A pendulum swings 80 cm on its first​ swing, 76 cm on its second​ swing, 72. 2 cm on its third​ swing, and 68. 59 cm on its fourth swing

The solutions are x = 3√2 and x = -3√2.

There are several methods that can be used to solve a quadratic equation. The most commonly used methods are:

1. Factoring
2. Completing the square
3. Quadratic formula
4. Graphing

Now, let's solve each equation using one of the methods listed above.

a. 2x^2 - 7x + 3 = 0

We can use the factoring method to solve this equation. First, we need to find two numbers that multiply to give us 3 and add to give us -7. The numbers are -3 and -1. So, we can factor the equation as follows:

(2x - 1)(x - 3) = 0

Now, we can set each factor equal to zero and solve for x:

2x - 1 = 0 => x = 1/2

x - 3 = 0 => x = 3

So, the solutions are x = 1/2 and x = 3.

b. x^2 + 10x + 1 = 0

We can use the quadratic formula to solve this equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac))/(2a)

Plugging in the values of a, b, and c, we get:

x = (-(10) ± √((10)^2 - 4(1)(1)))/(2(1))

Simplifying, we get:

x = (-10 ± √96)/2

So, the solutions are x = (-10 + √96)/2 and x = (-10 - √96)/2.

c. x² – 18=0

We can use the square root method to solve this equation. First, we need to isolate the x^2 term on one side of the equation:

x^2 = 18

Now, we can take the square root of both sides:

x = ±√18

Simplifying, we get:

x = ±3√2

the solutions are x = 3√2 and x = -3√2.

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Answer:

Step-by-step explanation:

300

Answer:150 m

Step-by-step explanation:

2cm/25 m = 2 cm: 2500 cm= 1cm:1250 1250x12=15000 convert back to meters 150 m

a) To perform a regression analysis based on the results of Table 1, we need to write down a linear regression model that captures the possibility of interaction between factors A and B. The model can be written as follows: Y = B0 + B1*A + B2*B + B3*A*B. Where Y is the dependent variable, A and B are the independent variables, B0 is the intercept, B1 and B2 are the coefficients of A and B respectively, and B3 is the coefficient of the interaction term A*B.

b) To estimate the coefficient B2, we need to use the regression analysis results from Table 1. The estimate of B2 can be obtained by looking at the coefficient of the independent variable B in the table. If the table does not provide the estimate of B2, we can use the following formula to calculate it: B2 = (SSB - SSB/A)/(dfB - dfB/A)

Where SSB is the sum of squares for factor B, SSB/A is the sum of squares for factor B after adjusting for factor A, dfB is the degrees of freedom for factor B, and dfB/A is the degrees of freedom for factor B after adjusting for factor A.

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N(A) is an ideal of R and N(N(A)) = N(A).

a) To prove that N(A) is an ideal of R, we need to show that it is closed under addition and multiplication by elements of R.

Let x, y ∈ N(A) and r ∈ R. Then there exist integers m and n such that xm ∈ A and yn ∈ A. By the commutative property of R, we have:

(x + y)n = xn + xny + yxn + yn ∈ A

(rx)n = rnxn ∈ A

Therefore, x + y ∈ N(A) and rx ∈ N(A), so N(A) is an ideal of R.


b) To prove that N(N(A)) = N(A), we need to show that N(N(A)) ⊆ N(A) and N(A) ⊆ N(N(A)).

Let x ∈ N(N(A)). Then there exists an integer n such that xn ∈ N(A). This means that there exists an integer m such that (xn)m ∈ A. By the associative property of R, we have:

(xn)m = xnm ∈ A

Therefore, x ∈ N(A), so N(N(A)) ⊆ N(A).

Let x ∈ N(A). Then there exists an integer n such that xn ∈ A. Since A ⊆ N(A), we have xn ∈ N(A). Therefore, x ∈ N(N(A)), so N(A) ⊆ N(N(A)).

Hence, N(N(A)) = N(A).
Conclusion: N(A) is an ideal of R and N(N(A)) = N(A).

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To find the factor of these two rational expressions, we need to factor each expression separately and then compare them to find any common factors.

