

⇒ 16/172 = 4/43

For plant B,

The ratio of green bottles to total bottles

⇒ 12/135 = 4/45

For plant C,

The ratio of green bottles to total bottles

⇒ 14/154 = 1/11

For plant D,

The ratio of green bottles to total bottles

⇒ 18/198 = 1/11

Thus, plants C and D have the same ratios of green bottles to total bottles recycled.

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The number of customers Alejandro will need in order to break even is 38 customers.

Given data:

Cost of business = 48 x + 500

Income = R = 65 x - 145

Let x is the number of customers.

Income equation  

R= 65x- 145

Substitute 48x + 500  for y

48x + 500 = 65x -145

Substract 48x from both side

500 = 17x - 145

Add 145 to each side

645 = 17x

Divide both side by 17x

x =645 /17

x = 37.9

x = 38 customers (Approximately)

Therefore the number of customers is 38 customers.

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A comparison of hot coffee and cold lake is, the heat from the coffee will be absorbed by the cold lake through convection method of heat transfer.

Heat is transferred through the convection, which is basically large-scale movement of molecules inside gases and the liquids. Conduction is basically used to move heat from one of the object to the fluid initially, but fluid motion is all responsible for the bulk of the heat transfer.

Conservation of energy

The principle of conservation of the energy states that the total energy of an isolated system is always in conserved state.

Heat lost by all the hot coffee = Heat absorbed by all the cold lake

Heat transfer process

Heat from the coffee will all be absorbed by the cold lake through convection method of the heat transfer.

Thus, the comparison of hot coffee and the cold lake is, the heat from the coffee will all be absorbed by the cold lake through convection method of the heat transfer.

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Answer:

12x+3

Step-by-step explanation:

distribute the 5 to both the 2x and -1

10x-5

do the same with the the 2. Distribute the 2 to the x and to the 4.

2x+8

10x-5+2x+8

add like terms: the 10x and the 2x are like terms and the -5 and 8 are like terms

12x+3 : the answer

The value x in the equation h(x) = 4x-4 is, x =1

Polynomial functions are functions of a single independent variable.

To solve for x

Since h(x) =0

substitute h(x) into equation  

h(x) = 4x - 4

0=4x - 4

Add 4 to both sides

4x = 4

Divide both side by 4

4x/4 = 4/4

x = 1

therefore, the value of X= 1

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The sampling distribution is t= 88 degrees of freedom.

To find the sampling distribution of the sample mean difference we need to do a two-sample t-test: equal variances.

When we do a two-sample t-test, we have to take into account the following assumptions:

1) the sample variances are normal.

2) the sample variances of both populations are equal.

Now, let the sample size be [tex]n_x[/tex] and [tex]n_y[/tex] respectively and the means be [tex]x^-[/tex] and [tex]y^-[/tex] respectively.

Now, we are assuming that the distribution of x and y are

[tex]x\sim N([/tex]mewx, [tex]\sigma x = \sigma[/tex])

[tex]y\sim N[/tex] (mewy, [tex]\sigma y = \sigma[/tex])

Accordingly, the system of hypothesis is defined by:

Null hypothesis: mewx=mewy

alternatively, it can also be mewx≠ mewy

Then, the random  sample t is distributed as [tex]t\sim t_n_x+ny-2[/tex] with the degrees of freedom given as df= [tex]n_x+n_y-2[/tex] =40+50-2= 88

So, in this case answer will be 88 degrees of freedom.

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The mean mark of the group would be 8.

Used the concept of mean that states,

Mathematicians find the mean by adding up all the numbers in a data collection, then dividing that total by the number of numbers.

Given that,

There is a group of students who were given a spelling test.

The table shows their marks.

By dividing the total number of students by the sum of each mark's frequency and each mark, you can determine the group's mean grade:

[tex]\text { Mean Mark } = [ (6 \times 5) + (7 \times 4) + (8 \times 7) + (9 \times 10) + (10 \times 4) ] \div30[/tex]

                    [tex]= [ 30 + 28 + 56 + 90 + 40 ] \div 30[/tex]

                    [tex]= 244 \div 30[/tex]

                    [tex]= 8.1333333333...[/tex]

                    [tex]= 8 (\text{ nearest whole number}})[/tex]

Therefore, the value of the mean is 8.

