The monthly demand function for a product sold by a monopoly is p = 2104 – (1/3)x^2 dollars, and the average cost is C = 1000 + 22x + x2 dollars. Production is limited to 1000 units and x is in hundreds of units. (a) Find the quantity (in hundreds of units) that will give maximum profit. (b) Find the maximum profit. (Round your answer to the nearest cent.)

The solution set for the equation 1/2 = 1/3 - |(x-3/6)+x| is {1/12, -1/12}.

To solve the equation 1/2 = 1/3 - |(x-3/6)+x|, we need to isolate the absolute value expression and solve for x in two cases.

Case 1: (x-3/6)+x is nonnegative, which means the absolute value can be removed.

1/2 = 1/3 - (x-3/6)-x

1/2 = 1/3 - 2x + 1/2

2x = 1/6

x = 1/12

Case 2: (x-3/6)+x is negative, which means the absolute value must be flipped.

1/2 = 1/3 + (x-3/6)+x

1/2 = 1/3 + 2x - 1/2

2x = -1/6

x = -1/12

Therefore, the solution set is {1/12, -1/12}.

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The derivative of the equation g(x) = 8x^2 – 4; h(x) = 1.6^x is f'(x) = 40.96x · 1.6^x - 15.040128 · 1.6^x

To find the derivative of f(x) = g(x) · h(x), we use the product rule of derivatives, which states that if f(x) = u(x) · v(x), then f'(x) = u'(x) · v(x) + u(x) · v'(x).

Using this rule, we can find the derivative of f(x) = g(x) · h(x) as follows:

f(x) = g(x) · h(x) = (8x^2 – 4) · (1.6^x)

f'(x) = g'(x) · h(x) + g(x) · h'(x) [applying the product rule]

To find g'(x), we take the derivative of g(x) = 8x^2 – 4, which is:

g'(x) = 16x

To find h'(x), we take the derivative of h(x) = 1.6^x, which is:

h'(x) = ln(1.6) · 1.6^x [using the chain rule and the fact that the derivative of a^x is ln(a) · a^x]

h'(x) ≈ 0.470004 · 1.6^x

Now we substitute these values into the product rule formula:

f'(x) = (16x) · (1.6^x) + (8x^2 – 4) ·0.470004 · 1.6^x

Simplifying this expression, we get:

f'(x) = 25.6^x + (12.8x^2 – 6.4) ·0.470004 · 1.6^x

f'(x) = 40.96x · 1.6^x - 15.040128 · 1.6^x
Therefore, the derivative of f(x) = g(x) · h(x) is:
f'(x) = 40.96x · 1.6^x - 15.040128 · 1.6^x

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The paper initial area of the paper is 90 square inches and needs to be folded at least 5 times before its area becomes less than 1 square inch.

a. When n=0, the expression for the area of the paper after folding becomes:

a = 90•(1/3)⁰ = 90•1 = 90 square inches

This means that the initial area of the paper before any folding is 90 square inches.

b. To find out how many folds are needed before the area is less than 1 square inch, we can set the expression for the area of the paper after folding to be less than 1 and solve for n:

a = 90•(1/3)ⁿ < 1

(1/3)ⁿ < 1/90

Taking the logarithm of both sides with base 1/3, we get:

n > log(1/90)/log(1/3)

n > 4.8

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The volume of the large box is 120[tex]s^3[/tex].

If Frank can fit 15 cube-shaped containers in each layer and stack 8 layers into one box, then the total number of containers he can fit in one box is:

15 containers/layer x 8 layers = 120 containers

Since each container is cube-shaped, we can assume that it has the same length, width, and height. Let's represent the length of one side of the container as "s". Then, the volume of one container is:

Volume of one container = [tex]s^3[/tex]

The volume of 120 containers that can fit in one box is:

Volume of 120 containers = 120 x Volume of one container

Substituting the expression for the volume of one container, we get:

Volume of 120 containers = 120[tex]s^3[/tex]

Therefore, the volume of the large box that can hold 120 cube-shaped containers with side length "s" is 120[tex]s^3[/tex].

