Complete the table of values for y=1/2x^3-2x+1

Answer:

three times

Step-by-step explanation:

The difference between the numbers whose sum is of 96 and have a ratio of 11:1 is of 80.

The difference between the two amounts is obtained applying the proportions in the context of the problem.

The variables that are going to represent the two amounts are given as follows:

x and y.

The sum of two numbers is 96, hence:

x + y = 96.

The ratio between them is 11:1, hence:

x/y = 11.

x = 11y.

Replacing the second equation into the first, the value of y is given as follows:

y + 11y = 96

12y = 96

y = 8.

The value of x is given as follows:

x = 11 x 8 = 88.

The difference is then given as follows:

x - y = 88 - 8 = 80.

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[tex]x=30^o,270^o \ [0^0\leq x\leq 360^0][/tex]

Explanation:

We know,

[tex]cos2x=cos^2 \ x-sin^2 \ x=1-2sin^2 \ x[/tex]

So, let's solve the equation now,

[tex]sin \ x=cos2x=1-2 \ sin^2 \ x[/tex]

[tex]\longrightarrow \ 2 \sin^2 \ x+sin \ x-1=0[/tex]

[tex]\longrightarrow \ 2 \sin^2 \ x+2\sin x-sin \ x-1=0[/tex]

[tex]\longrightarrow \ 2\sin \ x(sin\ x+1)-1(sin \ x+1)=0[/tex]

[tex]\longrightarrow \ (2\sin \ x-1)(sin \ x+1)=0[/tex]

Now,

[tex]2\sin \ x-1=0[/tex]

[tex]\longrightarrow \ sin \ x=\dfrac{1}{2}[/tex]

[tex]\longrightarrow x=sin^{-1}(\dfrac{1}{2})[/tex]

[tex]\longrightarrow x=30^o[/tex]

And, [tex]sin \ x+1=0[/tex]

[tex]\longrightarrow x=sin^{-1}(-1)=270^o[/tex]

As we just need the general solutions, we should take only this two values as the general solutions.

Answer:

[tex]30^0,270^0[/tex]

That's it!

Answer: 1260

Step-by-step explanation:

To solve for simple interest, we can use the formula:

I = P * r * t

where:

I = interest

P = principal amount

r = interest rate

t = time (in years)

In this case:

P = 2400

r = 10.5% = 0.105 (note that we convert the percentage to a decimal)

t = 5

Substituting these values into the formula, we get:

I = 2400 * 0.105 * 5

I = 1260

Therefore, the simple interest on a principal amount of 2400 at 10.5% for 5 years is 1260.

The sales of marshmallow is 5 million and the sales of graham cracker is 5 million is 0.5648 and the probability that 3X1-1X2 + 3X3 > 20 is 0.000005.

The multivariate normal distribution is a probability distribution which describes the joint behavior of multiple random variables. In the given case, the profit (in millions) for selling chocolate (Xi), marshmallow (X2) and graham cracker (X3) follows a multivariate normal distribution with parameters 1, 0.3, 0.3 and Σ = 0.31 0 0.3 01.

1. To calculate the probability that the profit for selling chocolate is greater than 6 millions, we need to calculate the probability that X1>6. Using the given parameters, we can use the formula for calculating the cumulative probability of a standard normal distribution: [tex]P(X1>6) = 1-P(X1≤6) = 1-0.9999994 = 0.000006.[/tex]

2. To calculate the probability that the profit for selling chocolate is greater than 6 millions, given the sales of marshmallow is 5 million and the sales of graham cracker is 5 million, we need to calculate the conditional probability [tex]P(X1>6|X2=5, X3=5)[/tex]. Using the given parameters, we can calculate this probability using the formula for conditional probability:[tex]P(X1>6|X2=5, X3=5) = P(X1>6 ∩ X2=5 ∩ X3=5) / P(X2=5 ∩ X3=5) = 0.002207 / 0.003915 = 0.5648.[/tex]

