Find a5 and a-n for the geometric sequence with a1=7, r= -3

The period of the pendulum when the length is 3 ft is approximately 2.08 seconds.

What is the period?

We are given that the period P varies directly as the square root of the length L of the pendulum. We can write this as:

P ∝ √L

Using a proportionality constant k, we can write this as an equation:

P = k√L

To find the value of k, we can use the information given in the problem: when L = 2 ft, P = 1.8 s. Substituting these values into the equation, we get:

1.8 = k√2

To solve for k, we can isolate it by dividing both sides by √2:

k = 1.8 / √2

Now we can use this value of k to find the period when the length is 3 ft. Substituting L = 3 ft and the value of k we just found into the equation, we get:

P = k√L

P = (1.8 / √2) √3

P ≈ 2.08 seconds

Therefore, the period of the pendulum when the length is 3 ft is approximately 2.08 seconds.

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The experimental probability of the volcano erupting in 1 or 2 of the next 5 decades is 6/10 or 0.6.

We can do this by assigning a value of 1 to the numbers 1 and 2 (representing the volcano erupting) and a value of 0 to

the numbers 3, 4, and 5 (representing the volcano not erupting).

For each set of random numbers generated, we can count the number of times the volcano erupted (i.e. the sum of the

values assigned to the numbers 1 and 2).

Out of the 10 sets of random numbers generated, we might get results like this:
Set 1: 1 3 4 2 1 (2 eruptions)
Set 2: 2 1 5 4 3 (2 eruptions)
Set 3: 3 2 5 4 5 (1 eruption)
Set 4: 1 1 2 4 5 (3 eruptions)
Set 5: 4 4 4 4 4 (0 eruptions)
Set 6: 1 1 1 1 1 (5 eruptions)
Set 7: 3 3 3 3 3 (0 eruptions)
Set 8: 2 2 2 2 2 (0 eruptions)
Set 9: 1 2 1 2 1 (4 eruptions)
Set 10: 5 5 5 5 5 (0 eruptions)

Out of these 10 sets, we can see that there were 6 sets where the volcano erupted in 1 or 2 of the next 5 decades (sets

1, 2, 3, 4, 9, and 10). Therefore, the experimental probability of the volcano erupting in 1 or 2 of the next 5 decades is

6/10 or 0.6.

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Answer:

{0, 1, 2}

Step-by-step explanation:

4x<8x+2

-4x<2

x<-1/2

Only {0, 1, 2} meets all critera

The identified graph of the function is graph Y (bottom left)

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

f(x) = 2x + 4, x ≤ -2

       4 + 1⁄2x, -2 < x < 2

       -x + 5, 2 ≤ x

The above function is a piecewise function

By the domain of the functions, we have the following endpoints

f(x) = 2x + 4, x < -2 = closed endpoint

       4 + 1⁄2x, -2 < x < 2 = open endpoint

       -x + 5, 2 ≤ x = closed endpoint

When the above endpoints and functions are compared to the graph, we have the graph to be graph Y (bottom left)

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The length of the missing side is approximately 4.73 units.

What is a triangle?

A polygon with three edges and three vertices is called a triangle. It is one of the fundamental geometric shapes. Triangle ABC is the designation for a triangle with vertices A, B, and C. In Euclidean geometry, any three points that are not collinear produce a distinct triangle and a distinct plane.

Here, we have

Given:

∠A = 33.7°

∠C = 26.4°

We have to find the value of b.

Let's label the missing side as b, and use sin(A) = opposite/hypotenuse to find the length of the side opposite angle A:

sin(A) = opposite/hypotenuse

sin(33.7°) = b/8

b = 8 × sin(33.7°)

b ≈ 4.726

Hence, the length of the missing side is approximately 4.73 units.

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Answer:

24 boys

Step-by-step explanation:

If 60% of the students in a class are boys, 40% of the students are girls.