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The limit of [tex]L_n[/tex] as n approaches infinity is 1/2, and it can be expressed as the definite integral of x from 0 to 1.

To express the limit of [tex]L_n[/tex] as n approaches infinity as a definite integral, we can use the fact that the limit of a Riemann sum is equal to the corresponding definite integral. Thus, we can rewrite [tex]L_n[/tex] as:

[tex]L_n[/tex] = 1/n * (0 + 1 + 2 + ... + (n-1))

This is a Riemann sum for the integral:

[tex]\int\limits^1_0 {x} \, dx[/tex]

with n subintervals of width 1/n. Therefore, we can write:

[tex]\lim_{n \to \infty} L_n = \lim_{n \to \infty} 1/n * (0 + 1 + 2 + ... + (n-1)) = \int\limits^1_0 {x} \, dx = [x^2/2] \ from \ 0 \ to \ 1 = 1/2[/tex]

So, the limit of Ln as n approaches infinity is 1/2, and it can be expressed as the definite integral of x from 0 to 1.

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First, we need to convert the mixed number 2 3/8 to an improper fraction:

2 3/8 = (2 x 8 + 3) / 8 = 19/8

To find out how many times Craig needs to fill a 1/3 cup measuring cup, we can divide 19/8 by 1/3:

(19/8) / (1/3) = (19/8) x (3/1) = 57/8

Therefore, Craig needs to fill a 1/3 cup measuring cup 57/8 times to measure 2 3/8 cups.

The sequence of transformations the triangle XYZ have gone through to produce figure X'Y'Z' is: D. Rotation 90° counterclockwise, reflection across the y-axis.

Transformation is a method required to change the orientation, or resize a given object to produce its image. Some of the types of transformation are: rotation, reflection, dilation, and translation.

Rotation requires turning a given object about a point at a certain angle.

Reflection implies flipping an object over a line as a reference point.

Dilation deals with either increasing or decreasing the dimensions of a given object.

Translation involves moving an object in a specific direction and given number of units.

Considering triangle XYZ and the figure produced, we can conclude that:

The sequence of transformations the triangle XYZ have gone through to produce figure X'Y'Z' is: D. Rotation 90° counterclockwise, reflection across the y-axis.

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The home repair crew worked for 9 days.

To determine the number of days the home repair crew worked for, we can use algebra. Let's assume that the number of days they worked for is represented by "d". We know that they charge $75 per day plus a $250 service fee, so we can set up the following equation:

75d + 250 = 925

Simplifying the equation, we get:

75d = 675

Dividing both sides by 75, we get:

d = 9

However, we need to keep in mind that the $250 service fee is a one-time charge, not a daily charge. So we need to subtract that from the total amount to get the actual amount charged for the days worked:

925 - 250 = 675

Dividing 675 by the daily rate of $75, we get:

675 / 75 = 9

Therefore, the home repair crew worked for 9 days.

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To determine the probability of the coin flip and the cube roll describing the pizza crust and topping combination made at Mario's Pizzeria, we need to consider the total possible outcomes of these events.

There are 2 types of crust, so the coin has 2 sides (heads for one crust type and tails for the other). For the toppings, there are 6 options, represented by the numbers 1 to 6 on the number cube.

First, we find the total possible outcomes by multiplying the number of crust options (2) by the number of topping options (6):
Total possible outcomes = 2 crust options * 6 topping options = 12 combinations

Since there is only one specific combination that Mario's Pizzeria just made, the probability of the coin flip and the cube roll describing this specific combination is:
Probability = 1 (specific combination) / 12 (total possible outcomes)

To express this as a decimal rounded to the nearest thousandth, divide 1 by 12:
Probability ≈ 0.083

So, the probability that the coin flipped and the cube rolled will describe the pizza crust and topping combination that Mario's Pizzeria just made is approximately 0.083, or 8.3%.