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Answer:

3 ,0

Step-by-step explanation:

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field \nabla f whose value at a point p is the "direction and rate of fastest increase". Wikipedia

Answer:

1/3

Step-by-step explanation:

We need two easy-to-read points on grid intersections.

Look at the point on the y-axis. It is (0, 3).

Now go up to the right until you find the next easy-to-read point.

It is (3, 4) which is on a grid intersection.

gradient = run/rise

rise = distance in y (vertical distance)

run = distance in x (horizontal distance)

We start at (0, 3) and go to (4, 4) by going only vertically and horizontally.

From (0, 3), go 1 unit up. That is a rise of 1.

Then go 3 units right. That is a run of 3.

gradient = rise/run = 1/3

Answer: 1/3

The value of the expression will be;

⇒ 56.5

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ 8.1p + 7.9

And, The values are,

p = 6 and r = 7

Now,

Since, The expression is,

⇒ 8.1p + 7.9

Substitute p = 6, we get;

⇒ 8.1 × 6 + 7.9

⇒ 48.6 + 7.9

⇒ 56.5

Thus, The value of the expression will be;

⇒ 56.5

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[tex]\frac{2x+3}{3}[/tex]

-3 + x - [tex]\frac{1}{3}[/tex]x + 4

-3 + x + [tex]\frac{-x}{3}[/tex] + 4

-3 + x + [tex]\frac{-x}{3}[/tex] + 4

1 + x + [tex]\frac{-x}{3}[/tex]

1 + x + [tex]\frac{-x}{3}[/tex]

[tex]\frac{3}{3}[/tex] + [tex]\frac{3x}{3}[/tex] + [tex]\frac{-x}{3}[/tex]

[tex]\frac{3}{3}[/tex] + [tex]\frac{3x}{3}[/tex] +[tex]\frac{-x}{3}[/tex]

[tex]\frac{3 + 3x - x}{3}[/tex]

[tex]\frac{3 + 3x - x}{3}[/tex]

[tex]\frac{3 + 2x}{3}[/tex]

[tex]\frac{3 + 2x}{3}[/tex]

[tex]\frac{2x+3}{3}[/tex]

[tex]\frac{2x+3}{3}[/tex]

I hope my answer helped you! If you need more information or help, comment down below and I will be sure to respond if I am online. Have a wonderful rest of your day!

The appropriate design for testing the significance of the difference between the means is called an independent-samples t-test.

The Independent Samples t- test analyzes the means of two separate groups to observe if there is statistical support that the population mean values are statistically substantially different.

A parametric test is said to be the Independent Samples t Test and the Independent t Test is another name for this test. Both examine the significance of a discrepancy between two means.

Independent-sample t tests compare scores on the same variable for the two different groups of cases, whereas paired-sample t tests compare scores on two different variables for the same group of instances.

When doing a t-test, it is usual to make the following assumptions: like the measuring scale, random sampling, normality of the data distribution, sufficiency of the sample size, and an equality of variance in standard deviation.

The appropriate design for testing the significance of the difference between the means is - b. independent-samples t-test.

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The measure of the angle with an initial ray in the 3 o'clock position and a terminal ray passing through the car is 0 or origin.

Terminal ray:

In geometry, terminal ray refers the ray in the terminal position, after the rotation, is called the terminal side of the angle.

Given,

Here we need to find if θ measures 0 radians, then the measure of the angle with an initial ray in the 3 o'clock position and a terminal ray passing through the car.

While we looking into the given question,

the value of θ = 0 radians

And the Slope of terminal ray is calculated by using the formula,

=> tan θ

Then Slope = tan 0

Now, by using radians calculator,

=> tan 0 = 0

Therefore, the slope of the terminal ray is  0 which means the origin of the line.

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The probability that mean blood pressure less than 118 is 0.9938.

We have given that,

blood pressure are normally distributed mean of 116

i.e. μ = 116

standard deviation σ = 4.8

n = 36

and we have to find probability that their mean blood pressure will be less than 118.

