If the manager of a bottled water distributor wants to estimate, 95% confidence, the mean amount of water in a 1-gallon bottle to within ±0.006 gallons and also assumes that the standard deviation is 0.003 gallons, what sample size is needed?

If a light bulb manufacturing company wants to estimate, with 95% confidence, the mean life of compact fluorescent light bulbs to within ±250 hours and also assumes that the population standard deviation is 900 hours, how many compact fluorescent light bulbs need to be selected?

If the inspection division of a county weighs and measures department wants to estimate the mean amount of soft drink fill in 2-liter bottles to within ± 0.01 liter with 95% confidence and also assumes that the standard deviation is 0.08 liters, what sample size is needed?

An advertising executive wants to estimate the mean amount of time that consumers spend with digital media daily. From past studies, the standard deviation is estimated as 52 minutes. What sample size is needed if the executive wants to be 95% confident of being correct to within ±5 minutes?

Answer:

29. 59.06°

30. 10.6

Step-by-step explanation:

29.
By using the Tangent angle rule, we can find the angle of elevation,

We know that

Tan Angle = opposite/adjacent

Tan x=AB/BC

Tan x=20/12

Tan x=5/3

[tex]x=Tan^{- }(\frac{5}{3})[/tex]

x=59.06°

30.

The law of sine is a formula that can be used to find the lengths of the sides of a triangle, or to find the angles of a triangle, when two sides and the angle between them are known. The formula is:

a / sin(A) = b / sin(B) = c / sin(C)

Here taking

a / sin(A) = c / sin(C)

here A=75°, C=180-75-50=55° and c -9 and

we need to find a,

substituting value

a/Sin(75°)=9/Sin(55°)

a=9*Sin(75°)/Sin(55°)

a=10.61

Therefore, the value of a is 10.6

Answer:

        tan (θ) = 20/12, and solve for θ

       We then get the equation tan (θ) =  20/12

        θ = arctan(5/3)

        Tan θ  =  opp/adj

        Tan θ  = 20/12

         θ  =  Tan^-1 (20/12)

          θ  =  59.03624346 degrees

           θ = 59.0 degrees

       Hence, the Angle of Elevation is ------->  59.0°

       m < C = 180 degrees - m<A - m<B

       m<C  = 180 degrees - 75 degrees  -  50 degrees

      m<C  =  55 degrees

       a/sin A  =  c/sin C

       a/sin 75 degrees  =  9/sin 55 degrees

        a  =  9 * sin 75 degrees/sin 55 degrees

        a  =  10.3

Therefore, The length of side A in --------> △ABC is approximately 10.3

Hope this helps you!

Geno read 36 pages during the first and second hour, and 1.5 times that, which is 54 pages, during the third hour.

Let's break down the information given:

Geno read 126 pages in 3 hours.

He read the same number of pages each hour for the first 2 hours.

Geno read 1.5 times as many pages during the third hour as he did during the first hour.

Let's solve this:

Let's assume that Geno read x pages during the first hour.

Since he read the same number of pages each hour for the first 2 hours, he also read x pages during the second hour.

During the third hour, Geno read 1.5 times as many pages as he did during the first hour, which is 1.5x pages.

To find the total number of pages he read, we can add up the pages from each hour: x + x + 1.5x = 126.

Combining like terms, we have 3.5x = 126.

Divide both sides of the equation by 3.5 to solve for x: x = 36.

Therefore, Geno read 36 pages during the first and second hour, and 1.5 times that, which is 54 pages, during the third hour.

In summary, Geno read 36 pages during each of the first two hours and 54 pages during the third hour, for a total of 126 pages in 3 hours.

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Anna ran the fastest with a speed of approximately 4.67 miles per hour.

To find the speed at which each person ran, we can use the formula: Speed = Distance / Time.