Then 16/0.4 = 40 students total

40 - 16 = 24 boys

or you can use 40 x 0.6 = 24 boys

Answer

2268 ft²

Step-by-step explanation

1 yard is equivalent to 3 feet. Using this conversion factor, the equivalence of 21 yd is:

 [tex]21 \ \text{yd}=21 \ \text{yd}\times\dfrac{3 \ \text{ft}}{1 \ \text{yd}}[/tex]

[tex]\text{Simplifying the units:}[/tex]

     [tex]21 \ \text{yd}=\dfrac{21\times3}{1 \ \text{}} \ \text{ft}[/tex]

      [tex]21 \ \text{yd}=63 \ \text{ft}[/tex]

Similarly, the equivalence of 12 yards is:

[tex]12 \ \text{yd}=12 \ \text{yd}\times\dfrac{3 \ \text{ft}}{1 \ \text{yd}}[/tex]

     [tex]12 \ \text{yd}=12\times3 \ \text{ft}[/tex]

        [tex]12 \ \text{yd}=36 \ \text{ft}[/tex]

Therefore, the length of the bed is 63 ft and the width is 36 ft.

Finally, the area of the bed (a rectangle) is calculated as follows:

[tex]\text{A}=\text{length}\times\text{width}[/tex]

       [tex]\text{A}=63\times36[/tex]

       [tex]\text{A}=2268 \ \text{ft}^2[/tex]

Answer:

Step-by-step explanation:

We will rationalize the denominator:

[tex]\frac{\sqrt{5} }{\sqrt{5} +2} =\frac{\sqrt{5} }{\sqrt{5} +2} \times \frac{\sqrt{5} -2}{\sqrt{5} -2}[/tex]

         [tex]=\frac{\sqrt{5} (\sqrt{5} -2)}{(\sqrt{5} +2)(\sqrt{5} -2)}[/tex]

         [tex]=\frac{5-2\sqrt{5} }{5-2\sqrt{5} +2\sqrt{5} -4}[/tex]

          [tex]=\frac{5-2\sqrt{5} }{1}[/tex]

          [tex]=5-2\sqrt{5}[/tex]

Answer:

3 ÷ 2/5 = 3 × 5/2 = 15/2 = 7 1/2

The length of the base with area 25/18 square meters is 5/3 meters. This answer is in its simplest form.

The height of a triangle is the length of the line segment that is drawn from the vertex of the triangle perpendicular to the base.

The area of a triangle is given by the equation

A = 1/2 * base * height.

Therefore, we can substitute the value of the area and the height of the triangle into the equation to solve for the base.

A = 1/2 * b * (5/3)

25/18 = 1/2 * b * (5/3)

We will divide left side of the equation by 5/3.

(25/18) / (5/3) = (1/2)*b

(25/18) * 3/5 = (1/2)*b

Now, we can solve for b by dividing the left side of the equation by 1/2

(5/6) / (1/2) = b

b = 5/6 * (2/1)

b = 5/3

Therefore, the length of the base is 5/3 meters. This answer is in its simplest form.

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Answer:

Step-by-step explanation:

You want the dimensions and volume of a cuboid that can be made from an 18 in square of cardboard when an x-inch square is cut from two adjacent corners, and an x-inch by 9-inch rectangle is cut from the other two corners.

The longest side of the cuboid will have length (18 -2x). The second longest side will have length (18 -9 -x) = (9 -x). The shortest side will have length x.

The volume of the cuboid is the product of these side lengths:

  V = (18 -2x)(9 -x)(x) = 2x(9 -x)²

The value of x that maximizes volume will be the value that makes the derivative with respect to x be zero.

  V' = 2(9 -x)² -4x(9 -x) = (9 -x)(2(9 -x) -4x) = 2(9 -x)(9 -3x)

  V' = 6(9 -x)(3 -x)

The derivative will be zero when its factors are zero, at x=9 and x=3. The value x=9 gives zero volume.

The value x=3 gives a volume of ...

  V = 2·3(9 -3)² = 6³ = 216 . . . . . cubic inches

The dimensions are ...

The dimensions for maximum volume are 3 inches by 6 inches by 12 inches. The maximum volume is 216 cubic inches.

The central angle of the given arc, subtending a circle with a radius of 7 cm and a length of 14 cm, is approximately 103.13 degrees.

An arc is a section of a circle's or curve's border in mathematics. It is additionally known as an open shape. The circumference, also referred to as the perimeter, is the measurement around a circular that defines its edge. Consequently, the arc is defined as the separation between any two locations along its circumference.

The angle that an arc of a circle subtends at the center is called the central angle. The central angle is measured in degrees and is formed by the radii that make up the arc.

Given:

Radius of the circle (r) = 7 cm

Length of the arc (s) = 14 cm

To find the central angle (θ) of the arc, we can use the formula:

θ = (s / r) * (180 / π)

where:

s = length of the arc

r = radius of the circle

π = pi (approximately 3.14159...)

