The table shows the results of a survey of male and female students about whether their school should adopt a dress code. What is the probability that a randomly selected male student says that the school should adopt a dress code? Give your answer as a decimal rounded to the nearest thousandths

No, the expression (x^3 - 1) / (x^2 - 1) does not simplify to x.

To simplify the expression, let's first factorize both the numerator and denominator.

The numerator can be factorized using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). So, we have (x^3 - 1) = (x - 1)(x^2 + x + 1).

The denominator is a difference of squares: a^2 - b^2 = (a - b)(a + b). Therefore, (x^2 - 1) = (x - 1)(x + 1).

Now, we can simplify the expression by canceling out the common factors in the numerator and denominator:

[(x - 1)(x^2 + x + 1)] / [(x - 1)(x + 1)]

The (x - 1) terms cancel out, leaving us with:

x^2 + x + 1 / (x + 1)

So, the simplified form of the expression is (x^2 + x + 1) / (x + 1), which is not equal to x.

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Answer:

Length of the rectangle is 8 in and the breadth is 8 in.

a. \(A=xy\)

b. \(y=16-x\)

Step-by-step explanation:

Length = x

Breadth = y

Area is given by \(A=xy\)

Perimeter of rectangle is given by

\(2(x+y)=32\\\Rightarrow x+y=16\\\Rightarrow y=16-x\)

The condition that x and must satisfy is \(y=16-x\)

So, area is

\(A(x)=x(16-x)\\\Rightarrow A(x)=16x-x^2\)

Differentiating with respect to x we get

\(A'(x)=16-2x\)

Equating with zero

\(0=16-2x\\\Rightarrow x=\dfrac{16}{2}\\\Rightarrow x=8\)

Double derivative of A(x)

\(A''(x)=-2\)

So \(A''(x)<0\) which means \(A(x)\) is maximum at \(x = 8\)

\(y=16-x=16-8\\\Rightarrow y=8\)

So length of the rectangle is 8 in and the breadth is 8 in.

The unit circle has the co-ordinates (-√2/2, -√2/2)

In mathematics, trigonometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. The six trigonometric functions are Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent.

Given here: The unit circle and the point makes an angle of 225

we know the co-ordinates of any point on the unit circle is given by

x=cost and y=sint where t is the angle that line passing through the origin containing the point makes with x-axis

Thus x=cos225

           =-√2/2

and y=sin225

        =-√2/2

Hence, The required point is given by co-ordinates (-√2/2, -√2/2)

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To find the length of arc LM, we need to use the formula for the circumference of a circle:

C = 2πr

In this case, the arc LM is a portion of the circumference of the circle. To find the length of the arc, we need to determine the angle at the center of the circle that corresponds to the arc.

Looking at the image you provided, we can see that angle LOM is a right angle (90 degrees). The angle at the center of the circle, LOCM, is half of the right angle, which is 45 degrees.

To calculate the length of arc LM, we need to find the fraction of the total circumference represented by the central angle LOCM. The fraction is given by:

Fraction of circumference = Angle at center / 360 degrees

Fraction of circumference = 45 degrees / 360 degrees

Fraction of circumference = 1/8

Now, we can calculate the length of arc LM:

Length of arc LM = Fraction of circumference * Circumference

Length of arc LM = (1/8) * (2πr)

Since the radius (r) is not given in the image, we cannot calculate the exact length of arc LM. However, we can provide the formula for the length of arc LM in terms of the radius:

Length of arc LM = (1/8) * (2πr) = πr/4

So, the length of arc LM is πr/4, where r is the radius of the circle.

If you have the value of the radius, you can substitute it into the formula to find the specific length of arc LM.

Answer:

See Explanation

Step-by-step explanation:

I think you mean how do you find linear equations. It is actually very simple when you memorize the formula. Linear equations are in the form of y = mx + b. m is the slope. b is the y-intercept.

ex. given: slope = 5, y-intercept = -3: our equation is y = 5x - 3

Slope is rise/run

y-intercept is (0,y) or you substitute.

Answer: In that picture the slope is increasing

Step-by-step explanation: positive rise and positive run (uphill from left to right)

The least amount of paint that is needed to paint the walls of a room with a rectangular prism shape is 1. 8 gallons.

