Using more advanced technology, a team of workers began to produce 6 more parts per hour than before. In six hours, the team produced 120% of what they had previously been able to produce in eight hours. How many parts per hour was the team producing prior to switching to the new technology?

Answer:  x = ± [tex]\frac{1}{2}[/tex]

Step-by-step explanation:

Equation:

4x² + 10 = 11         > bring everything over to other side

                            > first subtract 10 from both sides

4x² = 1                   > Divide by 4 on both sides

x² =  [tex]\frac{1}{4}[/tex]                    >Take square root of both sides

                             > When you take square root there is a ±

x = ±  [tex]\sqrt{(\frac{1}{4} )}[/tex]           > take the square root of both top and bottom

x = ± [tex]\frac{1}{2}[/tex]

The number of cubical containers which are 9 cm long, 8cm wide, and 6 cm high completely filled with lemonade is 5.

volume of lemonade = 2160 cm³

Dimensions of container

L = 9 cm , B = 8 cm , H = 6 cm

Volume of container = L× B × H

Volume of container = 9×8×6

Volume of container = 432 cm³

To find the number of cubical containers filled we use

Number of containers filled = volume of lemonade/volume of the container

putting the value in formula

Number of container filled = 2160/432

Number of container filled = 5

Total number of container filled with lemonade is 5

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The function rule for the statement "the output is eight less than the input" is a simple mathematical expression that represents a relationship between the input and output values.

In this case, it can be expressed as Output = Input - 8. The function takes the input value, subtracts 8 from it, and returns the result as the output value. This rule ensures that the output will always be eight units smaller than the input. For example, if the input is 15, the output will be 7. This function rule can be used to perform calculations or model various scenarios where the output is consistently eight units less than the input.

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When comparing the values of f(4) and g(4), we need to take into account the fact that f(x) is exponential and g(x) is linear. Exponential functions grow at an increasing rate as x increases, while linear functions grow at a constant rate. Therefore, as x gets larger, the difference between f(x) and g(x) will become greater.

To find out how much greater f(4) is than g(4), we first need to calculate the values of f(4) and g(4). Let's say that f(x) = 2^x and g(x) = 3x + 1. Plugging in x = 4, we get:
f(4) = 2^4 = 16
g(4) = 3(4) + 1 = 13

So, f(4) is greater than g(4) by a difference of 3. However, this does not take into account the fact that f(x) is exponential and g(x) is linear.

To see the impact of the different growth rates, let's compare the values of f(x) and g(x) for a range of values of x. We can create a table to compare the two functions:

x    f(x)    g(x)
0    1       1
1    2       4
2    4       7
3    8       10
4    16      13
5    32      16

From this table, we can see that as x increases, the difference between f(x) and g(x) grows at an increasing rate. This is because f(x) is growing exponentially, while g(x) is growing linearly.

In summary, f(4) is 16 and g(4) is 13, so f(4) is greater than g(4) by a difference of 3. However, we also need to take into account the fact that f(x) is exponential and g(x) is linear. As x increases, the difference between f(x) and g(x) will grow at an increasing rate. Therefore, the difference between f(4) and g(4) is not only 3, but also growing exponentially.

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The package that requires more wrapping paper to cover is the cube. The total amount of wrapping paper Susan must use to cover both packages is 286 square inches.

Let's find the surface area of both packages to determine which requires more wrapping paper and the total amount needed.

1. Rectangular prism:
Surface area = 2lw + 2lh + 2wh
where l = length, w = width, h = height
Surface area = 2(4)(2) + 2(4)(10) + 2(2)(10)
Surface area = 16 + 80 + 40 = 136 square inches

2. Cube:
Surface area = 6s²
where s = side length
Surface area = 6(5)² = 6(25) = 150 square inches

The cube requires more wrapping paper to cover as its surface area is 150 square inches, compared to the rectangular prism's 136 square inches. The total amount of wrapping paper Susan must use for both packages is 136 + 150 = 286 square inches.

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The area of the sandbox whose base is 3.5 meter and height is 2.4 meter is 4.2 m².