The formula to find the above probability is,

P(Z=X) = (x - μ) / (σ/[tex]\sqrt{n}[/tex] )

⇒ P(Z < 118) = (118 - 116) / 4.8/[tex]\sqrt{36}[/tex]

                   = 2 / (4.8/6)

                   = 2/0.8

  P(Z < 118) = 2.5

From the z-table the value of z at 2.5 is 0.9938,

⇒ P(Z < 118) = 0.9938.

Therefore the probability that their mean blood pressure will be less than 118 is 0.09938.

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The probabilities for the given hours are,

[tex](a) Pr(100 < x < 250) = 0.302440605\\(b) Pr(\frac{x > 300}{x > 200})=0.670320046\\ (c) Pr(x > 100) = 0.670320046[/tex]

What is probability?

The area of mathematics known as probability deals with numerical descriptions of how likely it is for an event to happen or for a proposition to be true. A number between 0 and 1 is the probability of an event, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes its certainty.

P.D.F , [tex]f(x) = \frac{1}{250}e^\frac{-x}{250} ; x \geq 0\\[/tex]

C.D.F ,  [tex]F(X) = 1-e^\frac{-x}{250} ;x \geq 0[/tex]

(a)

[tex]Pr(100 < x < 250) = F(250) - F(100)\\Pr(100 < x < 250) = (1-e^\frac{-250}{250})-(1-e^\frac{100}{250} )\\ Pr(100 < x < 250) = (e^\frac{-2}{5}-e^-^1) \\ Pr(100 < x < 250) = 0.670320046 - 0.367079441\\ Pr(100 < x < 250) = 0.302440605[/tex]

(b)

[tex]Pr(\frac{x > 300}{x > 200}) = \frac{Pr[(x > 300) intersection(x > 200)]}{Pr(x > 200)} \\ Pr(\frac{x > 300}{x > 200}) = \frac{Pr(x > 300)}{Pr(x > 200)} \\Pr(\frac{x > 300}{x > 200}) = \frac{1 - F(300)}{1-F(200)} \\Pr(\frac{x > 300}{x > 200}) = \frac{e^\frac{-300}{250} }{e^\frac{-200}{250} } \\Pr(\frac{x > 300}{x > 200}) = e^\frac{-2}{5} \\Pr(\frac{x > 300}{x > 200})= 0.670320046[/tex]

(c)

[tex]Pr(x > 100) = 1-Pr(x < 100) = 1- F(100) = e^\frac{-100}{250} = e^\frac{-2}{5} = 0.670320046[/tex]

Hence, the probabilities for the given hours are,

[tex](a) Pr(100 < x < 250) = 0.302440605\\(b) Pr(\frac{x > 300}{x > 200})=0.670320046\\ (c) Pr(x > 100) = 0.670320046[/tex]

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Complete question:

Complete question is attached below,

Answer: 1st point : 0,-1 second point : 1,0

Step-by-step explanation: the y intercept is -1. The slope is 2/1 rise over run. Two rise 1 over

There are three commonly used trigonometric identities.

Sin x = 1/ cosec x

Cos x = 1/ sec x

Tan x = 1/ cot x or sin x / cos x

Cot x = cos x / sin x

sin²x = (sec x - cos x) / sec x is,

sin²x + cos²x = 1

(proved)

There are three commonly used trigonometric identities.

Sin x = 1/ cosec x

Cos x = 1/ sec x

Tan x = 1/ cot x or sin x / cos x

Cot x = cos x / sin x

We have,

sin²x = (sec x - cos x) / sec x

We will prove that,

sin²x + cos²x = 1

sec x = 1/cos x

Now,

sec x - cos x

= 1/cos x - cos x

= (1 - cos²x) / cos x

Now,

sin²x = (1 - cos²x) / [(cos x) x 1/cos x]

sin²x = 1 - cos²x

sin²x + cos²x = 1

Proved

Thus,

sin²x = (sec x - cos x) / sec x is,

sin²x + cos²x = 1

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T is the linear transformation described by T(w,z) in L(C2). This indicates that the linear transformation T changes two-dimensional vectors of the type (w,z) to (-z,w).

To get T's eigenvalues, solve the equation T(v) = v, where is the eigenvalue and v is the eigenvector. By solving this equation, we discover that T's eigenvalues are λ1 = -1 and λ2 = 1.