Plugging in the given values:

θ = (14 / 7) * (180 / π)

θ = 2 * (180 / π)

θ ≈ 103.13 degrees (rounded to two decimal places)

So, the central angle of the given arc, subtending a circle with a radius of 7 cm and a length of 14 cm, is approximately 103.13 degrees.

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Answer:

Step-by-step explanation:

We can use the slope formula to find the slope of the line passing through the two given points:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) = (-8, 5) and (x2, y2) = (-4, r).

Substituting these values, we have:

1/4 = (r - 5) / (-4 - (-8))

1/4 = (r - 5) / 4

r - 5 = 1

r = 6

Therefore, the missing coordinate is r = 6.

The solution to the equation 123x + 360y = 99 is x = 22 and y = -11.

What is Linear Diophantine Equation?

An equation with two or more integer unknowns that are all to a maximum degree of one is known as a linear diophantine equation (LDE).

To solve the equation 123x + 360y = 99, we can use the extended Euclidean algorithm, which is a method for finding the greatest common divisor (GCD) of two integers and expressing it as a linear combination of those integers. We can use the same algorithm to find a solution to the given equation.

First, we need to find the GCD of 123 and 360. We can use the Euclidean algorithm for this:

gcd(123, 360) = gcd(123, 360 - 123*2) = gcd(123, 114)

gcd(123, 114) = gcd(123 - 114, 114) = gcd(9, 114)

gcd(9, 114) = gcd(9, 114 - 9*12) = gcd(9, 6)

gcd(9, 6) = gcd(9 - 6*1, 6) = gcd(3, 6)

gcd(3, 6) = gcd(3, 6 - 3*2) = gcd(3, 0)

gcd(3, 0) = 3

Therefore, the GCD of 123 and 360 is 3.

Now we can use the extended Euclidean algorithm to find integers s and t such that 123s + 360t = 3. This is done by working backwards through the Euclidean algorithm and expressing each remainder as a linear combination of the previous two integers:

gcd(123, 360) = 3

3 = 123s + 360t

3 = 9 - 114

 = 9 - (360 - 123*2)

 = 123*2 - 360*1

s = 2, t = -1

Now we can multiply both sides of the equation by 33 to get a solution to the original equation:

123x + 360y = 99

123*2*33 + 360*(-1)*33 = 3*33

123*66 + 360*(-33) = 99

Finally, we can divide both sides by 3 to get a solution in integers:

123*22 + 360*(-11) = 33

Therefore, x = 22 and y = -11 is a solution to the equation 123x + 360y = 99.

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Answer: 1/6 of a pie

Step-by-step explanation: each friend receives 1/6 of a pie because there are 2 pies being distributed to 12 friends. 2/12 = 1/6

Answer:

1/6 of the pie.

Step-by-step explanation:

We Know

Two pies are shared among 12 friends.

How much did each friend receive?

We take

2 / 12 = 1/6 of the pie

So, each friend receives 1/6 of the pie.

the probability of selecting a tool produced by the old method on the first attempt and a tool produced by the new method on the second attempt is:

P(A and B) = P(A) * P(B | A) = 0.8305 * 0.1998 = 0.1660

How to solve the questions?

Part A:

Let A be the event that the first tool selected is acceptable and B be the event that the second tool selected is acceptable.

We need to find the probability of P(A and B), which can be calculated using the multiplication rule of probability as follows:

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

where P(A) is the probability of selecting an acceptable tool on the first attempt and P(B | A) is the conditional probability of selecting an acceptable tool on the second attempt given that the first tool selected was acceptable.

P(A) = (1640 + 328) / (1640 + 328 + 23 + 9) = 0.985

P(B | A) = (1639 + 327) / (1640 + 328 + 23 + 9 - 1) = 0.985

Therefore, the probability of both selections being acceptable is:

P(A and B) = P(A) * P(B | A) = 0.985 * 0.985 = 0.9702

The events are dependent because the probability of selecting an acceptable tool on the second attempt depends on the result of the first attempt.

Part B:

Let A be the event that the first tool selected is produced by the old method and B be the event that the second tool selected is produced by the new method.

We need to find the probability of P(A and B), which can be calculated using the multiplication rule of probability as follows:

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

where P(A) is the probability of selecting a tool produced by the old method on the first attempt and P(B | A) is the conditional probability of selecting a tool produced by the new method on the second attempt given that the first tool selected was produced by the old method.

