SIxty meters of rope is used to make three sides of a rectangular camping area with a long wall used as the other side. The length of each side of the rectangle is a natural number. WHat is the largest area that can be enclosed by the rope and the wall?

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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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]

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:

{0, 1, 2}

Step-by-step explanation:

4x<8x+2

-4x<2

x<-1/2

Only {0, 1, 2} meets all critera

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.

Answer:

e

Step-by-step explanation:

Answer:

The volume is 60cm³

Step-by-step explanation:

(bxhxl)/2 = answer

(5x4x6)/2

5x4 = 20

20x6=120

120/2 = 60

Answer:

3 ÷ 2/5 = 3 × 5/2 = 15/2 = 7 1/2

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.

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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We need a sample size of 98 to estimate the mean with 90% confidence and an error of ±0.25 mpg, using the given range of fuel economy for the 2016 Toyota 4Runner.

Standard deviation is a measure of the amount of variation or dispersion in a set of data. It is a statistical measure that is used to quantify the amount of variation or spread of a set of data values relative to the mean or average value of the data.

(a) Using Method 3, the Empirical Rule for a normal distribution, we know that for a normal distribution, approximately 68% of the values fall within 1 standard deviation from the mean, 95% fall within 2 standard deviations from the mean, and 99.7% fall within 3 standard deviations from the mean.

Therefore, to find the standard deviation for the given range of 17 to 22 mpg, we can use the Empirical Rule as follows:

The range between the mean and the upper limit (22 mpg) is 3 standard deviations, so the standard deviation can be estimated as (22-17)/3 = 1.6667 mpg.

Thus, the estimated standard deviation using Method 3 is 1.6667 mpg, rounded to 4 decimal places.

(b) To find the sample size required to estimate the mean with 90% confidence and an error of ±0.25 mpg, we can use the following formula:

[tex]n = (z*(E))^2[/tex]

where n is the sample size, z is the z-score corresponding to the desired confidence level (90% = 1.645), σ is the estimated standard deviation from part (a), and E is the desired margin of error (±0.25 mpg).

Substituting the values, we get:

[tex]n = (1.645*(1.6667/0.25))^2[/tex]

n ≈ 97.15

Rounding up to the nearest whole number, we get the sample size of 98.

Therefore, we need a sample size of 98 to estimate the mean with 90% confidence and an error of ±0.25 mpg, using the given range of fuel economy for the 2016 Toyota 4Runner.

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Answer: x= -20

Step-by-step explanation:

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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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

The statement is not true, since 13 is not less than 6. Therefore, the point (3, 2) is not a solution of the linear inequality 3x + 2y < 6.

To determine if the point (3, 2) is a solution of the linear inequality 3x + 2y < 6, we substitute the values of x and y in the inequality and check if it is a true statement or not.

A linear inequality is an inequality in which the variables have a degree of one, and the inequality can be expressed as a linear equation with an inequality symbol. A linear inequality can be represented graphically as a line with a shaded region that either includes or excludes certain points in the coordinate plane.

For example, the inequality 2x + 3y > 6 is a linear inequality. To graph this inequality, we can first plot the line 2x + 3y = 6 by finding its x and y intercepts, which are (3, 0) and (0, 2), respectively. Then, we can choose a test point not on the line, such as (0, 0), and substitute its coordinates into the inequality to see if it is a solution.

3(3) + 2(2) < 6

9 + 4 < 6

13 < 6

The statement is not true, since 13 is not less than 6. Therefore, the point (3, 2) is not a solution of the linear inequality 3x + 2y < 6.

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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:

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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To solve the equation:

1/(x+1) + 9/(x+9) = 1

First, we can simplify the equation by finding a common denominator:

((x+9)+9(x+1))/(x+1)(x+9) = 1

Simplifying the numerator, we get:

(10x+18)/(x+1)(x+9) = 1

Cross-multiplying gives:

10x+18 = (x+1)(x+9)

Expanding the right side of the equation, we get:

10x+18 = x^2+10x+9

Simplifying and rearranging gives:

x^2-9 = 0

Using the quadratic formula, we solve for x:

x = (±sqrt(81))/1

x = ±9

Therefore, the solutions to the equation are x = -9 and x = 9.

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.

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•

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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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.

To distribute -3√3 to (2 + √6), we can use the distributive property:

-3√3 (2 + √6) = (-3√3 * 2) + (-3√3 * √6)

Next, we can simplify each term using the product rule of radicals:

(-3√3 * 2) = -6√3

(-3√3 * √6) = -3√(3

Therefore, when we distribute -3√3 to (2 + √6), we get:

-3√3 (2 + √6) = -6√3 - 9√2.

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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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:

  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.

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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