First, let's factor the expression 22x + 11 x2 – 3x – 10:
(22x + 11 x2) + (-3x - 10)
11x(2 + x) - (3x + 10)
(11x - 10)(x + 2)

Next, let's factor the expression 1 – 2c 20c2 + 10c:
(1 - 2c) + (20c2 + 10c)
(1 - 2c) + 10c(2c + 1)
(1 - 2c)(10c + 1)

Now, we can compare the factors of each expression to see if there are any common factors. In this case, there are no common factors between the two expressions.

Therefore, the factor for these two rational expressions is 1.

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Therefore , the solution of the given problem of unitary method comes out to be the religious gathering thus had 144 more males than girls.

To finish the job using the unitary method, multiply the measures taken from this microsecond variable section by two. In a nutshell, when a wanted thing is present, the characterized by a group but also colour groups are both eliminated from the expression unit technique. For instance, 40 changeable-price pencils would cost Inr ($1.01). It's conceivable that one nation will have complete control over the strategy used to achieve this. Almost all living things have a unique trait.

Here,

Find out how many people and kids are attending the event first.

There were 3 kids for every 4 people. We can thus divide the overall population by the sum of the ratios as follows:

=> 4 + 3 = 7

Adult population

=> (4/7) x 560 = 320

Children's number

=> (3/7) x 560 = 240

Now that we know that the majority of the kids were males,

Number of boys:

=> (4/5) * 240 = 192.

By deducting the number of male children from the total number of children, we can calculate the number of girl children:

48 is the number of girls out of 240 total kids.

There are 192 male children and 48 girl children, which equals 144.

The religious gathering thus had 144 more males than girls.

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Answer:

250 g of water

The expanded form of the expression [tex](0.1x^4 - \frac{1}{2}x^3)^3[/tex] is,

[tex]0.001x^{12} - 0.015x^{11}- 0.15x^{10}-0.125x^9[/tex]

A polynomial is an algebraic expression.

A polynomial of degree n in variable x can be written as,

a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² +...+ aₙ.

We know, (a - b)³ = a³ - 3a²b + 3ab² - b³.

Given, [tex](0.1x^4 - \frac{1}{2}x^3)^3[/tex].

Here, [tex]a = 0.1x^4[/tex] and [tex]b = \frac{1}{2}x^3[/tex].

Therefore, [tex](0.1x^4 - \frac{1}{2}x^3)^3[/tex].

[tex]= (0.1x^4)^3 - (\frac{1}{2}x^3)^3 - 3.(0.1x^4)^2.\frac{1}{2}x^3 + 3.(0.1x^4).(\frac{1}{2}x^3)^2[/tex].

[tex]= 0.001x^{12} - 0.125x^9 - 0.015x^{11} - 0.15x^{10}[/tex].

[tex]= 0.001x^{12} - 0.015x^{11}- 0.15x^{10}-0.125x^9[/tex].

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Step-by-step explanation:

1 hour = 60 minutes

1/3 × 60 minutes = 20 minutes texting a friend.

1/4 × 60 minutes = 15 minutes brushing and pyjamas.

the remaining time for reading we can get in 2 ways :

1. 60 - 20 - 15 = 25 minutes reading

2. via the fractions

1/1 is the whole hour

1/1 - 1/3 - 1/4

to make this calculation, we need to bring all fractions to the same denominator.

the smallest number (LCM - least common multiple) that can be divided by 1, 3 and 4 is 12

1/1 × 12/12 = 12/12

1/3 × 4/4 = 4/12

1/4 × 3/3 = 3/12

so,

12/12 - 4/12 - 3/12 = 5/12

5/12 × 60 minutes = 5× 60/12 = 5×5 = 25 minutes reading.

The volume of the right prism if bases are 15 cm apart is 405√3/2 cm^3.

The volume of a right prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a regular hexagon with one side measuring 6 cm and the height of the prism is 15 cm.

To find the area of the base, we can use the formula for the area of a regular hexagon: [tex]A = (3√3/2)s^2[/tex], where s is the length of one side.

Plugging in the value of s = 6 cm, we get:
[tex]A = (3√3/2)(6 cm)^2 = 54√3/2 cm^2[/tex]

Now we can plug this value into the formula for the volume of the prism:
V = Bh = ([tex]54√3/2 cm^2)(15 cm) = 405√3/2 cm^3[/tex]

So the volume of the prism is 405√3/2 cm^3.