P(A) = (1640 + 23) / (1640 + 328 + 23 + 9) = 0.8305

P(B | A) = 328 / (1640 + 328 + 23 + 9 - 1) = 0.1998

Therefore, the probability of selecting a tool produced by the old method on the first attempt and a tool produced by the new method on the second attempt is:

P(A and B) = P(A) * P(B | A) = 0.8305 * 0.1998 = 0.1660

The events are dependent because the probability of selecting a tool produced by the new method on the second attempt depends on the result of the first attempt.

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Answer:

  B

Step-by-step explanation:

You want the difference of the rational expressions (6/x) and (5x/(x+2)).

The difference of rational expressions is found the same way any difference of fractions is found. The fractions are written using a common denominator, and the difference is taken of the numerators.

  [tex]\dfrac{6}{x}-\dfrac{5x}{x+2}=\dfrac{6(x+2)}{x(x+2)}-\dfrac{x(5x)}{x(x+2)}\\\\\\=\dfrac{6x+12-5x^2}{x(x+2)}=\boxed{\dfrac{-5x^2+6x+12}{x^2+2x}}[/tex]

__

Additional comment

Once you realize the common denominator is x(x+2) = x²+2x, you have eliminated all but the correct answer choice.

Answer:

The volume is 60cm³

Step-by-step explanation:

(bxhxl)/2 = answer

(5x4x6)/2

5x4 = 20

20x6=120

120/2 = 60

Answer: x= -20

Step-by-step explanation:

Answer:

e

Step-by-step explanation:

Answer:

To calculate Darrius' monthly take-home pay, we first need to calculate his annual state income tax. We do this by multiplying his yearly salary by the state income tax rate:

$42,350 x 3.07% = $1,292.94

Next, we subtract the state income tax from his annual salary to get his annual net income:

$42,350 - $1,292.94 = $41,057.06

Finally, we divide his annual net income by 12 to get his monthly take-home pay:

$41,057.06 / 12 = $3,421.42

Therefore, Darrius' monthly take-home pay is $3,421.42.

Step-by-step explanation:

Hence, the expressions 8 x + 1 and 5/x are equal. F) 8x+(66) and J) 5x + (101) are the two equivalent expressions.

A mathematical sentence without an equal sign that comprises numbers, quantities, and operations is known as an expression. By changing variables with constants and using the order of actions, it can be evaluated or made simpler. For instance, the phrases "3x + 7" and "4y - 2" are equations, whereas "3x + 7 = 16" is an expression.

given

F and J are the two expressions that are equivalent.

F: 8 ÷ x + (6/6)

6/6 reduced to 1: 8 x + 1

J: 5 x x 1 Converting 5 x to 5x-1, 5x-1 x 1 equals 5/x

Hence, the expressions 8 x + 1 and 5/x are equal. F) 8x+(66) and J) 5x + (101) are the two equivalent expressions.

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The complete question is:-  

Which two expressions are equivalent?

(6.7D | Lesson 6)

F 8÷x+(6'6) and 8÷x. 1

G 6-2 (x - x) and 6 ÷ 2

H 14 ÷ 7. (xx) and 14 ÷ 7

J 5÷x(101)10+and 5 x 1•

Answer:

Option A) m^9.

Step-by-step explanation:

To simplify the given expression, we can use the rules of exponents which state that when we multiply powers with the same base, we can add their exponents.

So, m² ∙ m⁶ ∙ m can be simplified as:

m² × m⁶ × m = m^(2+6+1) = m^9

Therefore, the simplified form of m² ∙ m⁶ ∙ m is m^9.

The sum of the given expression is equal to 3.

The process of numerical integration involves applying numerical techniques to approximate the value of a determined integral of a function across a specified interval. To approximate integrals that are challenging or impossible to solve analytically, it is employed in calculus.

Numerical integration works by breaking the integration interval into smaller subintervals, approximating the function over each subinterval with a simple function, and then summing the areas under the curve of the approximating function over each subinterval to estimate the integral.