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Both the limits are equal to 0: a. lim (x→∞) f(x) = 0 b. lim (x→-∞) f(x) = 0

Know more about rational function,

First, let's rewrite the rational function f(x) more clearly and identify the two limits we need to find: f(x) =[tex](x + 6) / (x^2 + 20)[/tex]

a. lim (x→∞) f(x) b. lim (x→-∞) f(x)

To find these limits, we can use the following steps:

Step 1: Divide both the numerator and the denominator by the highest power of x in the denominator.

In this case, it is x^2. f(x) = [tex][(x + 6) / x^2] / [(x^2 + 20) / x^2][/tex]

Step 2: Simplify the function. f(x) =[tex][(1/x) + (6/x^2)] / [1 + (20/x^2)][/tex]

Step 3: Evaluate the limits.

a. lim (x→∞) f(x)

= [(0) + (0)] / [1 + (0)]

= 0 / 1 = 0

b. lim (x→-∞) f(x)

= [(0) + (0)] / [1 + (0)]

= 0 / 1 = 0

So, both the limits are equal to 0: a. lim (x→∞) f(x) = 0 b. lim (x→-∞) f(x) = 0

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Answer:

Triangles: 4(1/2)(12)(10) = 240 ft^2

Square: 10^2 = 100 ft^2

Total Surface Area: 340 ft^2

Answer:

[tex]-2a^{4}b^{6} c^{3}[/tex]

Step-by-step explanation:

(2a² b c³) (-a² b⁵)

distribute amongst each other

-2a²bc³a²b⁵

rearrange

-2a²a²bb⁵c³

combine like terms

[tex]-2a^{4}b^{6} c^{3}[/tex]

Hope this helps!

Answer:  -2[tex]a^{4}[/tex][tex]b^{6}[/tex]c³  

Step-by-step explanation:

Given:

(2a² b c³) (-a² b⁵)

Rules:

>Multiply numbers first

>If a number with an exponent is multiplied to another, you can combine if the bases are the same(lower number)  

>If they have the same base, you can add the exponents

Solution:

(2a² b c³) (-a² b⁵)       >multiply the - and 2

-2(a² b c³) (a² b⁵)      > combine the a's by adding the expnents 2+2

-2[tex]a^{4}[/tex]( b c³) ( b⁵)           > combine b, there is an invisible 1 by the first b 1+5

-2[tex]a^{4}[/tex][tex]b^{6}[/tex]c³                   >c stays as is because nothing to combine with

m > 8 is an inequality used to calculate the number of articles Mustafa could have written.

Let's assume that Mustafa has written m articles for the school newspaper.

Then, according to the given information:

Heloise has written 1/4 as many articles as Mustafa has, which means she has written 1/4 × m = m/4 articles.

Gia has written 3/2 as many articles as Mustafa has, which means she has written 3/2 × m = 3m/2 articles.

The combined total of articles written by all three is more than 22, so we can write:

m/4 + 3m/2 + m > 22

Simplifying and solving for m:

11m/4 > 22

m > 22 × 4/11

m > 8

Therefore, m > 8 is an inequality used to calculate the number of articles Mustafa could have written.

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After leveling the sand box, the height of the sand box is 1.57 in

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space.

The volume of a cuboid is expressed as;

V = l × w × h

V = 30 × 20 × 5

V = 3000 in³

After leveling, the volume decreases by 1680 in³, therefore the new volume of the sand box is

3000-1680 = 1320

Therefore the new height of the sand is calculated as;

1320 = 30 × 28 × h

1320 = 840h

divide both sides by 840

h = 1320/840

h = 1.57 in

therefore the height of the remaining sand is 1.57

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The evidence on a credit report that indicates good standing may vary depending on the individual's credit history and financial situation.