Let's calculate the speed for each person:

Noah:

Distance = 2 1/2 miles

Time = 3/5 hour

Speed = (2 1/2) / (3/5)

= (5/2) / (3/5)

= (5/2) [tex]\times[/tex] (5/3)

= 25/6 ≈ 4.17 miles per hour

Emily:

Distance = 3 3/4 miles

Time = 5/6 hour

Speed = (3 3/4) / (5/6)

= (15/4) / (5/6)

= (15/4) [tex]\times[/tex] (6/5)

= 9/2 = 4.5 miles per hour

Anna:

Distance = 3 1/2 miles

Time = 3/4 hour

Speed = (3 1/2) / (3/4)

= (7/2) / (3/4)

= (7/2) [tex]\times[/tex] (4/3)

= 14/3 ≈ 4.67 miles per hour

Based on the calculations, Noah ran at a speed of approximately 4.17 miles per hour, Emily ran at a speed of 4.5 miles per hour, and Anna ran at a speed of approximately 4.67 miles per hour.

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The number of membership passes that will contribute to deferred revenue in 2016 is: 1,000 (total passes sold) x 20% (passes utilized in 2017) = 200 passes.

To calculate the deferred revenue in 2016 for LuluYoga, we need to consider the membership passes that were sold but not yet utilized.

In 2016, LuluYoga sold 1,000 membership passes for $2,000 each. We know that 80% of the classes purchased were used in 2016, which means 20% of the classes will be utilized in 2017.

Therefore, the number of membership passes that will contribute to deferred revenue in 2016 is:

1,000 (total passes sold) x 20% (passes utilized in 2017) = 200 passes

The revenue generated from these 200 passes will be realized in 2017 when the classes are utilized. Therefore, the revenue from these passes should be deferred to the following year.

To calculate the deferred revenue, we need to multiply the number of passes by the price per pass:

200 (passes) x $2,000 (price per pass) = $400,000

Hence, the deferred revenue in 2016 for LuluYoga is $400,000.

Deferred revenue represents the amount of revenue that has been received but has not yet been earned. In this case, LuluYoga has received payment for the membership passes, but the revenue associated with the unused classes will be recognized in the subsequent year when the classes are actually utilized.

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Answer: none

Step-by-step explanation:

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Answer:

31.  m∠E = 56.1°

32.  c = 24.9 inches

Step-by-step explanation:

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all three sides of the triangle.

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]

Given values of triangle DEF:

To find m∠E, substitute the values into the Law of Sines formula and solve for E:

[tex]\dfrac{\sin D}{d}=\dfrac{\sin E}{e}[/tex]

[tex]\dfrac{\sin 81^{\circ}}{25}=\dfrac{\sin E}{21}[/tex]

[tex]\sin E=\dfrac{21\sin 81^{\circ}}{25}[/tex]

[tex]E=\sin^{-1}\left(\dfrac{21\sin 81^{\circ}}{25}\right)[/tex]

[tex]E=56.1^{\circ}\; \sf(nearest\;tenth)[/tex]

Therefore, the measure of angle E is 56.1°, to the nearest tenth.

See the attachment for the accurate drawing of triangle DEF.

[tex]\hrulefill[/tex]

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of triangle ABC:

To find the length of side c, substitute the values into the Law of Cosines formula and solve for c:

[tex]c^2=a^2+b^2-2ab \cos C[/tex]

[tex]c^2=13^2+15^2-2(13)(15) \cos 125^{\circ}[/tex]

[tex]c^2=169+225-390 \cos 125^{\circ}[/tex]

[tex]c^2=394-390 \cos 125^{\circ}[/tex]

[tex]c=\sqrt{394-390 \cos 125^{\circ}}[/tex]

[tex]c=24.8534667...[/tex]

[tex]c=24.9\; \sf inches\;(nearest\;tenth)[/tex]

Therefore, the length of side c is 24.9 inches, to the nearest tenth.

Answer:

The y-intercept of a function is the point where the graph of the function intersects the y-axis. To find the y-intercept of f(x), we can substitute x=0 into the equation for f(x):

f(0) = -3(1.02)^0 = -3

Therefore, the y-intercept of f(x) is -3. To find the y-intercept of g(x), we can look at the table and see that when x=0, g(x)=-3. Therefore, the y-intercept of g(x) is also -3.

