The diagonal of rectangle ABCD is 42. 3 cm, and it forms an angle of 53° with the shorter side AD of the rectangle

Domain: {x | x is a real number}

Option D, "{x | x is a real number}" accurately represents the domain of the function.

The domain of the step function f(x) = [2x] - 1, where [2x] represents the greatest integer less than or equal to 2x, can be determined by considering the restrictions on the input values.

In this case, the step function is defined for all real numbers. However, the greatest integer function imposes a restriction. Since the greatest integer function only outputs integers, the input values (2x) must be such that they produce integer outputs.

For any real number x, the greatest integer less than or equal to 2x will always be an integer. Therefore, the domain of the function f(x) = [2x] - 1 is:

Domain: {x | x is a real number}

Option D, "{x | x is a real number}" accurately represents the domain of the function.

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Answer:

When I work with function composition, I usually convert "( f o g)(x)" to the more intuitive " f (g(x))" form. This is not required, but I certainly find it helpful. In this case, I get:

(g o f )(1) = g( f(1))

This means that, working from right to left (or from the inside out), I am plugging x = 1 into f(x), evaluating f(x), and then plugging the result into g(x). I can do the calculations bit by bit, like this: Since f(1) = 2(1) + 3 = 2 + 3 = 5, and since g(5) = –(5)2 + 5 = –25 + 5 = –20, then (g o f )(1) = g( f(1)) = g(5) = –20. Doing the calculations all together (which will be useful later on when we're doing things symbolically), it looks like this:

(g o f )(1) = g( f (1))

  = g(2(   ) + 3)   ... setting up to insert the original input

  = g(2(1) + 3)

  = g(2 + 3)

  = g(5)

  = –(   )2 + 5    ... setting up to insert the new input

  = –(5)2 + 5

  = –25 + 5

  = –20

Note how I wrote each function's rule clearly, leaving open parentheses for where the input (x or whatever) would go. This is a useful technique. Whichever method you use (bit-by-bit or all-in-one), the answer is:

(g o f )(1) = g( f (1)) = –20

I just computed (g o f )(1); the composition can also work in the other order

Step-by-step explanation:

The value of a is 4.

Hence, a = 4

Here,

\(f(x) = (5 - x)^3 and, g(x) = a + x^2,\)

and \(f(g(x)) = (1 - x^{2} )^3\)

We have to find the value of a.

What is composite function?

Function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)).

Now,

\(f(x) = (5 - x)^3 and, g(x) = a + x^2,\)

and \(f(g(x)) = (1 - x^{2} )^3\)

⇒ \(f(g(x)) = f ( a + x^{2} )\)

⇒ \(f ( a + x^{2} ) = ( 5 - a - x^{2} )^3\)

And,

\(f(g(x)) = (1 - x^{2} )^3\)

\(( 5 - a - x^{2} )^3 = ( 1 - x^{2} )^3\\\)

Taking cube root both sides,

\(( 5 - a - x^{2} ) = ( 1 - x^{2} )\\ 5 - a - x^{2} = 1 - x^{2} \\a = 4\)

Hence, The value of a = 4.

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There is only 1 power of 7 contained in 10.

Students use powers of numbers to solve this problem and learn what is occurring to the numbers as a result.

Students also learn how to divide a seemingly vast and challenging equation into smaller, more accessible components. The pupils should understand that raising 7 to a power only produces a finite number of unit digits. Furthermore, as the power of 7 rises, these particular numbers "circle round." 7, 9, 3, and 1 make up this cycle.

The digit in the tens place is the same.

The total number of times we multiply a number is known as its exponent or power. For instance, 2 to the power 3 denotes a 3x3 multiplication of 2.

The largest power of 7 that is less than 10 is 7^1 = 7, therefore there is only 1 power of 7 contained in 10.

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There is only 1 power of 7 contained in 10.

Students use powers of numbers to solve this problem and learn what is occurring to the numbers as a result.

Students also learn how to divide a seemingly vast and challenging equation into smaller, more accessible components. The pupils should understand that raising 7 to a power only produces a finite number of unit digits. Furthermore, as the power of 7 rises, these particular numbers "circle round." 7, 9, 3, and 1 make up this cycle.