For each eigenvalue, the associated eigenspaces are the subspaces of C2 that are spanned by the eigenvectors of T. The eigenspace is the subspace covered by the eigenvector λ1= -1 is (1,0). The eigenspace is the subspace spanned by the eigenvector λ2= 1 is (0,1).

As a result, the subspaces covered by (1,0) and (1,1) are the generalized eigenspaces corresponding to the different eigenvalues of T. (0,1).

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A hole can be found at x = -3.

Area is the total amount of space that an object's shape or a flat (2-D) surface occupy.

Create a square on paper by using a pencil. Two dimensions make it up. A shape's area on paper is the space it takes up.

Imagine that your square is made up of smaller unit squares.

The area of a figure is equal to the number of unit squares required to completely cover the surface area of a particular 2-D shape. Square cms, square feet, square inches, square meters, etc. are a few common units for measuring area.

To get the area of the square figures presented below, draw unit squares with 1-centimeter sides. Therefore, the shape will be measured.

According to our question-

When the denominator is zero, the function will have a hole at the x-values and become undefined. In a language of mathematics:

There are two answers to this equation:

The function will have a gap at those x-values.

Hence, A hole can be found at x = -3.

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Answer: If A is true, then B must be true.

The closest distance the family will come to the landmark while on the road is 12.03 miles.

Here, the situation represented in the question can be depicted as shown in the image below.

The family moves from F1 to F2 as the angle changes from 62 to 38.

Now, using trigonometric ratios in the smaller triangle we have-

Cot 62 = X/Y

X = Y Cot 62

Similarly, Cot 38 = (X + 9)/ Y

Substituting the value of X in the above equation we get-

Y Cot 38 = (Y Cot 62 + 9)

Y Cot 38 - Y Cot 62  = 9

Y(Cot 38 - Cot 62) = 9

Y = 9/ (Cot 38 - Cot 62)

Y = 9/ (1.2799 - 0.5317)

Y = 9/ 0.7482

Y = 12.028

Thus, the closest distance the family will come to the landmark while on the road is 12.03 miles.

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Therefore, L has a relative and absolute minimum when x = 400 m

The area is as defined as the amount of space occupied by any shape in two dimensions. Whereas, Perimeter is the boundary or the outline of a flat shape.

Let x = length of the side parallel to the river

     y = length of each of the other two sides

Then xy = 80,000, so y = 80,000/x

Minimize:  L = length of the fence

L = x + 2y

  = x + 160,000/x, where x > 0

L' = 1 - 160,000/x2 = (x2 - 160,000)/x2

L' = 0 when x = 400 or -400

Since x can't be negative, x must be 400.

When 0 < x < 400, L' < 0.  So, L is decreasing.

When x > 400, L' > 0.  So L is increasing.

Therefore, L has a relative and absolute minimum when x = 400 m

                                                                                    y = 80,000/x = 200 m

The fence is shortest if the side parallel to the river has length 400 m and the other 2 sides each have length 200 m.  

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The relationship between RS and RT is RT is the base of this isosceles triangle.

We know two similar planer figures are congruent when we have sides or angles or both that are the same as the corresponding sides or angles or both.

We know the sum of all the interior angles in a triangle is 180°.

Given, A triangle RST,

So, ∠S + ∠T + ∠R = 180°.

50° + 65° + ∠R = 180°.

∠R = 65°.

This is an isosceles triangle with two similar sides RS and ST.

The relation between RS and RT is RT is the base of this isosceles triangle.

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The function will have a value of 0 at x = 1. In that case, the variable "k" will have a value of 2.

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences. A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a connection between inputs in which each input is connected to precisely one output.

Given f(x) = 3x³ + kx – 5, and the factor of the given function is (x – 1).

If the factor (x – 1) is zero. Then the value of the function will be zero.

x – 1 = 0

x = 1

At x = 1, the value of the function will be zero. Then the value of the variable 'k' will be calculated as,

f(1) = 3(1)³ + k(1) – 5

0 = 3 + k – 5

k = 5 – 3

k = 2

The function's value will be 0 at x = 1. The variable "k" will then have a value of 2.

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Answer:  18 cookies

Step-by-step explanation:

27 divided by 1.50 = 18

Answer:

Step-by-step explanation:

6 cookies