Comparing the y-intercepts of the two functions, we see that they are equal. Therefore, the correct answer is:

The y-intercept of f(x) is equal to the y-intercept of g(x).

Step-by-step explanation:

Answer:

The correct answer is A, the y-intercept of f(x) is equal to the y-intercept of g(x).

Step-by-step explanation:

First, note that the y intercept is what y is equal to when x is equal to 0.

The given function, f(x), is an exponential function. Exponential functions are written in the formula [tex]f(x) = a(1 + r)^x[/tex], where a = y-intercept!

a in the function f(x) is -3, so this means that the y intercept is -3.

In the given table, g(x), the y value is -3 when the x value is 0.

This means that in the g(x) table, the y-intercept is also -3.

Thus, A is correct and the y-intercept of f(x) is equal to the y-intercept of g(x).

In statistics, simulation practice is a method used to model and analyze real-world scenarios using a computer program. It involves creating a virtual representation of a system, situation, or process and performing experiments on it to generate data.

This method allows statisticians to investigate the potential outcomes of various scenarios without actually having to conduct real-world experiments.

Simulation practice is often used in statistical modeling, optimization, and decision-making. It can be applied to various fields, including finance, economics, engineering, and healthcare. Some examples of simulation practice include Monte Carlo simulation, agent-based modeling, and discrete-event simulation.

In conclusion, simulation practice is a valuable tool for statisticians and researchers as it enables them to gain insights into complex systems and make informed decisions based on data generated from virtual experiments.

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The approximate slope of the graph of the linear function is 0.15.

To find the approximate slope of the graph of the linear function, we can choose two points from the table and calculate the slope using the formula:

slope = (change in y) / (change in x)

Let's select the points (50, 7.5) and (250, 37.5) from the table.

Change in y = 37.5 - 7.5 = 30

Change in x = 250 - 50 = 200

slope = (change in y) / (change in x) = 30 / 200 = 0.15

Note: A linear function is a mathematical function that represents a straight line.

It can be written in the form:

f(x) = mx + b

where m is the slope of the line and b is the y-intercept (the point where the line intersects the y-axis).

The slope (m) determines the steepness or slant of the line.

A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line.

The slope represents the rate of change of the function.

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Answer:

to solve the equation you first need to bring it to factors and by doing that you first need to let the equation equal 0 hence you need to minus 2 on both sides of the equation therefore

x^2 + 10x + 25 - 2 =2 - 2

therefore

x^2 + 10x +23 = 0

now since the equation cannot be factored, we use the formula.

x= [tex]\frac{-b +- \sqrt{b^{2}-4ac } }{2a}[/tex]

where

a=1

b=10

c=23

note we use the coefficients only.

therefore x = [tex]\frac{-10 -+ \sqrt{10^{2}-4(1)(23) } }{2(1)}[/tex]

=[tex]\frac{-10-+\sqrt{100-92} }{2}[/tex]

=[tex]\frac{-10-+\sqrt{8} }{2}[/tex]

then we form two equations according to negative and positive symbols

x=[tex]\frac{-10+\sqrt{8} }{2} or x =\frac{-10-\sqrt{8} }{2}[/tex]

therefore x = [tex]-5+\sqrt{2}[/tex]   or x=[tex]-5-\sqrt{2}[/tex]

Answer:

[tex]\textsf{B.} \quad a^{12}[/tex]

Step-by-step explanation:

To simplify the given rational expression, we can apply the rule of exponents, which states that when dividing two powers with the same base, we subtract the exponents.

Using this rule:

[tex]\dfrac{a^{18}}{a^{6}}= a^{18-6} = a^{12}[/tex]

Therefore, the given rational expression is equivalent to a¹².

To determine whether there is a minimum or maximum value to the quadratic function h(t) = -8t² + 4t - 1 and find the minimum or maximum value of h, one has to follow the steps given below. So, the minimum or maximum value of h = -1/2.

Step 1: Write the quadratic function in standard form.

The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants.

h(t) = -8t² + 4t - 1 ... (1)

Step 2: Calculate the axis of symmetry of the parabola.

