1 Factorise completely.
(a) 2a+4+ap+2p
(b) 162 – 8t^2

Using Richardson extrapolation, the computed first derivative of f(x) at x = 2 is approximately -0.052.

The absolute error with respect to the true solution is 22.348.

To compute the accuracy gain using Richardson extrapolation, we'll use the central difference formula for the first derivative, which is given by:

f'(x) = [f(x + h) - f(x - h)] / (2h)

h₁= 0.02 and h₂ = 0.03, and then apply Richardson extrapolation.

Calculate the derivative using h₁ = 0.02

x = 2

h₁ = 0.02

f(x + h₁) = f(2 + 0.02) = f(2.02) = 5sin(10 × 2.02) + (2.02)³ - 2(2.02)² - 6(2.02) + 10

= 5sin(20.2) + 8.120808 - 8.162808 - 12.12 + 10

= -3.5799

f(x -h₁) = f(2 - 0.02) = f(1.98)

= 5sin(19.8) + 7.783608 - 7.844808 - 11.88 + 10

= -3.4758

f'(x) = [-3.5799 - (-3.4758)] / (2 × 0.02)

= -0.052

Calculate the derivative using h₂ = 0.03

x = 2

h₂  = 0.03

f(x + h₂ ) = f(2 + 0.03) = f(2.03) = 5sin(10 × 2.03) + (2.03)³ - 2(2.03)² - 6(2.03) + 10

= -3.5998

f(x - h₂) = f(2 - 0.03) = f(1.97)

= -3.3943

f'(x) = [-3.5998 - (-3.3943)] / (2 × 0.03)

= -0.052

Richardson extrapolation is given by the formula:

\(f'(x) = (2^p \times f'(x, h_2) - f'(x, h_1)) / (2^p - 1)\)

Here, p is the order of the method. Since we're using the central difference formula, which is second-order accurate, p = 2.

f'(x) = (2² × -0.052 - (-0.052)) / (2² - 1)

= (4 × -0.052 + 0.052) / 3

= -0.052

Therefore, using Richardson extrapolation, the first derivative of f(x) at x = 2 is approximately -0.052.

To compute the absolute error with respect to the true solution, we need to find the true derivative of f(x) and evaluate it at x = 2.

f(x) = 5sin(10x) + x³ - 2x² - 6x + 10

f'(x) = 50cos(10x) + 3x² - 4x - 6

f'(2) = 50cos(20) + 3(2)^2 - 4(2) - 6

= 50(-0.408) + 12 - 8 - 6

= -22.4

Absolute error = |(-22.4) - (-0.052)|

= |-22.4 + 0.052|

= 22.348

Therefore, the absolute error with respect to the true solution is  22.348.

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Answer:

they are complementary

Step-by-step explanation:

complementary means adding up to 90° and it's true Abt the two angles

By finding the roots we can rewrite our quadratic equation as:

3x² + 10x + 8  =  3*(x + 4/3)*(x + 2)

Here we have a quadratic function which is:

3x² + 6x + 4x + 8

To factorize this we need to simplify it first, we can add the two middle terms because these have the same power of x:

3x² + 6x + 4x + 8  = 3x² + 10x + 8

Now we need to find the zeros, to do so we need to solve:

3x² + 10x + 8  = 0

And using the quadratic formula we will get:

\(x = \frac{-10 \pm \sqrt{10^2 - 4*3*8} }{2*3} \\\\x = \frac{-10 \pm 2 }{6}\)

Then the roots are:

x = (-10 - 2)/6 = -2

x = (-10 + 2)/6 = -8/6 = -4/3

Then we can factorize the quadratic as:

3x² + 10x + 8  =  3*(x + 4/3)*(x + 2)

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The value of x which is hypotenuse is 4 units .

Using pythagoras theorem ,

( a ^ 2 ) + ( b ^ 2 ) = ( h ^ 2 )

where a is the base , b is a side and h is the hypotenuse of a right- angled triangle .