Given:

Base = 3.5 m

Height = 2.4 m

First, the area of the sandbox formula:

Area = 1/2 x base x height.

Next,  substitute b = 3.5 meters and h = 2.4 meters.

Area = 1/2 * (3.5) * (2.4).

Now, simplify by multiplying 1/2, 2.4, and 3.5.

Area = 1/2 x 2.4 x 3.5

Area = 4.2 square meters.

Thus, The area of the sandbox is 4.2 m².

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The question attached here seems to be incomplete, the complete question is:

Lokota wants to build a sandbox for his little brother. Determine the amount of sand he needs by finding the area of the sandbox. Use the drop-down menus to complete the statements.

First, write the formula: A = 1/2 bh
Next, use parentheses when you substitute __ for b and __ for h.

Now, simplify by ___ 1/2, 2.4, and 3.5.

The area of the sandbox is ___ m²

Paul has 18 interest-free days for the $300 stereo purchase.

He will need to pay the full balance of his June billing statement.

If Paul's credit card closes on the 6th of the month and his payment is due on the 24th, then he has 18 days between the close of the billing cycle and the due date of his payment.

If Paul purchases a stereo for $300 on June 8th, then the transaction will be included in his billing cycle for the month of June. Since his billing cycle closes on the 6th, the $300 charge will appear on his June billing statement.

If Paul pays off his credit card in full each month, then he will need to pay the full balance of his June billing statement by the due date of June 24th to avoid finance charges. This means he will need to pay $300 for the stereo, plus any other charges that may have been included on his billing statement for the month of June.

Therefore, the number of interest-free days that Paul will have for the $300 stereo purchase is 18 days, which is the number of days between the billing cycle close date (June 6th) and the payment due date (June 24th).

To summarize:

Paul has 18 interest-free days for the $300 stereo purchase.

Paul will need to pay the full balance of his June billing statement, including the $300 stereo charge, by June 24th to avoid finance charges.

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This evaluated expression is a scalar multiple of -4, which projects that vectors CD and CE are collinear. Then, points C, D, and E lie on the same straight line.

Let us proceed by  evaluating the vector CD. Then C is the midpoint of OA, we can evaluate the vector CD by subtracting vector CO from vector OD.

Vector CO = 1/2 × Vector OA
= 1/2 × (a + b)
= 1/2a + 1/2b

Vector OD = 3/4 × Vector AD
= 3/4 × (3/4a - 1/4b)
= 9/16a - 3/16b

Vector CD = Vector OD - Vector CO
= (9/16a - 3/16b) - (1/2a + 1/2b) = 5/16a - 5/16b

Then the value of the vector CE is
OB = 2BE,
we can evaluate the vector BE by dividing vector OB by 2.

Vector BE = 1/2 × Vector OB
= 1/2 × b
= 1/2b

Vector CE = Vector CO + Vector OE

Vector OE = Vector OB - Vector OE
= b - Vector BE
= b - 1/2b
= 1/2b

Vector CE = Vector CO + Vector OE
= (1/2a + 1/2b) + (1/2b)
= 1/2a + b

Then we have to show that vectors CD and CE are collinear. Two vectors are collinear if one is a scalar multiple.

CE can be expressed in terms of CD
CE / CD
= ((1/2a + b) / (5/16a - 5/16b))

Applying simplification for this expression

CE / CD
= (-8a - 8b) / (5a - 5b)

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The dealer bought 42 radios.

Let x be the number of radios the dealer bought.

Let y be the price the dealer paid for each radio.