There would be two walls with 10 x 16 dimensions so the area is :

= 2 x 10 x 16

= 320 square feet

Two walls with 20 x 10 dimensions :

= 2 x 20 x 10

= 400 square feet

The total area is :

= 400 + 320

= 720 square feet

One gallon of paint can cover 400 square feet so the number of gallons needed is:

= 720 / 400

= 1. 8 gallons

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Full question is:

One gallon of paint covers 400 square feet. What is the least amount of paint needed to paint the walls of a room in the shape of a rectangular prism with a length of 20 feet, a width of 16 feet, and a height of 10 feet? Write your answer as a decimal.

Answer:

(a) \(y = \frac{4}{x^2}\)

(b) \(x = \frac{2}{5}\)

Step-by-step explanation:

Given

Variation: Inverse proportional.

This is represented as:

\(y\ \alpha\ \frac{1}{x^2}\)

See attachment for table

Solving (a):

First convert variation to equation

\(y = k\frac{1}{x^2}\)

From the table:

\((x,y) = (1,4)\)

So, we have:

\(4 = k * \frac{1}{1^2}\)

\(4 = k * \frac{1}{1}\)

\(4 = k * 1\)

\(4 = k\)

\(k = 4\)

Substitute 4 for k in \(y = k\frac{1}{x^2}\)

\(y = 4 * \frac{1}{x^2}\)

\(y = \frac{4}{x^2}\)

Solving (b): x when y = 25.

Substitute 25 for y in \(y = \frac{4}{x^2}\)

\(25 = \frac{4}{x^2}\)

Cross Multiply

\(25 * x^2 = 4\)

Divide through by 25

\(x^2 = \frac{4}{25}\)

Take positive square roots of both sides

\(x = \sqrt{\frac{4}{25}\)

\(x = \frac{2}{5}\)

The volume of the solid of revolution formed by revolving the region bounded by f(x) = -3x² + 8 and g(x) = 3x² + 2 about the x-axis is 16π cubic units.

To find the volume of the solid of revolution formed by revolving the region bounded by the curves f(x) = -3x² + 8 and g(x) = 3x² + 2 about the x-axis, we can use the method of cylindrical shells and integrate.

First, let's find the points of intersection between the two curves by setting them equal to each other:

-3x² + 8 = 3x² + 2

Rearranging the equation, we get:

6x² = 6

x² = 1

x = ±1

So, the curves intersect at x = -1 and x = 1.

To calculate the volume, we'll integrate along the x-axis from x = -1 to x = 1. The volume of each cylindrical shell can be determined by multiplying the circumference (2πy) by the height (dx), where y represents the distance from the axis of revolution to the curve at a given x-value.

The radius of the cylindrical shell, y, can be obtained by subtracting the lower curve (f(x)) from the upper curve (g(x)). Therefore, y = (3x² + 2) - (-3x² + 8) = 6x² - 6.

The integral to compute the volume of the solid can be expressed as:

V = ∫[from -1 to 1] 2π(6x² - 6) dx

V = 2π ∫[from -1 to 1] (6x² - 6) dx

Simplifying and evaluating the integral, we get:

V = 2π [2x³ - 6x] [from -1 to 1]

V = 2π [(2(1)³ - 6(1)) - (2(-1)³ - 6(-1))]

V = 2π [2 - 6 - (-2 + 6)]

V = 2π [8]

V = 16π

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Answer:

Step-by-step explanation:

From the given information:

The present value of the house = 128000

interest rate compounded monthly r = 7.8% = 0.078

number of months in a year n= 12

duration of time t = 30 years

To find their regular monthly payment, we have:

\(PV = P \begin {bmatrix} \dfrac{1 - (1 + \dfrac{r}{n})^{-nt}}{\dfrac{r}{n}} \end {bmatrix}\)

\(128000 = P \begin {bmatrix} \dfrac{1 - (1 + \dfrac{0.078}{12})^{- 12*30}}{\dfrac{0.078}{12}} \end {bmatrix}\)

128000 = 138.914 P

P = 128000/138.914

P = $921.433

∴ Their regular monthly payment P = $921.433

To find the unpaid balance when they begin paying the $1400.

when they begin the payment ,

t = 30 year - 5years

t= 25 years

\(PV= 921.433 \begin {bmatrix} \dfrac{1 - (1 - \dfrac{0.078}{12})^{25*30}}{\dfrac{0.078}{12}} \end {bmatrix}\)

PV = $121718.2714

C) In order to estimate how many payments of $1400  it will take to pay off the loan, we have:

\(121718.2714 = \begin {bmatrix} \dfrac{1300 (1 - \dfrac{12.078}{12}))^{-nt}}{\dfrac{0.078}{12}} \end {bmatrix}\)

\(121718.2714 = 200000 \begin {bmatrix} (1 - \dfrac{12.078}{12}))^{-nt} \end {bmatrix}\)

\(\dfrac{121718.2714}{200000 } = \begin {bmatrix} (1 - \dfrac{12.078}{12}))^{-nt} \end {bmatrix}\)

\(0.60859 = \begin {bmatrix} (1 - \dfrac{12}{12.078}))^{nt} \end {bmatrix}\)

\(0.60859 = (0.006458)^{nt}\)

\(nt = \dfrac{0.60859}{0.006458}\)

nt = 94.238 payments is required to pay off the loan.

How much interest will they save by paying the loan using the number of payments from part (c)?

The total amount of interest payed on $921.433 = 921.433 × 30(12) years

= 331715.88

The total amount paid using 921.433 and 1300 = (921.433 × 60 )+( 1300 + 94.238)

= 177795.38

The amount of interest saved = 331715.88  - 177795.38

The amount of interest saved = $153920.5

The amount of the investment after 8 years will be, $4,357.64

Given, $500 is invested in an account earning 7% interest compounded quarterly.

We have to find the value of the invested amount after 8 years,

as, Amount = P(1 + r/100)^t

where, P is the amount of money invested, r is the rate of interest and t is the amount of time

Amount = 500(1 + 7/100)^(8×4)

Amount = 500(1.07)^32

Amount = 500×8.715

Amount = 4,357.635

So, the amount of the investment after 8 years will be, $4,357.64

Hence, the amount of the investment after 8 years will be, $4,357.64

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The formula for Pythagoras theorem is -

(hypotenuse)² = (base)² + (perpendicular)².

Given is to discuss the Pythagoras theorem.

Pythagoras theorem states that the square of the hypotenuse is equivalent to the square of the base and perpendicular.

We can write the formula for Pythagoras theorem as -

(hypotenuse)² = (base)² + (perpendicular)²

Therefore, the formula for Pythagoras theorem is -

(hypotenuse)² = (base)² + (perpendicular)².

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{Question in english -

What is the Pythagorean theorem? Formula}

Answer:

Train 1: 130 km/h

Train 2: 108 km/h

Step-by-step explanation:

Let train 1 be the faster train.

Train 1:

distance = d

speed = s

time = 2 hours

Train 2:

distance = 476 - d

speed = s - 22

time = 2 hours

speed = distance/time

distance = speed × time

Train 1:

d = 2s

Train 2:

476 - d = 2(s - 22)

d = 2s

476 - d = 2s - 44

d = 2s

476 - 2s = 2s - 44

520 = 4s

s = 130

s - 22 = 108

Train 1: 130 km/h

Train 2: 108 km/h

Jessica would have to drive 20 miles for the cost to rent a moving truck for one day to be the same at Rentals Plus and Speedy Move.

Let's assume the number of miles driven is represented by 'm'. The cost of renting a moving truck for one day at Rentals Plus can be calculated using the equation:

Cost at Rentals Plus = $5.24 + $4.46 * m

Similarly, the cost of renting a moving truck for one day at Speedy Move can be calculated using the equation:

Cost at Speedy Move = $85.24 + $0.46 * m

To find the number of miles that makes the costs equal, we can set up the equation:

$5.24 + $4.46 * m = $85.24 + $0.46 * m

By rearranging the equation, we can solve for 'm':

$4.46 * m - $0.46 * m = $85.24 - $5.24

$4 * m = $80

m = $80 / $4

m = 20

Therefore, Jessica would have to drive 20 miles for the cost to rent a moving truck for one day to be the same at Rentals Plus and Speedy Move.

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The Area of the shaded portion is: 72 square inches

The formula for the area of a triangle is given by the formula:

Area = ¹/₂ * base * height

Now, to get the area of the shaded portion, we will get the area of the larger triangle and subtract the area of the smaller one from it.

Thus:

Area of larger triangle = ¹/₂ *  17 * 13 = 110.5 square inches

Area of smaller triangle =  ¹/₂ * 11 * 7 = 38.5 square inches

Thus:

Area of shaded portion = 110.5 square inches - 38.5 square inches

Area of shaded portion = 72 square inches

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The perimeter of the shape in simplified radical form is 6√5 + 10√3 or 2(3√5 + 5√3).