The digit in the tens place is the same.

The total number of times we multiply a number is known as its exponent or power. For instance, 2 to the power 3 denotes a 3x3 multiplication of 2.

The largest power of 7 that is less than 10 is 7^1 = 7, therefore there is only 1 power of 7 contained in 10.

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The expression after simplification is 6x - 6.

An expression in math is a statement with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction, multiplication, or division.

The given expression is -4 + 2x + 4x - 2.

\(\sf 2x+4x-4-2\)

\(\sf 6x-4-2\)

\(\bold{6x-6}\)

Hence, the simplification of the expression -4 + 2x + 4x - 2 is 6x - 6.

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The work required to compress the spring an additional 3 inches is 16.875 foot-pounds

The work done by the winch in winding up one-third of the chain is 133.4 foot-pounds.

Seven and one-half foot-pounds of work is required to compress a spring 2 inches from its natural length.

W = (1/2)kx²where W is the work done, k is the spring constant, and x is the displacement of the spring from its natural length.

Let's first find the spring constant:k = (2W)/(x²) = (2 * 7.5)/(2²) = 3.75

The work required to compress the spring an additional 3 inches can be found using the same formula as follows:

W = (1/2)kx² = (1/2)(3.75)(3²) = 16.875 foot-pounds

The work required to compress the spring an additional 3 inches is 16.875 foot-pounds.

8. Consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level.

we need to find the weight of one-third of the chain and the distance it is lifted.

The work done is given by the formula W = Fd, where F is the force applied and d is the distance moved.

Let's first find the weight of the chain:weight of chain = (20 ft)(3 lb/ft) = 60 lb

weight of one-third of chain = (1/3)(60 lb) = 20 lb

The winch needs to lift this weight a distance of one-third of the chain's length, which is (1/3)(20 ft) = 6.67 ft.

The work done by the winch can be found using the formula W = Fd:W = (20 lb)(6.67 ft) = 133.4 foot-pounds

The work done by the winch in winding up one-third of the chain is 133.4 foot-pounds.

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Answer:

y = -3x + 1

Step-by-step explanation:

First, we can find the slope of the line.

The slope is (change in y)/(change in x).

We can use the points that are marked in orange: (0,1) and (1,-2)

The change in y is 1 to -2, which is -3.

The change in x is 0 to 2, which is 1.

The slope will be -3/1, which is also: -3.

We have y = -3x + b currently.

The b is the y-intercept, meaning (the point that touches the y axis, the vertical line).

The point (0,1) touches the y-intercept, so b will be 1.

We can finish our equation as: y = -3x + 1

Answer: look at the explanation

Step-by-step explanation:

It goes down by 40%

Answer:

17/25

Step-by-step explanation:

sorry if I'm wrong

lay them out on.a piece of paper and circle every multiple of 2

With a speed of 0.419 m/s, the rider fast is a rider rising when his seat is 16m above ground level

The formula for tangential velocity is 2*pi*r/T --> 2*pi*10/120 --> pi/6.

It is 6 meters above the center of the wheel while the rider is 16 meters above the ground.

The horizontal portion of this distance must be 8 meters as its separation from the wheel's center is 10 meters (depending on the radius).

This indicates that the velocity and seat both make angles of 36.87 degrees off vertical and arctan(6/8) over the origin, respectively.

We will take into account the vertical component of the tangential velocity since the rate at which the seat is rising is what we're interested in.

Pi/6 * cos(arctan(6/8)) yields a value of 0.419 m/s.

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The aspect ratio represents the ratio of the width to the height of a television.  

The size of a TV is calculated using Pythagoras theorem. Assume the length ratio of the older 25-inch \(4:3\) TV is x.