First triangle's hypotenuse = \(\sqrt{2}\)

Second triangle's hypotenuse = \(\sqrt{3}\)

Third triangle's hypotenuse = 2

Therefore ,

Last triangle's hypotenuse = \(\sqrt{15}\)

Hence ,

( \(\sqrt{15}\) ^ 2 ) + [ ( 1 ) ^ 2 ] = x ^ 2

15 + 1 = x ^ 2

16 = x ^ 2

x = 4

Therefore , the value of x which is hypotenuse is 4 units .

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if h(x) = 2x² - 5x + 7

h(-3) = 2(-3)² - 5(-3) + 7

= 18 + 15 + 7

= 40

Answer:

1300

Step-by-step explanation:

add the S numbers than add the T number than there is your answer

y=-5/3x-4

is the linear equation in standard form

Answer:

all of them

Step-by-step explanation:

it is given that the expression is,

y = 5x + 23

here , y = number of people

x = number of flyers,

so, the given expression is representing that,

for every extra Flyer used in advertising, the attending increases by 5 people ​

thus, the correct answer is option D

Answer:

Step-by-step explanation:

\(cot(\frac{\pi }{2} -\theta)=tan \theta=-0.87\)

or second method

\(\cot(\frac{\pi }{2} -\theta)=\frac{1}{\tan(\frac{\pi }{2} -\theta)} =\frac{1}{\frac{\tan\frac{\pi }{2} -\tan \theta}{1+\tan \frac{\pi }{2 } \tan \theta} } =\frac{1+\tan\frac{\pi }{2} \tan \theta}{\tan\frac{\pi }{2} -\tan \theta} \\(divide~the~numerator~and~denominator~by~\tan\frac{\pi }{2} )\)

\(=\frac{\cot\frac{\pi }{2} +\tan\theta}{1-\cot\frac{\pi }{2} \tan\theta} \\=\frac{0+\tan \theta }{1-0*\tan\theta} \\=\tan \theta\\=-0.87\)

Answer: 6x

Step-by-step explanation:

so consider x as the number,

twice of x= 2x

2x tripled= 2x*3

               = (2*3)x

               = 6x

Answer: \(\Large\boxed{Point~B'~=~(-4,~-2)}\)

Step-by-step explanation:

Given information

Point B = (-2, -4)

Reflection line: y = x

Concept:

When reflecting over the line y = x, the position of y and x in the original coordinate switches places.

This is because it is " y = x ", assuming that they could interchange

Find the coordinate of Point B'

Original (x, y) = (-2, -4)

Point B' = (x', y') = (y, x) = \(\Large\boxed{(-4,~-2)}\)

Please refer to the attachment below for a graphical understanding

Hope this helps!! :)

Please let me know if you have any questions

The coordinates of B' after the reflection over the line y = x are (-4, -2).

Given that a point B = (-2, -4) has been reflected over the line x = y, we need to find the coordinates of the reflected point B'.

To find the coordinates of point B' after a reflection over the line y = x, we need to understand what a reflection over this line means.

The line y = x is a diagonal line that passes through the points where the x-coordinate and the y-coordinate are equal.

It has a 45-degree angle with both the x-axis and the y-axis. When you reflect a point over this line, you essentially swap its x and y coordinates.

Let's take point B(-2, -4) as an example. To reflect this point over the line y = x, we swap its x and y coordinates:

New x-coordinate of B' = Old y-coordinate of B = -4

New y-coordinate of B' = Old x-coordinate of B = -2

So, the coordinates of B' will be (y, x) = (-4, -2).

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Answer:

Step-by-step explanation:

The equation is y=7x

x      y          Calculation

5     35         y=7x, y=7*5, y=35

6     42         y=7x, y=7*6, y=42

7      49       y=7x, y=7*7, y=49

8     56          y=7x, y=7*8, y=56

Answer:

The value of x is 25.

Step-by-step explanation:

Given that M is the midpoint of LN. So the distance of LM is equals to MN.

\(lm = mn\)

\(43 = 2x - 7\)

Then you can solve it :

\(2x = 43 + 7\)

\(2x = 50\)

\(x = 25\)

Answer:

74.42

Step-by-step explanation:

(150-86):86*100 =

(150:86-1)*100 =

174.41860465116-100 = 74.42

Answer:

576

Step-by-step explanation:

-67+ r7  

The composition of functions in this case is not commutative.To find f(g(4)), we first need to evaluate g(4) and then substitute the result into f(x).

g(4) = 4 - 2(4) = 4 - 8 = -4

Now we substitute -4 into f(x):

f(g(4)) = 3(-4) + 4 = -12 + 4 = -8

So, f(g(4)) = -8.