The symbol √ is used to represent or show that a number is a radical. Radical expression is defined as any expression containing a radical (√) symbol.

We shall simplify the exact perimeter of the shape with side lengths, √125 cm, 2√12 cm, √5 cm, and √108 cm as follows;

√125 cm = √(5 × 25) cm

√125 cm = √5 × √25) cm

√125 cm = √5 × 5 cm

√125 cm = 5√5 cm

2√12 cm = 2√(3 × 4) cm

2√12 cm = 2√3 × √4 cm

2√12 cm = 2√3 × 2 cm

2√12 cm = 4√3 cm

√5 cm

√108 cm = √(3 × 36) cm

√108 cm = √3 × √36 cm

√108 cm = √3 × 6 cm

√108 cm = 6√3 cm

perimeter of the shape = 5√5 cm + √5 cm + 4√3 cm + 6√3 cm

perimeter of the shape = 6√5 cm + 10√3 cm

perimeter of the shape = (6√5 + 10√3) cm

perimeter of the shape = 2(3√5 + 5√3) cm

Therefore, (6√5 + 10√3) cm or 2(3√5 + 5√3) cm is the simplified radical form for the perimeter of the shape.

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Answer:

B. 54 chairs

Step-by-step explanation:

9*4=36

9*5=45

9*6=54

9*7=63

The number of chairs for the 6 tables is 54 in the given table so, option B is correct.

The fundamental concept of making the same number additions repeatedly is represented by the action of multiplication. The results of multiplying two or more integers are known as the products, and the factors that are used in the multiplication are referred to as the factors.

Given:

Number of Tables 4, 5, 6, 7

Number of Chairs 36, 45, ? 63

As you can see that the number of chairs can be given as,

Number of chairs = 9 × Number of table

Calculate the number of chairs for 6 tables as shown below,

Number of chairs = 9 × 6

Number of chairs = 54

Thus, the number of chairs for 6 tables is 54.

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Answer:

40+20=72-12

60=60

Step-by-step explanation:

4*10+4*5=6*12-6*2

40+20=72-12

60+60

Answer:

The exact value of the other leg is 10.

Step-by-step explanation:

Finding the exact value of x given a 45 degree angle and a side length of 10 can be done using trigonometry. In a 45-45-90 right triangle, the two legs are congruent and the hypotenuse is equal to the square root of 2 times the length of the legs.

Therefore, if one leg is 10, the hypotenuse is 10 times the square root of 2. To find the length of the other leg, we can use the Pythagorean theorem:

c^2 = a^2 + b^2, where c is the hypotenuse, a and b are the legs of the right triangle.

Substituting known values, we get:

(10√2)^2 = 10^2 + b^2

200 = 100 + b^2

b^2 = 100

b = 10

Therefore, the exact value of the other leg is 10.

Answer: To calculate the average cost per widget, we need to consider both the fixed costs and the variable costs.

Fixed costs: $34,000

Variable costs per widget: $5.00

Total costs = Fixed costs + (Variable costs per widget × Number of widgets)

Total costs = $34,000 + ($5.00 × 7,000)

Total costs = $34,000 + $35,000

Total costs = $69,000

Average cost per widget = Total costs / Number of widgets

Average cost per widget = $69,000 / 7,000

Average cost per widget ≈ $9.86

Therefore, the average cost per widget of producing 7,000 Wild Widgets is approximately $9.86.

Step-by-step explanation: :)

According to the given charges, the mean of the sum would be as follows $1,692.5 and the difference of the amount of money earned by Jennie and Paul would be $-232.5

Mean:

In statistics, mean refers the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers

Given,

As Jennie charges $200 per passenger for a sunset sailing tour on her catamaran, while Paul charges $250 per passenger. we can define the amount of money each person earns per cruise as a random variable.

Here we need to find the mean and the difference of the amount of money earned by Jennie and Paul.

Let us consider x be the amount earned by Jennie and y be the amount earned by Paul.

While we looking into the given question , we have identified the value of the following,

=> Money earned by Jennie(A) = 200x

=> Money earned by Paul(B) =  250 X

Here the value of μ = $730 and σ = $ 192.62 for Jennie

And the value of Paul is μ = $962.5 and σ = $253.425

Then the mean of the sum is calculated as,

=> 730 + 962.5 = 1292.5

Similarly, the difference of the amount of money is calculated as,

=> -232.5

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Answer:

D) On average, Jennie makes $232.50 less than Paul in a typical cruise.