The axis of symmetry of the parabola is given by x = -b/2a, where a and b are the coefficients of x² and x, respectively. Therefore, the axis of symmetry of the parabola given by h(t) = -8t² + 4t - 1 is given by: t = -b/2a = -4/(2 * (-8)) = 4/16 = 1/4

Step 3: Calculate the vertex of the parabola.

The vertex of the parabola is given by (h, k), where h and k are the coordinates of the vertex. Therefore, the coordinates of the vertex of the parabola given by h(t) = -8t² + 4t - 1 are given by: (1/4, h(1/4))

Substituting t = 1/4 in Equation (1), we have: h(1/4) = -8(1/4)² + 4(1/4) - 1h(1/4) = -8/16 + 4/4 - 1h(1/4) = -1/2 + 1 - 1h(1/4) = -1/2

Therefore, the vertex of the parabola given by h(t) = -8t² + 4t - 1 is given by the point(1/4, -1/2)

Step 4: Determine the nature of the extrema of the functionThe coefficient of the x² term in Equation (1) is -8, which is negative. Therefore, the parabola is downward-facing and the vertex represents a maximum value. Thus, the maximum value of the function h(t) = -8t² + 4t - 1 is given by h(1/4) = -1/2. Answer: Thus, the minimum or maximum value of h = -1/2.

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Step-by-step explanation:

24-10=14. So the girls are 14 the ratio is 10:14 =5:7

The equation of the line of fit is y = 15x + 30.

To determine the equation of the line of fit, we can use the given data points (0,30) and (2,60). We can use the slope-intercept form of a linear equation, which is y = mx + b, where m represents the slope and b represents the y-intercept.

Using the two data points, we can calculate the slope (m) as the change in y divided by the change in x:

m = (60 - 30) / (2 - 0) = 30 / 2 = 15

Now that we have the slope, we can substitute one of the data points into the equation to solve for the y-intercept (b). Let's use the point (0,30):

30 = 15(0) + b

30 = 0 + b

b = 30

Therefore, the equation of the line of fit is y = 15x + 30. This means that for every additional hour of study time (x), the grade percent (y) increases by 15, and the line intersects the y-axis at 30.

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The correct answer is c) =, because the fraction four eighths is equal to the fraction two fourths.

To determine which symbol should go in the box to make the equation true, let's analyze the fractions given and compare their values.

The fraction "two fourths" can be simplified to "one-half" since both the numerator and denominator can be divided by 2. Therefore, "two fourths" is equal to "one-half."

Now, let's look at the fraction "four eighths." We can simplify this fraction by dividing both the numerator and denominator by 4, which gives us "one-half" as well. So, "four eighths" is also equal to "one-half."

Now, based on the given fractions, we have the equation:

(one-half) [BOX] (one-half)

We need to determine the correct symbol to fill in the box.

Looking at the values of the fractions, we see that both "two fourths" and "four eighths" are equivalent to "one-half." Therefore, the correct symbol to make the equation true is the equality symbol (=).

Hence, the correct answer is:

c) =, because the fraction four eighths is equal to the fraction two fourths.

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Point B: Electric field strength due to sphere X = 2073.6 NC⁻¹ and Electric field strength due to sphere Y = -2073.6 NC⁻¹.

data: Spheres X and Y are 4.0 m apart. The charge on sphere

X = + 72 mC = 72 × 10⁻³ C.

The charge on sphere

Y = -72 mC = -72 × 10⁻³ C.

The constant of proportionality = 9 × 10⁹ Nm²/C².