To find g(f(4)), we first need to evaluate f(4) and then substitute the result into g(x).

f(4) = 3(4) + 4 = 12 + 4 = 16

Now we substitute 16 into g(x):

g(f(4)) = 4 - 2(16) = 4 - 32 = -28

So, g(f(4)) = -28.

Now, let's check if the composition of functions is commutative.

Commutative property states that changing the order of the functions should not affect the result of the composition.

In this case, we have f(g(4)) = -8 and g(f(4)) = -28, which are not equal.

Therefore, the composition of functions in this case is not commutative.

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y=x+5 this the the answer

The correct statements are:

b) Ordering costs decrease with respect to lot size,

c) It is good to have fixed interval ordering systems for products that have independent demand,

e) Taxes and insurance costs can be considered as carrying costs of inventory,

f) Costs incurred for defective products identified after the products are shipped are classified as internal failure costs,

h) Costs of repairing faulty products under warranty are limited to external failure costs, and j) Under EOQ inventory model, there is an assumption which states that there is no possibility of inventory stockout to occur.

b) Ordering costs decrease with respect to lot size: Ordering costs involve the expenses incurred when placing orders for inventory, such as administrative costs, processing fees, and transportation costs.

When the lot size increases, the frequency of placing orders decreases, resulting in lower ordering costs per unit. This is because the fixed costs associated with ordering are spread over a larger quantity of inventory.

c) It is good to have fixed interval ordering systems for products that have independent demand: Fixed interval ordering systems involve placing orders at regular intervals, regardless of the inventory level.

This approach is suitable for products with independent demand, where the demand is unpredictable or sporadic. By setting fixed intervals, the company can ensure a consistent replenishment schedule and avoid stockouts while optimizing inventory levels.

e) Taxes and insurance costs can be considered as carrying costs of inventory: Carrying costs are the expenses associated with holding and storing inventory.

They include costs such as storage fees, insurance premiums, taxes on inventory, and the opportunity cost of tying up capital in inventory. Taxes and insurance costs directly impact the financial burden of inventory holding and are considered as components of carrying costs.

f) Costs incurred for defective products identified after the products are shipped are classified as internal failure costs: Internal failure costs are expenses incurred due to quality issues discovered within the company's operations.

In the context of inventory, costs related to defective products identified after they are shipped, but before reaching the customer, fall under internal failure costs. These costs include rework, scrap, and any necessary corrective actions taken to address the quality problems.

h) Costs of repairing faulty products under warranty are limited to external failure costs: External failure costs are the expenses incurred due to quality issues discovered by customers after the products have been delivered.

When faulty products are repaired under warranty, the costs associated with the repairs are considered external failure costs. This includes the direct costs of repairing or replacing the faulty products and any associated customer service expenses.

j) Under EOQ inventory model, there is an assumption which states that there is no possibility of inventory stockout to occur: The Economic Order Quantity (EOQ) model is a widely used inventory management technique.

It assumes that demand for the product is constant and known with certainty, there are no quantity discounts or price variations, and there is no possibility of stockouts occurring.

The assumption of no stockouts simplifies the calculations and ensures that the optimal order quantity obtained from the model will meet the demand without interruptions.

However, in real-world scenarios, stockouts can occur due to unforeseen factors, variability in demand, or supply chain disruptions.

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Answer: No solution.

Step-by-step explanation:

2(4a + 8) = 8a − 4

8a + 16 = 8a − 4

16 = − 4

Since 16 = -4 is false, there is no solution.

Answer:

Step-by-step explanation:

In this problem we are required to find the cost of the laptop when 9.25% of the cost is added as tax

we are given that the tax rate is 9.25% of the initial cost

and the initial cost is $703.98

 let us calculate 9.25% of  $703.98

(9.25/100)* 703.98= 0.0925*703.98= $65.12

Hence the charges for tax is $65.12

The total cost of the laptop when tax is included is

the initial cost Plus the tax charges= $703.98+$65.12= $769.098

$769.10

The possible polynomial functions with x-intercepts at (-2,0), (-3,0), and (5,0) will be one of the second degree (quadratic) polynomials. Thus, the correct answer is: Option (d) 2 possible polynomial functions.