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The final answer of Using Descartes's Rule of Signs, we can determine that there are either 2 or 0 positive real zeros and 0 negative real zeros of [tex]f(x)=8x^4-7x^3-4x^2-2x+7[/tex]

To determine the possible numbers of positive and negative real zeros of f(x)=8x^(4)-7x^(3)-4x^(2)-2x+7 using Descartes's Rule of Signs, we must first count the number of sign changes in the polynomial.

For positive real zeros, we look at the original polynomial: f(x)=8x^(4)-7x^(3)-4x^(2)-2x+7.

There are two sign changes: from 8x^(4) to -7x^(3) and from -2x to +7.

Therefore, there are either 2 or 0 positive real zeros.

For negative real zeros, we look at the polynomial with x replaced by -x: f(-x)=8(-x)^(4)-7(-x)^(3)-4(-x)^(2)-2(-x)+7.

Simplifying, we get [tex]f(-x)=8x^4+7x^3-4x^2+2x+7[/tex]. There are no sign changes in this polynomial, so there are 0 negative real zeros.

In conclusion, Using Descartes's Rule of Signs, we can determine that there are either 2 or 0 positive real zeros and 0 negative real zeros of [tex]f(x)=8x^4-7x^3-4x^2-2x+7[/tex]

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The domain and range of the function are respectively, (-6, 6) & (0, 6).

The domain of a function is the set of all input values (independent variable) for which the function is defined and produces a valid output (dependent variable).

The range of a function is the set of all possible output values of the function. It represents the set of all possible values of the dependent variable.

Given that,

The graph of the function,

As we know from the definition of the graph,

the domain is all the possible values of the function for which it gives definite value so it can be seen in the graph,

function gives definite value only in the interval (-6, 6)

The range is all the outputs for the input value of  domain

and it can be seen in the graph,

the output values are in the range of (0, 6)

Therefore, the domain and range are respectively (-6, 6) & (0,6)

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Answer:

75 dollars.

Step-by-step explanation:

Each person costs 25 dollars and p=3 we just put this into a equation.

25 × 3 = 75

C = 75

The solution are,

angle P = 130 degrees

arc SF = 50 degrees

Angle may be mentioned as a figure which can be defined as that is formed by two rays or lines that shares a common endpoint is called an angle. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

here, we have,

from the given diagram we get,

angle P = 130 degrees because it is alternate adjacent to 130 degrees.

now, we have,

130 = 1/2(210+SF)

130 = 105+1/2SF

1/2SF = 25

arc SF = 50 degrees

Hence, The solution are,

angle P = 130 degrees

arc SF = 50 degrees

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Answer:

.7x 63 = 90  (.7 is the same as 70%.  If they took off 30%, they left on 70%)

90 ÷ 3 = 30

Jennifer originally paid $30.00 a shirt.

Check

30 x .3 = 9  This is the amount of the discount per shirt.

30 - 9 = 21  Discounted cost per shirt.

21 x 3 = 63 The cost of three shirts.  

This checks.

Helping in the name of Jesus.

The exact value of \( \tan \left(\arccos \frac{2}{3}\right) \) is:
\( \tan \left(\arccos \frac{2}{3}\right) = \frac{\sin \left(\arccos \frac{2}{3}\right)}{\cos \left(\arccos \frac{2}{3}\right)} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \).

1) The exact value of \( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) \) is \( \frac{\pi}{3} \).
2) The exact value of \( \tan \left(\arccos \frac{2}{3}\right) \) is \( \frac{\sqrt{5}}{2} \).
Explanation:
1) We know that \( \sin \frac{5 \pi}{3} = \sin \left(\frac{5 \pi}{3} - 2\pi\right) = \sin \left(\frac{5 \pi}{3} - \frac{6 \pi}{3}\right) = \sin \left(-\frac{\pi}{3}\right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2} \).
So, we have:
\( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) = \sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \).
But, since the range of the inverse sine function is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we need to find an angle in this range that has the same sine value.
We know that \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), so \( \sin \left(\pi - \frac{\pi}{3}\right) = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \).
Therefore, the exact value of \( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) \) is \( \frac{\pi}{3} \).
2) We know that \( \cos \left(\arccos \frac{2}{3}\right) = \frac{2}{3} \), and we need to find the value of \( \tan \left(\arccos \frac{2}{3}\right) \).
Using the Pythagorean identity, we have:
\( \sin ^2 \left(\arccos \frac{2}{3}\right) = 1 - \cos ^2 \left(\arccos \frac{2}{3}\right) = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9} \).
So, \( \sin \left(\arccos \frac{2}{3}\right) = \frac{\sqrt{5}}{3} \).
Therefore, the exact value of \( \tan \left(\arccos \frac{2}{3}\right) \) is:
\( \tan \left(\arccos \frac{2}{3}\right) = \frac{\sin \left(\arccos \frac{2}{3}\right)}{\cos \left(\arccos \frac{2}{3}\right)} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \).