The ratio is represented as:

\(Width : Height = 4:3\)

Using Pythagoras theorem, we have:

\(Width^2 + Height^2 = 25^2\)

\((4x)^2 + (3x)^2 = 25^2\)

\(16x^2 + 9x^2 = 625\)

\(25x^2 = 625\)

Divide both sides by 25

\(x^2 = 25\)

Take positive square roots

\(x = 5\)

So, the height of the image is:

\(Height = 3x\)

\(Height = 3 \times 5 = 15\)

While the width is

\(Width = 4x = 4 \times 5 = 20\)

Hence, the height of the image is 15 inches

The height of each black bar is calculated as follows:

\(Width : Height = 2.35 : 1\)

Where

\(Width = 20\)

So, we have:

\(20 : Height = 2.35 : 1\)

Express as fraction

\(\frac{Height}{20} = \frac{1}{2.35}\)

Make Height the subject

\(Height = \frac{20}{2.35}\)

\(Height = 8.51\)

The height of each black bar at the top and at the bottom is 8.51 inches

The percentage of the screen’s area that is occupied by the image is calculated using

\(\% Screen = \frac{A_1}{A_2} \times 100\%\)

Where

\(A_1 \to\) Area of black bars

\(A_2 \to\) Area of the 25-inch 4:3 television

So, we have:

\(A_1 =8.51 \times 20 = 170.2\)

\(A_2 =15 \times 20 = 300\)

\(\% Screen = \frac{A_1}{A_2} \times 100\%\) becomes

\(\% Screen = \frac{170.2}{300} \times 100\%\)

\(\% Screen = \frac{170.2}{3} \%\)

\(\% Screen = 56.73 \%\)

Hence, the percentage of the screen area occupied is 56.73%

Consumers moved to TV with a wide screen because a TV with wide screen shows bigger pictures than smaller dimension TV screens,

When the pictures are bigger, the consumers will have a clearer view of what they're watching.

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Answer:

see explanation

Step-by-step explanation:

Given

4x³ + x² - 8x - 2 ( factor the first/second and third/fourth terms )

= x²(4x + 1) - 2(4x + 1) ← first option on list

Finishing the factoring by factoring out (4x + 1) from each term

= (4x + 1)(x² - 2)

Answer:

x2(4x + 1) – 2(4x + 1)

Step-by-step explanation:

it's A on edg

Answer:

answer for a. a^2

answer for b. c

Answer:

A

Step-by-step explanation:

-3×6x=-18x -3×-2=6 -18x+6

The transformation of the graph f(x) = x² to the graph g(x) = -3(x+5)² + 12 involves a horizontal shift, a vertical stretch, and a vertical translation.

The graphs of f(x) = x² and g(x) = -3(x+5)² + 12 are the parabola.

The transformation of the graph is following as:

Firstly, the parabola has been shifted horizontally by 5 units to the left, which is reflected in the expression (x+5)².

Secondly, the parabola has been stretched vertically by a factor of -3, which is reflected in the coefficient in front of (x+5)².

Finally, the parabola has been raised up to 12 units. This means that the vertex of the parabola has been shifted upwards from the origin to the point (−5,12).

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Answer:

Total wrapping paper need = 960.84 inch²

Step-by-step explanation:

Given:

Height of cylinder box = 8 inch

Radius of cylinder box = 9 inch

Find:

Total wrapping paper need

Computation:

Total wrapping paper need = Total surface area of cylinder

Total surface area of cylinder = 2πr(h + r)

Total wrapping paper need = 2[3.14][9][8+9]

Total wrapping paper need = 2[3.14][9][17]

Total wrapping paper need = [6.28][9][17]

Total wrapping paper need = [56.52][17]

Total wrapping paper need = 960.84 inch²

Answer:

(-6, 7)

Step-by-step explanation:

Therefore , the solution to this problem of resistance comes out to be  R = 50 ohms.

Resistance is the degree to which a substance obstructs or opposes an electric current. To better understand resistance, imagine how difficult it would be for a person to move from one shop to another in a crowded market.

Here,

Given: current = 2 amperes

and Voltage = 100 Volts

Using Ohm's Law,

V = I * R

At 100 v, current = 2 A

To calculate resistance

100 =  2 * R

R = 50 Ω

Resistance of material, R = 50 Ω

Therefore , the solution to this problem of resistance comes out to be  R = 50 ohms.

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The lengths of the sides of the triangle with vertices A(2,-1,4), B(-2,3,9), and C(6,4,8) are approximately 10.63, 7.07, and 7.81 units.