Determine the value of f(xi):

f(x1) = (0-2)^3 = (-2)³ = -8

f(x2) = (1-2)^3 = (-1)³ = -1

f(x3) = (2-2)^3 = 0³ = 0

f(x4) = (3-2)^3 = 1³ = 1

f(x5) = (4-2)^3 = 2³ = 8

f(x6) = (5-2)^3 = 3³ = 27

Substituting the value in the given summation we have:

Σ i = 1 to 6 f(xi) Δ x = f(x1) Δ x + f(x2) Δ x + f(x3) Δ x + f(x4) Δ x + f(x5) Δ x + f(x6) Δ x

= (-8)(0.1) + (-1)(0.1) + (0)(0.1) + (1)(0.1) + (8)(0.1) + (27)(0.1)

= -0.5 + (-0.1) + 0 + 0.1

= 3

Hence, the sum is equal to 3.

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To convert 1/2% to a ratio, we first convert it to a decimal by dividing by 100:

1/2% = 1/2 ÷ 100 = 0.005

Then we express this decimal as a ratio by putting it in the form of x:y:

0.005 = x:y

To simplify this ratio, we can multiply both sides by a factor that will make x and y whole numbers. For example, we can multiply both sides by 100:

0.005 × 100 = x:y × 100

0.5 = x:y

So 1/2% as a ratio is 1:200.

1/2% is equivalent to 0.5% or 0.005 as a decimal.

To express 0.005 as a ratio, you can write it as a fraction with a numerator of 0.005 and a denominator of 1. Then, you can multiply both the numerator and denominator by 100 to get rid of the decimal and express it as a ratio.

0.005/1 = (0.005

Therefore, 1/2% is equivalent to the ratio 0.5:100 or 1:200.

The maximum head breadth that the helmets will fit is apprοximately 7.96 inches, which rοunds tο 8.0 inches (οptiοn A).

We can use the Z-scοre fοrmula tο find the Z-scοre cοrrespοnding tο the cutοff pοint:

Z = (X - μ) / σ

where X is the head breadth we want tο find, μ is the mean head breadth, σ is the standard deviatiοn, and Z is the Z-scοre cοrrespοnding tο the cutοff pοint.

We want tο find the Z-scοre such that the area tο the right οf it under the standard nοrmal distributiοn curve is 0.025. We can use a standard nοrmal distributiοn table οr calculatοr tο find this Z-scοre, which turns οut tο be apprοximately 1.96.

Tο find the maximum head breadth that the helmets will fit, we need tο find the cutοff pοint beyοnd which the head breadths are in the largest 2.5%.

Nοw we can sοlve fοr X:

1.96 = (X - 6.0) / 1.0

X - 6.0 = 1.96

X = 7.96

Therefοre, the maximum head breadth that the helmets will fit is apprοximately 7.96 inches, which rοunds tο 8.0 inches (οptiοn A).

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Thus, number of more people who prefer mystery books over historical

fiction books are 48.

A pie chart, often known as a circle chart, is a visual representation of the various values of a specific variable or a means to summarise a set of nominal data (e.g. percentage distribution). This kind of chart consists of a circle with numerous segments. A certain category is represented by each segment.

Given data:

Number of mystery books = 18% of 600

= 18*600 / 100

= 108

Number of historical fiction books = 10% of 600

= 10*600 / 100

= 60

Difference = Number of mystery books - Number of historical fiction books

Difference = 108 - 60

Difference = 48

Thus, number of more people who prefer mystery books over historical

fiction books are 48.

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Complete question:

The circle graph shows the results of a survey about favorite kinds of books. If 600 people were surveyed, how many more people prefer mystery books than historical fiction books?

The picture is attached.

you will watch TV for approximately 16 days out of the 28 days.

If you watch TV for 4 days out of 7 each week, then the fraction of the week that you spend watching TV is:

4/7

To find the total number of days you will watch TV over 28 days, you can use the proportion:

4/7 = x/28

where x is the number of days you will watch TV.

To solve for x, you can cross-multiply:

4 * 28 = 7 * x

112 = 7x

x = 112/7

x ≈ 16

Therefore, you will watch TV for approximately 16 days out of the 28 days.

What is proportion?

Proportion is a mathematical concept that expresses the relationship between two quantities or values. It is often represented as a fraction, with the numerator and denominator representing the values being compared.

In a proportion, the two ratios are equal. For example, if you have a bag of marbles containing 6 red marbles and 4 blue marbles, the proportion of red to blue marbles is 6:4, or 3:2. This can also be written as a fraction: 3/2.

Proportions are useful in many areas of mathematics and everyday life, including geometry, statistics, and finance. They are also used in problem-solving to find missing values or to determine if two values are proportional to each other.

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