The polynomial functions are functions that consist of terms made up of constants, coefficients, and variables that are combined using the four fundamental mathematical operations: addition, subtraction, multiplication, and division.

Here, we are given three x-intercepts at (-2,0), (-3,0) and (5,0).X-intercepts are also known as zeros or roots of a polynomial function.

When a function intersects with the x-axis, it is said to have an x-intercept.In general, any polynomial function of degree n can have n zeros, which are the values of x for which f(x) = 0. So, as we have three x-intercepts, so there will be 3 zeros of the polynomial function.

Therefore, the possible polynomial functions with x-intercepts at (-2,0), (-3,0), and (5,0) will be one of the second degree (quadratic) polynomials. Thus, the correct answer is: Option (d) 2 possible polynomial functions.

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Kristin will pay 21.20 for the jeans after the discount and sales tax are applied.

The first step is to calculate the discount Kristin will receive:

Discount = 20% of 25 = 0.20 x 25 = 5

Now subtract the discount from the original price to find the sale price:

Sale price = 25 - 5 = 20

Next, calculate the amount of sales tax Kristin will have to pay on the sale price:

Sales tax = 6% of 20 = 0.06 x 20 = 1.20

Finally, add the sale price and sales tax to find the total cost:

Total cost = 20 + 1.20 = 21.20

Therefore, Kristin will pay 21.20 for the jeans after the discount and sales tax are applied.

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Using Pythagorean theorem, the side of the ladder will reach 22.45 ft far up now.

Pythagorean theorem describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two other side.

c² = a² + b²

If a 45 foot ladder is set up against the side of a house so that it reaches up 27 feet, then the distance between the base of the ladder and the base of the side of the house is the difference between the square of the two values.

c = 45 ft

a = 27 ft

b² = c² - a²

b² = (45 ft)² - (27 ft)²

b² = 2025 - 729

b² = = 1296

b = 36 ft

If Latanya grabs the ladder at its base and pulls it 3 feet farther from the house, then the distance becomes b + 3 = 39 ft.

Using Pythagorean theorem, determine how far up the ladder is on the side of the house.

c = 45 ft

b = 39 ft

a² = c² - b²

a² = (45 ft)² - (39 ft)²

a² = 2025 - 1521

a² = 504

a = 6√14

a = 22.45 ft

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Answer:

(b+1)(5a+3)

Step-by-step explanation:

Notice how (b+1) appears on both sides. That means we can "factor it out" by moving it to the very left -

(b+1)

Now we know the form of the answer we can continue doing it.

For the next step, I take the terms 5a and 3 (what's left) and move them to the right as shown:

(b+1)(5a+3)

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Best regards

The solutions to the quadratic equation \(2x^2 + 6x - 1 = 0\), obtained by completing the square, are x = -3 + √10 and x = -3 - √10.

To solve the quadratic equation\(2x^2 + 6x - 1 = 0\)by completing the square, follow these steps:

Ensure that the coefficient of \(x^2\) is 1. In this case, it is already 2, so we don't need to make any changes.

Move the constant term to the other side of the equation. Add 1 to both sides:

\(2x^2 + 6x = 1\)

Divide the coefficient of x by 2 and square it. In this case, (6/2)^2 = 9.

Add the result from step 3 to both sides of the equation:

\(2x^2 + 6x + 9 = 1 + 9\)

Simplifying, we get:

\(2x^2 + 6x + 9 = 10\)

Write the left side of the equation as a perfect square trinomial. In this case, it is \((x + 3)^2.\)

\((x + 3)^2 = 10\)

Take the square root of both sides:

\(√[(x + 3)^2] = ±√10\)

Simplifying:

\(x + 3 = ±√10\)

Solve for x by subtracting 3 from both sides:

x = -3 ± √10

So, the solutions to the quadratic equation \(2x^2 + 6x - 1\)= 0, obtained by completing the square, are x = -3 + √10 and x = -3 - √10.

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