As an AI language model, I do not have access to specific credit reports or personal information of individuals. However, I can provide you with general information on what constitutes good standing on a credit report and some examples of evidence that may indicate good standing.

A good standing on a credit report typically means that a person has been responsible with their credit and has not defaulted on their payments. This may be indicated by a high credit score, a long credit history, and a low debt-to-credit ratio.

Some examples of evidence that may indicate good standing on a credit report include:

A high credit score: A credit score is a numerical representation of a person's creditworthiness. A high credit score, typically above 700, indicates that a person is in good standing and is likely to be approved for credit or loans. A high credit score may be achieved through a consistent history of making on-time payments, a low debt-to-credit ratio, and a diverse mix of credit accounts.

A low debt-to-credit ratio: A debt-to-credit ratio is the amount of debt a person has compared to their available credit limit. A low debt-to-credit ratio, typically below 30%, indicates that a person is not overextended and is able to manage their credit responsibly. This may be achieved through regular payments and not maxing out credit cards.

Overall, the evidence on a credit report that indicates good standing may vary depending on the individual's credit history and financial situation.

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If the probability of picking out a basketball card is 2/5, and the probability of picking out a football card is 1/3, the probability of Justice randomly picking out a soccer card is 4/15.

To find the probability of Justice picking out a soccer card, we need to know the total probability of all three types of cards adding up to 1. Since there are only three types of cards, we can subtract the probability of picking a basketball card and the probability of picking a football card from 1 to find the probability of picking a soccer card.

Let P(S) be the probability of picking a soccer card.

We know that P(B) = 2/5 and P(F) = 1/3.

Therefore, the total probability of picking one of the three cards is:

P(B) + P(F) + P(S) = 1

Substituting the values we know, we get:

2/5 + 1/3 + P(S) = 1

Simplifying the equation, we get:

6/15 + 5/15 + P(S) = 1

11/15 + P(S) = 1

P(S) = 1 - 11/15

P(S) = 4/15

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Determine which proportion is correct, we will compare the cross products of each proportion. The correct proportion will have equal cross products. The correct proportion is 1/2 = 3/6.

1. 4/10 = 3/6
To check this proportion, we'll calculate the cross products:
(4 * 6) = (10 * 3)
24 = 30
Since 24 ≠ 30, this proportion is incorrect.
2. 1/2 = 7/8
To check this proportion, we'll calculate the cross products:
(1 * 8) = (2 * 7)
8 = 14
Since 8 ≠ 14, this proportion is incorrect.
3. 1/2 = 3/6
To check this proportion, we'll calculate the cross products:
(1 * 6) = (2 * 3)
6 = 6
Since 6 = 6, this proportion is correct.
4. 4/10 = 7/8
To check this proportion, we'll calculate the cross products:
(4 * 8) = (10 * 7)
32 = 70
Since 32 ≠ 70, this proportion is incorrect
So, the correct proportion is 1/2 = 3/6.

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Answer:

True

Step-by-step explanation:

Parallelogram: A parallelogram is 4-sided shape that has parallel opposite sides

Square: A square is 4-sided shape where all sides are congruent and has 4 right angles.  A square also has 2 pairs of opposite parallel sides.

Based on these definitions, we can see that not ALL parallelograms can be squares because a parallelogram can have no right angles, or can have all 4 angles be right angles.

All squares have to be parallelograms though because a square and a parallelogram both have 2 pairs of opposite parallel sides.

Hope this helps :)

5.) The dimensions of the banner whose perimeter is given is listed below:

length = 134in

width = -59in

To calculate the dimensions of the rectangular banner, the formula for the perimeter of rectangle should be used and it's given below;

Perimeter of rectangle = 2(length+width)

length = 16-2a

width = a

perimeter = 160in

That is;

160 = 2(16-2a+a)

160 = 32-4a+2a

160 = 42-2a

2a = 42-160

2a = -118

a = 118/2

= -59

Length = 16-2(-59)

= 16+118

= 134in

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When multiplying a whole number with a fraction, the first factor will be reduced to a fraction or a decimal, depending on the problem.