The formula to calculate the electric field strength due to a point charge is

E = k q / r²

where E is the electric field strength, k is the Coulomb's constant (= 9 × 10⁹ Nm²/C²), q is the magnitude of the charge, and r is the distance from the charge.The electric field due to sphere X at point A is

EaX = [tex]k q / r²where r = 4.0 m, q = + 72 × 10⁻³ CSo, EaX = 9 × 10⁹ × 72 × 10⁻³ / (4.0)²EaX = 9 × 9 × 2 × 2 × 2 × 2 / 10[/tex]EaX = 2592 / 10EaX = 259.2

NC⁻¹The electric field due to sphere Y at point A is

[tex]EaY = k q / r²where r = 4.0 m, q = -72 × 10⁻³ CSo, EaY = 9 × 10⁹ × 72 × 10⁻³ / (4.0)²EaY = -9 × 9 × 2 × 2 × 2 × 2 / 10EaY = -2592 / 10EaY = -259.2[/tex]

NC⁻¹The electric field due to sphere X at point B is

[tex]EbX = k q / r²where r = 4.0 m, q = + 72 × 10⁻³ C + 72 × 10⁻³ C = 144 × 10⁻³ C.So, EbX = 9 × 10⁹ × 144 × 10⁻³ / (4.0)²EbX = 9 × 9 × 4 × 4 × 4 × 4 / 10EbX = 20736 / 10EbX = 2073.6[/tex]

NC⁻¹The electric field due to sphere Y at point B is

[tex]EbY = k q / r²where r = 4.0 m, q = -72 × 10⁻³ C - 72 × 10⁻³ C = -144 × 10⁻³ C. So, EbY = 9 × 10⁹ × -144 × 10⁻³ / (4.0)²EbY = -9 × 9 × 4 × 4 × 4 × 4 / 10EbY = -20736 / 10EbY = -2073.6 NC⁻¹[/tex]

Therefore, the electric field strength at points A and B due to each sphere's presence are: Point A: Electric field strength due to sphere X = 259.2 NC⁻¹ and Electric field strength due to sphere Y = -259.2 NC⁻¹.

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The journal entries for the given transactions are as follows:

Cash A/c Dr. 40,000

Furniture A/c Dr. 5,000

Motor Car A/c Dr. 12,000

Stock A/c Dr. 20,000

To Capital A/c 77,000

Bank A/c Dr. 15,000

To Cash A/c 15,000

Purchase A/c Dr. 9,000

To Sen's A/c 9,000

Basu's A/c Dr. 6,000

To Sales A/c 6,000

Stationery A/c Dr. 200

To Cash A/c 200

Cash A/c Dr. 2,000

To Dalal's A/c 2,000

Travelling Expenses A/c Dr. 600

To Cash A/c 600

Drawings A/c Dr. 1,000

To Bank A/c 1,000

Cash A/c Dr. 3,000

To Bank A/c 3,000

Sen's A/c Dr. 8,800

To Bank A/c 8,800

Freight and Clearing A/c Dr. 400

To Cash A/c 400

Bank A/c Dr. 6,000

To Basu's A/c 6,000

Journal entries for the given transactions are as follows:

Pater commenced business with 40,000 cash and also brought into business furniture worth ₹5,000; Motor car valued for ₹12,000 and stock worth ₹20,000.

Cash A/c Dr. 40,000

Furniture A/c Dr. 5,000

Motor Car A/c Dr. 12,000

Stock A/c Dr. 20,000

To Capital A/c 77,000

Deposited ₹15,000 into State Bank of India.

Bank A/c Dr. 15,000

To Cash A/c 15,000

Bought goods on credit from Sen for ₹9,000.

Purchase A/c Dr. 9,000

To Sen's A/c 9,000

Sold goods to Basu on Credit for ₹6,000.

Basu's A/c Dr. 6,000

To Sales A/c 6,000

Bought stationery from Ram Bros. for Cash ₹200.

Stationery A/c Dr. 200

To Cash A/c 200

Sold goods to Dalal for ₹2,000 for which cash was received.

Cash A/c Dr. 2,000

To Dalal's A/c 2,000

Paid ₹600 as travelling expenses to Mehta in cash.

Travelling Expenses A/c Dr. 600

To Cash A/c 600

Patel withdrew for personal use ₹1,000 from the Bank.

Drawings A/c Dr. 1,000

To Bank A/c 1,000

Withdrew from the Bank ₹3,000 for office use.

Cash A/c Dr. 3,000

To Bank A/c 3,000

Paid to Sen by cheque ₹8,800 in full settlement of his account.