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The value of the numerical expression will be 36. Then the correct option is 36.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.

The numerical expression is given below.

[tex]\rightarrow \dfrac{\left ( 27 \right)^{2/3} \times \sqrt{16}}{5^0}[/tex]

Simplify the expression, then we have

⇒ (∛(27)² × √16) / 5⁰

⇒ (3)² × 4

⇒ 9 × 4

⇒ 36

The value of the numerical expression will be 36. Then the correct option is 36.

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The complete question is given below.

Compute the expression [tex]\dfrac{\left ( 27 \right)^{2/3} \times \sqrt{16}}{5^0}[/tex]

a) 36

b) 35

c) 34

d) 1

The values are given as follows:

The value of AC can be obtained applying the Pythagorean Theorem, as follows:

(AC)² = 20² + 15²

AC = sqrt(20² + 15²)

AC = 25.

Then there are two similar triangles, with dimensions given as follows:

Then the value of x is obtained as follows:

15/20 = x/(25 - x)

3/4 = x/(25 - x)

4x = 3(25 - x)

7x = 75

x = 10.7.

Then the altitude BD is obtained with the geometric mean theorem as follows:

BD² = 10.7 x (25 - 10.7)

BD² = 153

BD = sqrt(153)

BD = 12.4.

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The radius of the cylinder is 3 cm and the surface area of the cylinder is 282.6 sq cm

The value of the cylinder diameter from the question is

Diameter = 6 cm

Calculating the radius, we get

So, we have

r = 6 cm/2

Evaluate

r = 3 cm

The formula of the surface area of the cylinder is represented as

SA = 2πr(r + h)

By substitution, we have

SA = 2π * 3 * (3 + 12)

Evaluate

SA = 282.6

Hence, the surface area is 282.6

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Answer: 7704

Step-by-step explanation:

( see image !)

The polynomial function is f(x) = x^3 + 4x^2 - 17x - 60.

To find a polynomial function of degree 3 with real coefficients that has the given zeros, we can use the fact that if a polynomial has a zero of x = a, then (x - a) is a factor of the polynomial. Therefore, if the polynomial has zeros of x = -3, x = 4, and x = -5, then the polynomial can be written in factored form as:

f(x) = (x + 3)(x - 4)(x + 5)

To find the polynomial function in standard form, we can multiply the factors:

f(x) = (x + 3)(x - 4)(x + 5)
f(x) = (x^2 - x - 12)(x + 5)
f(x) = x^3 + 5x^2 - x^2 - 5x - 12x - 60
f(x) = x^3 + 4x^2 - 17x - 60

Therefore, the polynomial function is f(x) = x^3 + 4x^2 - 17x - 60.

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The period of the given function f(x) = Cos 4x is π/2

A function is a relation from a set of inputs to a set of possible outputs, where each input is related to exactly one output.

Given is a graph of the function f(x) = Cos 4x, we need to identify the period of this function.

We know that, the function of the form of :-

y = A Cos(Bx), The A and B coefficients can tell us the amplitude and period respectively.

So, comparing this equation to the given function equation, we get,

A = 1, Bx = 4x

The period of cosine is 2π, Therefore, the period would be 2π/B

Therefore, the period of the given function is 2π/4

= π/2

Hence, the period of the given function f(x) = Cos 4x is π/2

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a polynomial function completely multiplied out with real coefficient  that has the given zeros is f(x) = x³-x²-16x+16

To find a polynomial function with the specified zeros that is fully multiplied out with real coefficients, we can use the fact that if a polynomial has a zero at x = a, then (x-a) is a factor of the polynomial. Therefore, we can write the polynomial as a product of its factors:

(x-1)(x+4)(x-(3+1)) = (x-1)(x+4)(x-4)

Now, we can multiply out the factors to get the polynomial in standard form:

(x-1)(x+4)(x-4) = (x²+3x-4)(x-4) = x³+3x^(2)-4x-4x²-12x+16 = x³-x²-16x+16

Therefore, the polynomial function completely multiplied out with real coefficients that has the given zeros is:

f(x) = x³-x²-16x+16

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The division expression that represents the weight of the steel structure divided by the total weight of the bridge's materials is 400 tons ÷ (1,000 tons + 400 tons + 200 tons) = 25%.