To find the lengths of the sides of the triangle, we can use the distance formula in three-dimensional space. The distance formula is derived from the Pythagorean theorem, where the distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by:

d(PQ) = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

Applying this formula to our triangle, we can calculate the lengths of the sides as follows:

1. Side AB:

  AB = √((-2 - 2)² + (3 - (-1))² + (9 - 4)²)

     = √((-4)² + (4)² + (5)²)

     ≈ √(16 + 16 + 25)

     ≈ √57

     ≈ 7.55 units (rounded to two decimal places)

2. Side BC:

  BC = √((6 - (-2))² + (4 - 3)² + (8 - 9)²)

     = √((8)² + (1)² + (-1)²)

     ≈ √(64 + 1 + 1)

     ≈ √66

     ≈ 8.12 units (rounded to two decimal places)

3. Side CA:

  CA = √((6 - 2)² + (4 - (-1))² + (8 - 4)²)

     = √((4)² + (5)² + (4)²)

     ≈ √(16 + 25 + 16)

     ≈ √57

     ≈ 7.55 units (rounded to two decimal places)

Therefore, the lengths of the sides of the triangle ABC are approximately 7.55, 8.12, and 7.55 units.

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Answer: Undefined, or 0

Step-by-step explanation:

    We will use the formula for slope. Let (-8, 6) be point 1 and (-8, 2) be point 2.

\(\displaystyle \frac{y_{2} -y_{1} }{x_{2} -x_{1} }\)

\(\displaystyle \frac{2-6 }{-8--8}\)

\(\displaystyle \frac{2-6 }{-8+8}\)

\(\displaystyle \frac{-4 }{0}\)

  You cannot divide by 0. This means the line is straight up/down, or vertical, and has an undefined slope.

The expression which represents the volume of the prism, in cubic units, is one-half x³+ x².

Volume of prism is the amount of quantity, which is obtained in it in the 3 dimensional space. The volume of a prism is the product of the area of its base and height.

\(V=A_b\times h\)

Here, (\(A_b\)) is the area of the base and (h) is the height of the prism.

The side of the base of the triangular prism is x units. Thus the area of the base is,

\(A_b=\dfrac{x\times x}{2}\\A_b=\dfrac{x^2}{2}\)

The height of this prism is (x+2) units long. Thus, the volume of the prism is,

\(V=\dfrac{x^2}{2}\times (x+2)\\V=\dfrac{1}{2}x^3+x^2\)

Thus, the expression which represents the volume of the prism, in cubic units, is one-half x³+ x².

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Answer:

1. A- 1/2x^3

2. 168

3. 927

4A. 48

4B. 336

5. A D

6. 1 3 3

7. 12 cm

8. 8 , 4

9. B

10. A C D

Step-by-step explanation:

whole thing right on edge

No, the ratios 10:12 and 1:21 are not form a proportion.

A relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.

Given that;

The ratios are,

⇒ 10 : 12 and 1 : 21

We know that;

Two ratio are form a proportional when both are in same quantity or in a a same ratio.

Here, The ratios are,

⇒ 10 : 12 and 1 : 21

Clearly, We get;

10 / 12 = 5 / 6

1 / 21 = 1  21

Thus,

1 / 21 ≠ 5 /6

1 : 21 ≠ 10 : 12

Therefore, The ratios 10:12 and 1:21 are not form a proportion.

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Answer:

the most logical answer is

n^2-1

20th term is then 399

Step-by-step explanation:

The rule behind the sequence could be anything, but it is very likely that we're supposed to notice that the terms are all squares minus 1

0 = 1*1 - 1

3 = 2*2 - 1

8 = 3*3 - 1

15 = 4*4 - 1

24 = 5*5 - 1

So we expect that the explicit formula is n^2-1 and the 20th term is 399

Answer:

-5-2

Step-by-step explanation:

10*+8=8

-5-2

big one is a great place to be written and then we are going to be able hi I love the way your house for me either ir you are you excited hi I love thet are ready for the good of the good

Considering the definition of permutation, in 95,040 different ways he can arrange homes on the lots.

Permutation is placing elements in different positions. So, permutations of m elements in n positions are called the different ways in which the m elements can be arranged occupying only the n positions.