Sen's A/c Dr. 8,800

To Bank A/c 8,800

Paid ₹400 in cash as freight and clearing charges to Gopal.

Freight and Clearing A/c Dr. 400

To Cash A/c 400

Received a cheque for ₹6,000 from Basu.

Bank A/c Dr. 6,000

To Basu's A/c 6,000

These journal entries represent the various transactions and their effects on different accounts in the accounting system.

They serve as the initial records of the financial activities of the business and provide a basis for further accounting processes such as ledger posting and preparation of financial statements.

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The predicted population of Knox, rounded to the nearest whole number, after 8 years is 32,599.

To predict the population of Knox after 8 years, we can use the given information that the population is currently 42,000 and it is declining at a rate of 3.2% per year.

To calculate the population after 8 years, we need to apply the rate of decline for each year. Let's break down the calculation step by step:

Calculate the population after the first year:

Population after 1 year = 42,000 - (3.2% of 42,000)

= 42,000 - (0.032 * 42,000)

= 42,000 - 1,344

= 40,656

Calculate the population after the second year:

Population after 2 years = 40,656 - (3.2% of 40,656)

= 40,656 - (0.032 * 40,656)

= 40,656 - 1,299.71

= 39,356.29

Continue this process for each year up to 8 years, applying the 3.2% rate of decline each time.

After performing these calculations for each year, we arrive at the population after 8 years:

Population after 8 years ≈ 32,599

Therefore, the predicted population of Knox, rounded to the nearest whole number, after 8 years is 32,599.

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To find the experimental probability of rolling a number greater than 10, we need to determine the frequency of rolling a number greater than 10 and divide it by the total number of rolls.

Looking at the table, we can see that the frequency for rolling a number greater than 10 is the sum of the frequencies for rolling 11 and 12.

Frequency for rolling a number greater than 10 = Frequency of 11 + Frequency of 12

Frequency for rolling a number greater than 10 = 15 + 17 = 32

The total number of rolls is given as 200.

Experimental Probability of rolling a number greater than 10 = Frequency for rolling a number greater than 10 / Total number of rolls

Experimental Probability of rolling a number greater than 10 = 32 / 200

Experimental Probability of rolling a number greater than 10 = 0.16 or 16%

Therefore, the experimental probability of rolling a number greater than 10 is 16%.

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The shaded fraction of the shape is 2/3.

To determine the fraction of the shape that is shaded, we need to compare the shaded area to the total area of the shape.

1. Identify the shaded region in the shape. In this case, we have a shape with some part shaded.

2. Calculate the area of the shaded region. Given the dimensions provided, the area of the shaded region is determined by multiplying the length and width of the shaded part. In this case, the dimensions are 18 mm and 10 mm, so the area of the shaded region is (18 mm) × (10 mm) = 180 mm².

3. Calculate the total area of the shape. The total area of the shape is determined by multiplying the length and width of the entire shape. In this case, the dimensions are 18 mm and 12 mm, so the total area of the shape is (18 mm) × (12 mm) = 216 mm².

4. Determine the fraction. To find the fraction, divide the area of the shaded region by the total area of the shape: 180 mm² ÷ 216 mm². Simplifying this fraction gives us 5/6.

5. Convert the fraction to its simplest form. By dividing both the numerator and denominator by their greatest common divisor, we get the simplified fraction: 2/3.

Therefore, the fraction of the shape that is shaded is 2/3.

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Answer:

4/3

Step-by-step explanation:

1/3:(1/4) = (1/3)/(1/4) = 1/3*4 = 4/3

Answer:

c=25

Step-by-step explanation:

Since you are given [tex]x^{2}[/tex]+10x+c

We know that in an equation of [tex]ax^{2}+bx+c[/tex], when a = 1, c can be found by [tex](\frac{b}{2})^{2}[/tex]

So c = [tex](10/2)^{2}[/tex]=[tex]5^{2}[/tex]=25

(a) Chin had 93.1 marbles after the game.

(b) The three children had a total of 271.44 marbles altogether.

Let's break down the problem step by step to find the answers:

Initial marbles

Before the game:

Let's assume Afiq had x marbles.