In both of the given problems, we are multiplying a whole number with a fraction. When we multiply a whole number with a fraction, we can simply multiply the whole number with the numerator of the fraction and keep the denominator as it is. So, in the first problem, we have 10 multiplied by 1/2. Multiplying 10 with 1 gives us 10, and then we divide it by 2, which gives us 5. Therefore, the first factor, which is 10, will be reduced to 5 in this problem.

In the second problem, we have 15 multiplied by 7/2. Multiplying 15 with 7 gives us 105, and then we divide it by 2, which gives us 52.5. Therefore, the first factor, which is 15, will be reduced to 52.5 in this problem.

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a) Where the information about triangle ABC is given above, the measure of angle B is 60°

Using the Cos Rules,

(15√3)² = 30² + 15² - 2(15 x 30) cos B

⇒ 657 = 1125 - 900 cosB

⇒ 2 Cos B = 1 Cos B = 1/2

So

B = Cos ⁻¹(1/2)

Hence,

B = 60°

B) Given the above,

Now, we can use the tangent function to find tan B:

tan B = sin B / cos B

  = sin (π/3) / cos (π /3)

  = (√3 / 2) / 0.5

tan B = √3

C ) the area of the rriangle is given as

s = (a + b + c) /2

Substituting

s = (15 + 15√ 3 + 30)/2

s = 30 + 15√3

Area = √ [ (30 + 15√3)(15√3) (15)(30  - 15 - 15√3))

Simplifying  we can say

Area = √(30 + 15√3  )( 15√3)(15)(15 - 5√3 )]

= √[3(10 + 5√3)(15)(3√3 - 1)]

  = √[2250 - 1125√3]

Hence,

Area ≈ 17.36in

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The mean and the median of the data set is given as the median, 4, is greater than the mean, 3. The correct answer is option a.

To find the mean of the data set, we add up all of the numbers and divide by the total number of days. So, the mean is (1+4+5+5+1+5+5)/6 = 3.

To find the median, we first need to put the numbers in order from least to greatest: 1, 1, 4, 5, 5, 5. The median is the middle number, which in this case is 4.

Now we need to determine which value is greater. The options are:

A. The median, 4, is greater than the mean, 3.
B. The mean, 3, is greater than the median, 4.
C. The mean, 4, is greater than the median, 2.
D. The median, 5, is greater than the mean, 0.

We can eliminate options B and D because they don't make sense. The mean cannot be less than 0, and the median cannot be less than the mean.

Option A is correct. The median, 4, is greater than the mean, 3. This makes sense because there are two 1's in the data set, which bring down the mean. However, the median is not affected by outliers like this, so it remains higher.

Option C is incorrect because the mean is not 4, it is 3.

In summary, the mean of the data set is 3 and the median is 4. The median is greater than the mean, which is represented by option A.

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The amount of time it would take for the ball to reach its maximum height is 2.1825 seconds.

Based on the information provided, we can logically deduce that the height (h) in feet, of this ball above the​ ground is related to time by the following quadratic function:

Next, we would determine the maximum height of this ball by taking the first derivate in order to determine the time (t) it takes as follows;

h(t) = -16t² + 90t

h'(t) = -32t + 90

90 = 32t

t = 90/32 = 2.1825 seconds.

h(2.1825) = -16(2.1825)² + 90(2.1825)

h(2.1825) = 120.21 feet.

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The rate of change for the linear function represented in the table is 3/5.

In Mathematics and Geometry, the gradient, rate of change, or slope of any straight line can be determined by using the following mathematical equation;

Rate of change (slope) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Rate of change (slope) = rise/run

Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)

By substituting the given data points into the formula for the slope of a line, we have the following;

Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)

Rate of change (slope) = (69 - 66)/(5 - 0)

Rate of change (slope) = 3/5

Based on the table, the rate of change is the change in y-axis with respect to the x-axis and it is equal to 3/5.