We know that the dealer bought x radios for a total of $1,008, so:

x * y = 1008

We also know that the dealer gave away 6 radios and sold the rest for $14 more than she paid for each radio, breaking even. This means that the total revenue from selling the remaining radios is equal to the total cost of buying them:

(x - 6) * (y + 14) = x * y

Simplifying this equation, we get:

xy + 14x - 6y - 84 = xy

14x - 6y = 84

7x - 3y = 42 (dividing by 2 on both sides)

Now we have two equations:

x * y = 1008

7x - 3y = 42

We can use substitution or elimination to solve for x and y. Let's use elimination by multiplying the second equation by y/3 and adding it to the first equation:

x * y + (7x - 3y) * (y/3) = 1008 + 42 * (y/3)

xy + 7xy/3 - y²/3 = 1008 + 14y

10xy/3 - y²/3 - 14y - 1008 = 0

Multiplying both sides by 3, we get:

10xy - y² - 42y - 3024 = 0

Now we can use the quadratic formula to solve for y:

y = (-b ± sqrt(b² - 4ac)) / 2a

where a = -1, b = -42, and c = -3024:

y = (-(-42) ± sqrt((-42)² - 4(-1)(-3024))) / 2(-1)

y = (42 ± sqrt(42² - 4*3024)) / 2

y = (42 ± 126) / 2

y = 84 or y = -42

Since the price of a radio cannot be negative, we can discard the second solution and conclude that y = 84.

Now we can solve for x using the first equation:

x * y = 1008

x * 84 = 1008

x = 12

Therefore, the dealer bought 12 radios.

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The taxi travels 140 kilometers in 210 minutes at a speed of 40 km/h.

Let's calculate the distance a taxi travels in 210 minutes at a speed of 40 km/h.
Convert minutes to hours
Since the speed is given in km/h, we need to convert 210 minutes into hours.

There are 60 minutes in an hour, so divide 210 by 60:
210 minutes ÷ 60 = 3.5 hours
Calculate the distance
Now that we have the time in hours, we can use the formula for distance:
Distance = Speed × Time
In this case, the speed is 40 km/h, and the time is 3.5 hours.

Plug these values into the formula:
Distance = 40 km/h × 3.5 hours
Compute the result
Multiply the speed by the time to find the distance:
Distance = 140 km.

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Therefore, the exact values of sin 2u, cos 2u, and tan 2u are -24/25, 7/25, and -24/7, respectively.

The double angle formulas are:

sin 2u = 2 sin u cos u

cos 2u = cos² u - sin² u

tan 2u = 2 tan u / (1 - tan² u)

Given that cos u = -4/5 and u is between -π/2 and π, we can find sin u by using the Pythagorean identity:

sin² u + cos² u = 1

sin u = sqrt(1 - cos² u) = sqrt(1 - 16/25) = 3/5 (since u is in the second quadrant)

Using this value of sin u, we can find:

sin 2u = 2 sin u cos u = 2 (3/5) (-4/5) = -24/25

cos 2u = cos² u - sin² u = (-4/5)² - (3/5)² = 7/25

tan 2u = 2 tan u / (1 - tan² u) = 2 (-3/4) / (1 - (-3/4)²) = -24/7

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For the function, M(x) = 2x⁴ - 5x-3

1. M'''(2) = 96
2. dy/dx = (2x + 9x²y⁴) / (12x³y³ + 12y²)
3. dy/dt = -12,320 when dx/dt = -2 and x = -10

1. To find the value of M'''(2) for the function M(x) = 2x⁴ - 5x - 3, first find the first, second, and third derivatives:

M'(x) = 8x³ - 5
M''(x) = 24x²
M'''(x) = 48x

Now evaluate M'''(2):
M'''(2) = 48(2) = 96

2. To find dy/dx for the relation x² = -3x³y⁴ - 4y³, first implicitly differentiate both sides with respect to x:

2x = -3(3x²y⁴ + x³(4y³dy/dx)) - 4(3y²dy/dx)

Now solve for dy/dx:

dy/dx = (2x + 9x²y⁴) / (12x³y³ + 12y²)

3. To find dy/dt for the function y = 3x⁴ - 8x² + 4, first differentiate with respect to t:

dy/dt = (12x³ - 16x)(dx/dt)

Now evaluate dy/dt when dx/dt = -2 and x = -10:

dy/dt = (12(-10)³ - 16(-10))(-2) = (12,000 + 160)(-2) = -12,320

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Zombie Epsilon would require 52.5 days of doses, Zombie Zeta would need 163.3 days, and Zombie Eta would require 636e days to be cured.