Bala had 1/6 fewer marbles than Afiq, so Bala had (x - 1/6x) marbles.

Chin had 3/5 as many marbles as Bala, so Chin had (3/5)(x - 1/6x) marbles.

After the game

After the game, Bala lost 20% of his marbles to Chin, so he has 80% (or 0.8) of his initial marbles remaining.

Afiq lost 2/3 of his marbles to Chin, so he has 1/3 (or 0.33) of his initial marbles remaining.

Calculating the marbles

(a) How many marbles did Chin have after the game?

To find Chin's marbles after the game, we add the marbles gained from Bala to Chin's initial marbles and the marbles gained from Afiq to Chin's initial marbles.

Chin's marbles = Initial marbles + Marbles gained from Bala + Marbles gained from Afiq

Chin's marbles = (3/5)(x - 1/6x) + 0.8(x - 1/6x) + 0.33x

Chin's marbles = (3/5)(5x/6) + 0.8(5x/6) + 0.33x

Chin's marbles = (3/6)x + (4/6)x + 0.33x

Chin's marbles = (7/6)x + 0.33x

We are given that Chin gained 105 marbles, so we can equate the equation above to 105 and solve for x:

(7/6)x + 0.33x = 105

(7x + 2x) / 6 = 105

9x / 6 = 105

9x = 105 * 6

x = (105 * 6) / 9

x = 70

Substituting the value of x back into the equation for Chin's marbles:

Chin's marbles = (7/6)(70) + 0.33(70)

Chin's marbles = 10(7) + 0.33(70)

Chin's marbles = 70 + 23.1

Chin's marbles ≈ 93.1

Therefore, Chin had approximately 93.1 marbles after the game.

(b) After the game, the 3 children each bought another 40 marbles. To find the total number of marbles the 3 children have altogether, we need to sum up their marbles after the game and the additional 40 marbles for each.

Total marbles = Afiq's marbles + Bala's marbles + Chin's marbles + Additional marbles

Total marbles = 0.33x + 0.8(x - 1/6x) + (7/6)x + 40 + 40 + 40

Total marbles = 0.33(70) + 0.8(70 - 1/6(70)) + (7/6)(70) + 120

Total marbles = 23.1 + 0.8(70 - 11.7) + 81.7 + 120

Total marbles = 23.1 + 0.8 × 58.3 + 201.7

Total marbles = 23.1 + 46.64 + 201.7

Total marbles = 271.44

The three children had a total of 271.44 marbles altogether.

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Question

Afiq. Bala and Chin played a game of marbles. Before the game, Bala had 1/ 6 fewer marbles than Afig and Chinhad 3/5 as many marbles as Bala.

After the game, Balahad lost 20% of his marbles to Chinwhile Afig had lost

2/3 of his marbles to Chin. Chingained 105 marbles at the end of the

game.

(a)How many marbles didChinhave after the game?

(b)After the game, the 3 children each bought another 40 marbles. How manymarbles did the 3 children have altogether?

The derivative dy/dx of the implicit equation[tex]x - 2xy + x^2y + y = 10[/tex] is given by[tex]\frac{(2y - 2 + 2xy)}{(-2x + x^2 + 1)}[/tex]

To find the derivative dy/dx of the implicit equation [tex]x - 2xy + x^2y + y =[/tex]10, we will use the implicit differentiation technique.

Step 1: Differentiate both sides of the equation with respect to x.

For the left-hand side:

[tex]d/dx (x - 2xy + x^2y + y) = d/dx (10)[/tex]

Taking the derivative of each term separately:

[tex]d/dx (x) - d/dx (2xy) + d/dx (x^2y) + d/dx (y) = 0[/tex]

Step 2: Apply the chain rule to the terms involving y.

The chain rule states that if we have y = f(x), then dy/dx = dy/du * du/dx, where u = f(x).