Step-by-step explantion

If the given function is f(x)=x², the value of g(x) will be (3x)².

Each value in the domain is connected to exactly one value in the range according to the function, which is described as a specific form of relationship. They have a predetermined range and domain.

It is given that the coordinate values on the graph g (x) are (1, 9).

Now, For the given function,

g (x) = (3x)²

At point (1, 9)

g (1) = (3(1))²

g(1) = 3²

g( 1) = 9

Since, g(x) = (3x)² satisfies the given coordinate values it will be the obtained result.

Thus, the obtained function will be g(x)= (3x)².

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According to the given data we have the following:

Debra started washing at 8:33 AM

finished at 9:22 AM

One easy way to find out the amount of time it took to her washing her clothes would be the following:

Debra started washing at 8:33 AM, so from 8:33 am to 9 o clock 27 minutes passed.

So, if she finished at 9:22 AM, that means that we would have to add to the 27 minutes another 22 minutes, so 27+22=49 minutes.

Therefore, it took to her 49 minutes to wash her clothes.

Answer:

x^2 + 3x + 9 = (x + 3)(x + 3)

When x = 2, we have:

(2 + 3) * (2 + 3)

5 * 5

25

To solve this problem, we need to use Charles's law, which states that, at constant pressure, the volume of a sample of gas is directly proportional to its temperature.

The law can be expressed mathematically as:

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{ \frac{V_1}{T_1}=\frac{V_2}{T_2} } \end{gathered}$} }\)

Where:

Now we obtain the data to continue solving:

Now, we will convert the temperatures to Kelvin:

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{T_1=220 \ ^{\circ}C+273=493 \ Kelvin} \end{gathered}$} }\)

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{T_2=40 \ ^{\circ}C+273= 313 \ Kelvin} \end{gathered}$} }\)

Now we solve for V₂:

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V_2=\frac{V_1T_2}{T_1 } } \end{gathered}$} }\)

Where:

Now, we substitute the values in the formula:

\(\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V_2=\frac{(0.5 \ L\times313\not{k} )}{493\not{k} } } \end{gathered}$} }\)

\(\boxed{\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V_2\approx0.32 \ Liters } \end{gathered}$} }}\)

Aircraft A has 312 seats.

Let's assume that Aircraft B has x seats.

According to the given information, Aircraft A has 105 more seats than Aircraft B. So, the number of seats in Aircraft A can be expressed as x + 105.

The total number of seats in both aircraft is 519, which can be represented by the equation:

x + (x + 105) = 519

Simplifying this equation, we have:

2x + 105 = 519

Subtracting 105 from both sides, we get:

2x = 414

Dividing both sides by 2, we find:

x = 207

Therefore, Aircraft B has 207 seats.

To find the number of seats in Aircraft A, we substitute the value of x back into the expression x + 105:

Aircraft A = 207 + 105 = 312

Hence, Aircraft A has 312 seats.

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Using translation concepts, we have that:

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.

For function f(x), the multiplication by -1 of the function is in the output, meaning that it was a reflection over the y-axis. For function g(x), the multiplication is in the input, the domain, hence the reflection was over the x-axis.

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Case 1: a = 0; b = 0 --> Real Number

Case 2: a = 0; b ≠ 0 --> Non-Real Complex Number

Case 3: a ≠ 0; b = 0 --> Real Number

Case 4: a ≠ 0; b ≠ 0 --> Non-Real Complex Number

For each case, we can determine whether the expression a + bi represents a real number or a non-real complex number based on the values of a and b.

Case 1: a = 0; b = 0

In this case, both a and b are zero. The expression a + bi simplifies to 0 + 0i, which is equal to 0. Therefore, the expression represents a real number.

Case 2: a = 0; b ≠ 0

Here, a is zero, but b is nonzero. The expression a + bi becomes 0 + bi, where b is a nonzero real number multiplied by the imaginary unit i. Since the expression contains a nonzero imaginary part, it represents a non-real complex number.

Case 3: a ≠ 0; b = 0

In this case, a is nonzero, but b is zero. The expression a + bi simplifies to a + 0i, which is equal to a. As there is no imaginary part in the expression, it represents a real number.

Case 4: a ≠ 0; b ≠ 0

Here, both a and b are nonzero. The expression a + bi contains both a real part (a) and an imaginary part (bi). Thus, it represents a non-real complex number.

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Answer: 24

Step-by-step explanation: quick brain math