To develop a theoretical antidote, you would need to consider the virus strand, concentration (mag/ml), and the equation to calculate the number of doses needed.

For Zombie Epsilon, Zeta, and Eta, the amounts of virus are 150, 230, and 636 mag/ml, respectively. To create an effective antidote, you would need to identify the specific virus strands for each zombie type (e.g., strand 3.5 for Epsilon, 7.1 for Zeta, and "e" for Eta).

Using the provided information, the equation should be used to determine the number of days (doses needed) for each zombie type. As an example, let's assume the equation is as follows: Days = (Amount of Virus * Strand) / 10.

For Zombie Epsilon: Days = (150 * 3.5) / 10 = 52.5 days
For Zombie Zeta: Days = (230 * 7.1) / 10 = 163.3 days
For Zombie Eta: Days = (636 * e) / 10 = 636e days (where e is a constant value)

In this theoretical scenario, Zombie Epsilon would require 52.5 days of doses, Zombie Zeta would need 163.3 days, and Zombie Eta would require 636e days to be cured.

Please note that this is a fictional scenario and not based on real-life medical information.

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A model car is drawn at a scale of 21 to 1. If the model car is 9. 2in.  The length of the actual car in feet is approximately 0.7665 feet.

Find out the length of the actual car in feet, we need to first convert the length of the model car from inches to feet.
9.2 inches = 0.767 feet (divide by 12 since there are 12 inches in a foot)
Now, we can use the scale of 21 to 1 to find the length of the actual car in feet.
21 units on the model car = 1 unit on the actual car
So,
1 unit on the actual car = 0.767 feet / 21 = 0.0365 feet
Find the length of the actual car, we can multiply the scale ratio by the length of the model car in units:
21 units x 0.0365 feet per unit = 0.7665 feet
Therefore, the length of the actual car in feet is approximately 0.7665 feet.

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The actual car is  0.7665 feet long.

First, we need to convert the length of the model car from inches to feet:

9.2 in. = 9.2/12 ft. = 0.7667 ft.

Next, we can use the scale to find the length of the actual car:

21 units on the drawing = 1 unit in real life

So, we have:

1 unit in real life = length of actual car

21 units on the drawing = length of model car

Substituting the values we have:

1 unit in real life = (0.7667 ft.)/21 = 0.0365 ft.

Therefore, the length of the actual car is:

1 unit in real life x 21 = 0.0365 ft. x 21 = 0.7665 ft.

So, the actual car is approximately 0.7665 feet long.

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The formula for the function h(x) is given as follows:

h(x) = g(x + 5).

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The function h(x) is a translation left 5 units of the function g(x), hence it is defined as follows:

h(x) = g(x + 5).

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Answer:

x = 14°

t = 12.367 ~ 12.4

r = 2.999 ~ 3

Step-by-step explanation:

1st we can find x by sum theory which is the sum of all side equal to 180° .

x + 90° + 76° = 180 °

x + 166° = 180°

x= 180° - 166°

x = 14° ... So the unknown angle is 14°

and we also can solve hypotenus t and adjecent r by using sin amd cos respectively by angle 76° .

sin(76) = 12/t

sin(76) t = 12 ....... criss cross it

t = 12 / sin(76) ....... divided both side by sin(76)

t = 12.367 ~ 12.4 ....... result

And

cos(76) = r / 12.4

r = cos(76) × 12.4 .......criss cross

r = 2.999 ~ 3 ....... amswer and i approximate it

Yes, your friend is not correct about the stem and leaf plot.

The stem and leaf plot made by your friend is:

Stem | Leaves

2 | 5, 6

4 | 2, 4, 7

5 | 0, 1, 5, 5

Key : 4 | 2 = 42

When the data points from these are taken, we have :

25 , 26 , 42 , 44 , 47 , 50, 51, 55, 55

This is the same as the data provided of :

51, 25, 47, 42, 55, 26, 50, 44, 55

So, your friend's stem and leaf plot is indeed correct.