For the term 2xy, we have y = f(x) = xy. Applying the chain rule, we get:

[tex]d/dx (2xy) = d/dx (2xy) * dy/dx[/tex]

= 2y + 2x * dy/dx

Similarly, for the term x^2y, we have [tex]y = f(x) = x^2y.[/tex]Applying the chain rule:

[tex]d/dx (x^2y) = d/dx (x^2y) * \frac{dx}{dy} \\= 2xy + x^2 * \frac{dx}{dy}[/tex]

Step 3: Substitute the derivatives back into the equation.

[tex]d/dx (x) - (2y + 2x * dy/dx) + (2xy + x^2 * dy/dx) + d/dx (y) = 0[/tex]

Simplifying the equation:

[tex]1 - 2y - 2x * \frac{dx}{dy} + 2xy + x^2 * \frac{dx}{dy} + \frac{dx}{dy} = 0[/tex]

Step 4: Group the terms involving dy/dx together and solve for dy/dx.

Combining the terms involving dy/dx:

[tex]-2x * \frac{dx}{dy} + x^2 * \frac{dx}{dy} + dy/dx = 2y - 1 + 2xy - 1[/tex]

Factoring out dy/dx:

[tex](-2x + x^2 + 1) * \frac{dx}{dy} = 2y - 1 + 2xy - 1[/tex]

[tex]dy/dx = \frac{(2y - 2 + 2xy)}{(-2x + x^2 + 1)}[/tex]

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The probability of event A occurring given that event B has not occurred (P(A\B)) is 0.

To find P(A\B), we need to calculate the probability of event A occurring given that event B has not occurred. In other words, we want to find the probability of A happening when B does not happen.

The formula to calculate P(A\B) is:

P(A\B) = P(A∩B') / P(B')

Where B' represents the complement of event B, which is the event of B not occurring.

Given that P(A∩B) = 3/7 and P(B) = 7/8, we can find P(A∩B') and P(B') to calculate P(A\B).

To find P(B'), we subtract P(B) from 1, since the sum of the probabilities of an event and its complement is always equal to 1.

P(B') = 1 - P(B)

      = 1 - 7/8

      = 1/8

Now, to find P(A∩B'), we need to subtract P(A∩B) from P(B'):

P(A∩B') = P(B') - P(A∩B)

        = 1/8 - 3/7

        = 7/56 - 24/56

        = -17/56

Since the probability cannot be negative, we can conclude that P(A∩B') is 0.

Finally, we can calculate P(A\B) using the formula:

P(A\B) = P(A∩B') / P(B')

      = 0 / (1/8)

      = 0

Therefore, the probability of event A occurring given that event B has not occurred (P(A\B)) is 0.

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The solutions to the absolute value equations are:

1. b. x = 1, -19.      2. a. no solution.     3. c. x = 5, -9.

1. |3x+9| = 30

To solve this equation, we isolate the absolute value expression by considering two cases: 3x+9 = 30 and -(3x+9) = 30. Solving both equations, we find x = 7 and x = -19, respectively. Thus, the answer is b. x = 1, -19.

2. |2x+11| = -13

An absolute value cannot be negative, so there is no solution to this equation. The answer is a. no solution.

3. |x + 2| + 4 = 11

To solve this equation, we isolate the absolute value expression by subtracting 4 from both sides, resulting in |x + 2| = 7. Considering two cases: x + 2 = 7 and -(x + 2) = 7, we solve for x and find x = 5 and x = -9, respectively. Thus, the answer is c. x = 5, -9.

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Answer:

  (c)  61°

Step-by-step explanation:

You want the measure of the external angle formed by a tangent and secant that intercept arcs of 73° and 195° of a circle.

The measure of the angle at F is half the difference of intercepted arcs HE and EG.

  (195° -73°)/2 = 122°/2 = 61°

The measure of angle F is 61°.

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The result of the row operation on the matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The matrix in this problem is defined as follows:

[tex]\left[\begin{array}{cccc}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The row operation is given as follows:

[tex]R_1 \rightarrow \frac{1}{2}R_1[/tex]

The first row of the matrix is given as follows:

[2 0 0 16]

The meaning of the operation is that every element of the first row of the matrix is divided by two.

Hence the resulting matrix is given as follows:

[tex]\left[\begin{array}{cccc}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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