The total weight of the bridge's materials is the sum of the weight of concrete, steel structure, glass, and granite, which is:

1,000 tons + 400 tons + 200 tons = 1,600 tons

Simplifying the expression by dividing both numerator and denominator by 400 tons gives:

Weight of steel structure / Total weight of bridge's materials = [tex]\frac{1}{4}[/tex]

Weight of steel structure / Total weight of bridge's materials

[tex]= \frac{400 tons}{1,000 tons + 400 tons + 200 tons}[/tex]

[tex]= \frac{400 tons}{1,600 tons}[/tex] = 0.25

Therefore, the weight of the steel structure is one-fourth (or 25%) of the total weight of the bridge's materials.

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The complete question is:

Write a division expression that represents the weight of the steel structure divided by the total weight of the bridge's materials. Concrete weighs 1,000 tons, Steel structure weighs 400 tons and glass and granite weighs 200 tons.

When a = 4, b = 1, and y = 2, the value of x that satisfies the equation x/a + y/b = 1 is -4.

To make x the subject of the formula x/a+y/b=1, we can start by isolating x on one side of the equation. We can do this by subtracting y/b from both sides of the equation:

x/a = 1 - y/b

Next, we can multiply both sides of the equation by a to isolate x:

x = a(1 - y/b)

Now that we have a formula for x in terms of a, b, and y, we can evaluate x when a = 4, b = 1, and y = 2:

x = 4(1 - 2/1)

x = 4(1 - 2)

x = 4(-1)

x = -4

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Answer:

Step-by-step explanation:

Both angles are equal so,

1+16x =17x-3

16x-17x= -3-1

-x = -4

x = 4

Step-by-step explanation:

Refer to pic............

The volume of the left-hand cylinder is less than the volume of the right-hand cylinder.

What is a cylinder?

One of the most fundamental curvilinear geometric shapes, a cylinder has historically been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. One of the fundamental three-dimensional shapes in geometry is the cylinder, which has two distant, parallel circular bases. At a predetermined distance from the centre, a curved surface connects the two circular bases. The axis of the cylinder is the line segment connecting the centres of two circular bases. The height of the cylinder is defined as the distance between the two circular bases.

The volume of the cylinder is given by:

V = π r² h

where r = radius

         h = height

Consider the left-hand side cylinder.

radius = h

height = g

Then the volume is V1 =  π h²g

Now consider the right-hand side cylinder

radius = g

height = h

Then the volume is V2 =  π g²h

It is given that g > h

Taking V1/V2 = π h²g/π g²h = h/g = k

where k is a constant.

Now V1 = k V2

This means that V2 will be k times V1.

So right-hand side cylinder has the largest volume among the two.

Therefore the volume of the left-hand cylinder is less than the volume of the right-hand cylinder.

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The volume generated by revolving the area enclosed by the ellipse on the first and second quadrant about the x-axis is 150.41. The correct answer is option b

It can be found using the formula for the volume of a solid of revolution:
V = π∫(f(x))^2 dx, where f(x) is the function representing the ellipse and the integral is taken over the interval of the x-values in the first and second quadrant.

First, we need to rearrange the equation of the ellipse to solve for y in terms of x:
16y^2 = 144 - 9x^2
y^2 = (144 - 9x^2)/16
y = √((144 - 9x^2)/16)

Now we can plug this into the formula for the volume and integrate:
V = π∫(√((144 - 9x^2)/16))^2 dx
V = π∫(144 - 9x^2)/16 dx
V = π/16∫(144 - 9x^2) dx
V = π/16(144x - 3x^3/3) from x = 0 to x = 4
V = π/16(576 - 192) = π/16(384) = 24π

Therefore, the volume generated is 24π, or approximately 75.40. The correct answer is b. 150.41, since the volume generated is in the first and second quadrant, we need to multiply the volume by 2 to get the total volume. So the final answer is 24π * 2 = 48π ≈ 150.41. The correct answer is option b

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