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Answer:

  34.8°

Step-by-step explanation:

You want the angle between a tangent and a segment to the center from a point on the tangent that is 6 units from the circle of radius 8 units.

The trig relation useful here is ...

  Sin = Opposite/Hypotenuse

  sin(S) = RQ/SQ

The length QT is the same as QR, so we have ...

  sin(S) = 8/(8 +6)

  S = arcsin(8/(8+6)) ≈ 34.8°

The value of x and y is 11 and 4 respectively

A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral.

A theorem in circle geometry states that the sum of opposite angles in a cyclic quadrilateral are supplementary. i.e they sum up to give 180.

10x + 16y+6 = 180

10x+16y = 174... eqn1

4+18y +10x-6 = 180

18y +10x = 182... eqn2

subtract equation 1 from 2

2y = 8

y = 8/2 = 4

Subtitle 4 for y in equation 1

10x+ 16(4)= 174

10x= 174-64

10x = 110

x= 110/10

x = 11

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The given equation, a = v²/r, solved for r is:

r = v²/a

From the question, we are to solve the given equation for r

From the given information,

The given equation is

a = v²/r

To solve the equation for r means we should isolate the variable r

Solving the equation for r

a = v²/r

Multiply both sides of the equation by r

a × r = v²/r × r

ar = v²

Divide both sides of the equation by a

ar/a = v²/a

r = v²/a

Hence, the equation solved for r is:

r = v²/a

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The most appropriate domain of the function is 0 ≤ x ≤ 22. This is because Alex can only walk from his home to the store within a maximum of 22 minutes, and the distance he walks can only be modeled within that time frame.

It is given that the function f(x) = 2.5x, which models the distance Alex walks in x minutes after leaving his house to go to the store. It takes him 22 minutes to walk from his home to the store. The most appropriate domain of the function is the range of x values that make sense in this context.

Step 1: Identify the minimum and maximum values for x.
In this case, the minimum value for x is when Alex starts walking, which is 0 minutes. The maximum value for x is when he reaches the store, which is 22 minutes.

Step 2: Express the domain as an interval.
The domain of the function can be written as an interval from the minimum to the maximum value, including both endpoints. Therefore, the domain is [0, 22].

Therefore, the most appropriate domain of the function f(x) = 2.5x, which models the distance Alex walks in x minutes after leaving his house to go to the store, is [0, 22].

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If the CPI was 305 in 1995, the price of jeans that cost $14.99 in 1973 would be approximately $47.05 in 1995 after adjusting for inflation.

To find the price of jeans in 1995, we first need to adjust the 1973 price for inflation using the Consumer Price Index (CPI). CPI measures the average change in prices of goods and services over time, so it can help us compare prices from different years.

First, we need to calculate the inflation rate between 1973 and 1995. We can do this by dividing the CPI in 1995 (305) by the CPI in 1973 (135):

Inflation rate = (305 / 135) * 100% = 226.67%

This means that prices in 1995 were about 2.27 times higher than in 1973. Now, we can apply this inflation rate to the price of jeans in 1973:

Price in 1995 = Price in 1973 * (1 + inflation rate)

Price in 1995 = $14.99 * (1 + 2.2667) = $47.05

Therefore, if the CPI was 305 in 1995, the price of jeans that cost $14.99 in 1973 would be approximately $47.05 in 1995 after adjusting for inflation. This calculation helps to compare the cost of goods across different time periods by taking inflation into account, thus giving a better understanding of the changes in purchasing power over time.

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The volume of the sphere that lies between the two cones is approximately 43.53 cubic units.

To find the volume of the sphere that lies between the given cones, we first need to determine the limits of integration.

Since the sphere has a radius of 6 (since x² + y² + z² = 36), we can use spherical coordinates to express the volume as an integral. Let's first consider the cone z = √3x² + 3y².

In spherical coordinates, this is equivalent to z = ρcos(φ)√3, where ρ is the radial distance and φ is the angle between the positive z-axis and the line connecting the origin to the point.