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The image of P under a 270° counterclockwise rotation about the origin is (2, -8) .

Transformations are changes done in the shapes on a coordinate plane by rotation, reflection or translation.

Translation, rotation, reflection, and dilation are the four common types of transformations.

Rotation simply means turning. If a point P, whose position vector is (x,y), is rotated  270° counterclockwise rotation about the origin, the position of the image P' is (y, -x).

We have: P(8, 2)

Thus, the image of P will be:

P'(2, -8)

Therefore, the image of P under a 270° counterclockwise rotation about the origin is (2, -8).

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If a big sheet of white paper has a red dot in the center, the red dot is the  figure or object, and the white space is the ground or background.

In visual perception, the figure-ground relationship is the process by which our brains distinguish an object( the figure) from its surroundings( the ground).

This relationship is essential in our capability to fete and make sense of the visual world around us. The figure is the object of interest or focus, while the ground is the background against which it stands out.

The red dot becomes the focal point or center of attention, while the white space around it provides environment and contrast, making the dot more visible and commanding.

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The fraction is half of the sweaters cost $ 50 or less.

After placing the given set of numbers in order, the median is the value that falls in the middle of the group.

From the given box plot,

The value of the median is $ 50

We can clearly see that the median is $ 50 this means, 50% of the clothes are below $ 50 and 50% of the clothes are above $ 50.

The percentage of $ 50 or less sweaters is now 50%.

So, Required fraction = 50 / 100

= 5 / 10

= 1 / 2

Therefore, the fraction is half of the sweaters cost $ 50 or less.

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Given question is incomplete, complete question is below

Bella works at a popular clothing store. last winter ,her manager asked her to track the sweater sales. this box plot show the results.

question: what fraction of the sweater cost $50 or less?

30 km far has the athlete run.

A rectangle may be a geometric shape that is characterized by its four sides, where opposite sides are parallel and break even within the length. It has four sides the longer side is named length and the shorter side is named as breadth.

An Athlete runs around a rectangular field = 10 times

The Length of the rectangle = [tex]1.08 km[/tex]

The Breadth of the rectangle = [tex]0.42 km[/tex]

 [tex]1km = 1000 m\\= 420 /1000 m\\= 0.42 km[/tex]

Therefore, perimeter of the rectangle = 2 (length + breadth)

                                                              =  [tex]2 ( 1.08 + 0.42)[/tex]

                                                              = [tex]2 (1. 50)[/tex]

                                                              = [tex]3 km[/tex]

So, the athlete runs around a rectangular housing 10 times = [tex]3[/tex]×[tex]10[/tex]

                                                                                                   = [tex]30 km[/tex]

Therefore, the athlete runs 30 km far.

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Answer:the two sides still equal.

Step-by-step explanation:To solve 2x + 3 = 5, Sylvia first subtracted 5 from both sides. What did she do wrong? Because Sylvia did the same operation to both sides, the equation is still correct: the two sides still equal.

Answer:

1

Step-by-step explanation:

or, 2x=5-3

or, 2x=2

or, x=2/2

x=1

If Mr Louis gets a 6. 5% discount on the price of the oil, Mr. Louis gets a discount of £65.41 on this purchase.

First, let's find out how many litres of oil Mr. Louis needs to fill his tank:

Maximum capacity of the tank: 1,400 litres
Current oil in the tank: 150 litres
Litres needed to top up: 1,400 - 150 = 1,250 litres

Next, we'll calculate the original price of the oil without the discount:

Price per litre: £0.805
Cost of oil needed: 1,250 litres * £0.805 = £1,006.25

Now, let's calculate the discount amount:

Discount percentage: 6.5%
Discount amount: £1,006.25 * 0.065 = £65.41

So, Mr. Louis gets a discount of £65.41 on this purchase.

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