In other words, permutations are ways of grouping elements of a set in which:

To obtain the total of ways in which m elements can be placed in n positions, the following expression is used:

\(m Pn=\frac{m!}{(m-n)!}\)

where "!" indicates the factorial of a positive integer, which is defined as the product of all natural numbers before or equal to it.

In this case, you know:

To obtain the total of ways in which he can arrange homes on these lots, you use the permutation. This is, 12 elements o models can be arranged occupying only the 5 positions:

\(12P5=\frac{12!}{(12-5)!}\)

Solving:

\(12P5=\frac{12!}{7!}\)

12P5= 479,001,600 ÷ 5,040

12P5=95,040

Finally, in 95,040 different ways he can arrange homes on the lots.

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Answer:

5)a. π(14²)x = 4,116π

b. x = 4,116/196 = 21

c. The height is 21 feet.

Answer:

An exponential function is a function of the form

f(x)=bx

where b≠1 is a positive real number. The domain of an exponential function is (−∞,∞) and the range is (0,∞).

Solve the equation: 52x−3=752x−3=7.

Since we can’t easily rewrite both sides as exponentials with the same base, we’ll use logarithms instead. Above we said that logb(x)=ylogb⁡(x)=y means that by=xby=x. That statement means that each exponential equation has an equivalent logarithmic form and vice-versa. We’ll convert to a logarithmic equation and solve from there.

52x−3log

⎛⎝⎜

⎞⎠⎟=7=2x−352x−3=7log5

⁡(7

)=2x−3

From here, we can solve for xx directly.

2xx=log5(7)+3=log5(7)+32

A logarithmic function is a function defined as follows

logb(x)=ymeans thatby=xlogb⁡(x)=ymeans thatby=x

where b≠1b≠1 is a positive real number. The domain of a logarithmic function is (0,∞)(0,∞) and the range is (−∞,∞)(−∞,∞).

Solve the equation:

log3(2x+1)=1−log3(x+2).log3⁡(2x+1)=1−log3⁡(x+2).

With more than one logarithm, we’ll typically try to use the properties of logarithms to combine them into a single term.

log3(2x+1)log3(2x+1)+log3(x+2)log3((2x+1)(x+2))log3(2x2+5x+2)2x2+5x+22x2+5x−1=1−log3(x+2)=1=1=1=3=0log3⁡(2x+1)=1−log3⁡(x+2)log3⁡(2x+1)+log3⁡(x+2)=1log3⁡((2x+1)(x+2))=1log3⁡(2x2+5x+2)=12x2+5x+2=32x2+5x−1=0

Let’s use quadratic formula to solve this.

x=−5±52−4⋅2⋅−1−−−−−−−−−−−√2⋅2=−5±

−−−−−−−−⎷4.x=−5±52−4⋅2⋅−12⋅2=−5±33

4.

What happens if we try to plug x=

11 cars needed to take 53 girls to mach if each car can take only 5 girls .

Answer:

Step-by-step explanation:

53 : 5 = 10.6

53:5=10,6

5        

 3 (rest)      

 0      

 30      

 30      

   0      

10 cars carry 5 girls (10 x 5 = 50) and one car carries 3, so you need 11 cars

To the nearest whole number, the probability of rolling a 2 and a 4 is 2.8%.

A 2 rolling up on the first throw has a 1/6 chance of happening. Given that a 2 was rolled on the first throw, the likelihood of rolling a 4 on the second throw is equally 1/6. We will have to multiply the probabilities to get the probability of both the events happening,

= (1/6)x(1/6)

= 1/36.

We get 2.8% after converting to a percentage and rounding to the next full amount. So, the chance of rolling a 2 and then a 4 is found to be 2.8% chance.

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Based on the definition of congruent triangles, we can state that: C. They are not congruent because only one pair of corresponding sides is congruent.

Congruent triangles refer to a pair of triangles that possess identical shape and size. If two triangles are congruent, it implies that their corresponding sides and angles are precisely equal.

The image that is attached below shows the diagram of triangles FGH and FJH. We see that only one pair of sides are congruent (FH = FH).

This is not enough to conclude that both triangles are congruent. Therefore, we can conclude that:

C. They are not congruent because only one pair of corresponding sides is congruent.

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