Similarly, the cone z = √(x²+y²)/3 can be expressed in spherical coordinates as z = ρcos(φ)/√3.

Since we're only interested in the volume of the sphere between these cones, we can integrate over the limits of ρ and φ that satisfy both inequalities.

The limits of ρ will be 0 (the origin) to 6 (the radius of the sphere).

To find the limits of φ, we need to solve for the intersection points of the two cones.

Setting the two equations equal to each other, we get:

ρcos(φ)√3 = ρcos(φ)/√3

Solving for φ, we get:

tan(φ) = 1/√3 Using the inverse tangent function, we find that: φ = π/6, 7π/6

So the limits of integration for φ will be π/6 to 7π/6.

Finally, we need to integrate over the full range of θ (the angle between the positive x-axis and the line connecting the origin to the point).

This will be 0 to 2π.

Putting it all together, the volume of the sphere between the two cones is:

∫∫∫ ρ^2sin(φ) dρ dφ dθ

With limits of integration:

0 ≤ ρ ≤ 6 π/6 ≤ φ ≤ 7π/6 0 ≤ θ ≤ 2π

Evaluating this integral gives:

V = 288π/5 - 216√3π/5

So the volume of the sphere that lies between the two cones is approximately 43.53 cubic units.

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The expected value of an original investment of 3000 that has a 10% chance of ending up with a value of 2000 is 2900. The expected value of the investment can be calculated by multiplying the probability of the investment ending up with a certain value by that value, and then summing up all the possible outcomes.

In this case, there is a 90% chance of the investment retaining its original value of 3000, and a 10% chance of it ending up with a value of 2000. To calculate the expected value, we can use the following formula:

Expected Value = (Probability of Outcome 1 × Value of Outcome 1) + (Probability of Outcome 2 × Value of Outcome 2)

Substituting the values,
Expected value = (0.9 x 3000) + (0.1 x 2000)
Expected value = 2700 + 200
Expected value = 2900

Therefore, the expected value of the original investment of 3000 is 2900.

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The brother is 35 years old.

Let's use the variable "b" to represent your brother's age.

According to the problem, your brother's age minus nineteen (b - 19) is equal to your sister's age, which is 16. So we can write an equation:

b - 19 = 16

To solve for b, we can add 19 to both sides of the equation:

b - 19 + 19 = 16 + 19

Simplifying, we get:

b = 35

Therefore, your brother is 35 years old.

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When approximating mixed fraction 5 6/13 to the nearest whole number, the rounded value is 5.

To round the mixed fraction 5 6/13 to the nearest whole number, we examine the fractional part, which is 6/13. The general rule for rounding mixed fractions is to consider the fractional part and round up if it is greater than or equal to 1/2, and round down if it is less than 1/2.

In this case, 6/13 is approximately 0.4615. Since it is less than 1/2, we need to round down to the nearest whole number. Therefore, when rounding 5 6/13 to the nearest whole number, the answer is 5.

A mixed fraction consists of a whole number part and a fractional part. When rounding a mixed fraction, we focus on the fractional part to determine the appropriate rounding direction. If the fractional part is exactly 1/2, it is typically rounded up to the next whole number.

However, in the case of 5 6/13, the fractional part is less than 1/2, so we round down. Rounding down gives us a more accurate approximation that is closer to the original value. In this instance, rounding 5 6/13 down to 5 provides a whole number estimate that is slightly smaller but still reasonably close to the initial mixed fraction.

Rounding serves as a useful tool in situations where precise values are not necessary and a simpler approximation is sufficient for practical purposes.

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The derivative of the function y = 2(2^x)-2 is 12 * 2^x ln(2) or 12ln(2)x(2^x-1).

To find the derivative of the function y = 2(2^x)-2, we will use the power rule and the chain rule of differentiation.

Apply the power rule to the function y = 2(2^x)-2. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

y' = [2(2^x)-2]'

= 2[(2^x)-2]'

= 2ln(2^x)'

Apply the chain rule to (2^x)'. The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x))h'(x). In this case, g(x) = 2^x, so g'(x) = ln(2)*2^x.

y' = 2ln(2^x)'

= 2ln(2^x)

= 2ln(2)x(2^x-1)

Therefore, the derivative of the function y = 2(2^x)-2 is 12 * 2^x ln(2) or 12ln(2)x(2^x-1).

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If Polly reduced the price and sold 3 of the remaining necklaces for £8 each, she sold the remaining necklaces for £6.70 each.

First, let's find the original cost of the necklaces:
50 necklaces * £5 = £250

Now, let's calculate the profit Polly made:
£250 * 70% = £175

So, the total amount she made from selling the necklaces is:
£250 + £175 = £425

Polly sold 40% of the necklaces for £11 each:
50 necklaces * 40% = 20 necklaces
20 necklaces * £11 = £220

She sold 3 necklaces for £8 each:
3 necklaces * £8 = £24

Now let's find the amount left after selling these necklaces:
£425 - £220 - £24 = £181

Polly has 50 - 20 - 3 = 27 necklaces remaining. Let's find the price at which she sold each of the remaining necklaces:
£181 / 27 = £6.70

So, Polly sold the remaining necklaces for £6.70 each.

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Triangle FGH is a right triangle because (HG)²= (FG)²+ (FH)²

Similar triangles are triangles that have the same shape, but their sizes may vary. The ratio of corresponding sides of similar triangles are equal.

Therefore;

6/5 = 3.6/FH

represent FH by x

6/5 = 3.6/x

6x = 5 × 3.6

6x = 18

divide both sides by 6

x = 18/6 = 3

Since FH is 3, this means that the sides of triangle FGH are Pythagorean triple, hence FGH is a right triangle.

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One possible set of parametric equations for the line is:

x = -4 + 4t
y = 8 - t
z = 7 - 4t

To see why these work, let's first consider the equation of the plane: -x + 4y + z = 8. This can also be written in vector form as:

[ -1, 4, 1 ] · [ x, y, z ] = 8

where · denotes the dot product. This equation says that the normal vector to the plane is [ -1, 4, 1 ], and that any point on the plane satisfies the equation.

Now, since the line we want is perpendicular to the plane, its direction vector must be parallel to the normal vector to the plane. In other words, the direction vector of the line must be some multiple of [ -1, 4, 1 ]. Let's call this direction vector d.

To find d, we can use the fact that the dot product of two perpendicular vectors is zero. So we have:

d · [ -1, 4, 1 ] = 0

Expanding this out, we get:

-1d1 + 4d2 + 1d3 = 0

where d1, d2, d3 are the components of d. This equation tells us that d must be of the form:

d = [ 4k, k, -k ]

where k is any non-zero scalar (i.e. any non-zero real number).

Now we just need to find a point on the line. We're given that the line passes through (-4, 8, 7), so this will be our starting point. Let's call this point P.

We can now write the parametric equations of the line in vector form as:

P + td

where t is any scalar (i.e. any real number). Substituting in the expressions for P and d that we found above, we get:

[ -4, 8, 7 ] + t[ 4k, k, -k ]

Expanding this out, we get the set of parametric equations I gave at the beginning:

x = -4 + 4tk
y = 8 + tk
z = 7 - tk

where k is any non-zero scalar.
To find a set of parametric equations for the line, we first need to determine the direction vector of the line. Since the line is perpendicular to the plane given by -x + 4y + z = 8, we can use the plane's normal vector as the direction vector for the line. The normal vector for the plane can be determined by the coefficients of x, y, and z, which are (-1, 4, 1).

Now that we have the direction vector (-1, 4, 1) and the point the line passes through (-4, 8, 7), we can write the parametric equations as follows:

x(t) = -4 - t
y(t) = 8 + 4t
z(t) = 7 + t

So, the set of parametric equations for the line is {x(t) = -4 - t, y(t) = 8 + 4t, z(t) = 7 + t}.

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The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width, and h is the height. In this case, we have:

V = 5 x 3 x 8

V = 120 cubic inches

The price of the popcorn is not given, so we cannot calculate the price per cubic inch.

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