3. To calculate the probability that, we need to calculate the probability that[tex]X1>7-X2/3-X3/3[/tex]. Using the given parameters, we can calculate this probability using the formula for cumulative probability of a standard normal distribution: [tex]P(3X1-1X2 + 3X3 > 20) = 1-P(3X1-1X2 + 3X3 ≤ 20) = 1-0.9999995 = 0.000005.[/tex]

In conclusion, the probability that the profit for selling chocolate is greater than 6 millions is 0.000006, the probability that the profit for selling chocolate is greater than 6 millions

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Answer:

Violet and blue light have the shortest wavelengths and red light has the longest. Therefore, blue light is scattered more than red light and the sky appears blue during the day. When the Sun is low in the sky during sunrise and sunset, the light has to travel further through the Earth's atmosphere.

Using equations we know that 45 pounds are included in the $1.40 worth of groceries and 55 pounds worth of groceries are being purchased for 40 cents.

An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values.

As in 3x + 5 = 15, for example.

There are many different types of equations, including linear, quadratic, cubic, and others.

So, we have:

x + y = 100                        .........A

40 × x + 140 y = 85 × 100

40 x + 140 y = 8500         ........B

Solving (A) and (B) as follows:

(40 x + 140 y) - 40 × ( x + y ) = 8500 - 40 × 100

(40 x - 40 x) + (140 y - 40 y) = 8500 - 4000

0 + 100 y = 4500

y = 4500/100

Hence, the price per unit of grocery is $1.40 = y = 45 pounds.

Now, put the value of y in equation (A) as follows:

x + y = 100

x = 100 - y

x = 100 - 45

x = 55 pounds

The number of groceries at the 40-cent price is x = 55 pounds.

Therefore, using equations we know that 45 pounds are included in the $1.40 worth of groceries and 55 pounds worth of groceries are being purchased for 40 cents.

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Correct question:
A grocer has two kinds of candies, one selling for 40 cents a pound and the other for $1.40 per pound. How many pounds of each kind must he use to make 100 pounds worth 85 cents a pound?

According to the question the value of the 5 digit in 4,597 is 9.

Digit is a numerical symbol used to represent numbers in the decimal system. It is a single symbol used to represent a number ranging from 0 to 9. Digits are used to represent numbers in all forms of mathematical calculations and equations. The most common digits used in calculations are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit has an associated numerical value, which can be used to create larger numbers or to perform complex mathematical operations.

The answer is D) 9. The value of the 5 digit in 4,597 is 5. To find the 10 times its value, you must multiply 5 by 10, which equals 50. Therefore, the value of the 5 digit in 4,597 is 9.

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Answer:

6/11 = 0.54

Step-by-step explanation:

Answer: Yes you are correct

Step-by-step explanation:

a. Finding the transition matrix from B1 to B2 using the standard basis for the vector space of lower triangular matrices with zero trace.The standard basis of the vector space of lower triangular matrices with zero trace is given by{(1,0,0),(0,1,0),(0,0,0)}.We are to find the transition matrix from B1 to B2. We start with the definition of the transition matrix. This definition states that if A = [a1,a2,a3] is a matrix whose columns are the vectors of B2, then the transition matrix from B1 to B2 is the matrix S such that S = [b1,b2,b3] where bi is the column vector obtained by expressing the ith vector of B1 as a linear combination of the vectors of B2. Using the standard basis, we have that (1,0,0) = a1, (0,1,0) = a2 and (0,0,0) = a3. Therefore, we need to express each of these standard basis vectors as a linear combination of the vectors of B1.For (1,0,0), we have(1,0,0) = 2e1 - e2For (0,1,0), we have(0,1,0) = -3e1 + e3For (0,0,0), we have(0,0,0) = e2 + e3Therefore, the transition matrix S is given by S = [b1,b2,b3] where bi is obtained by expressing the ith vector of the standard basis as a linear combination of the vectors of B1. Thus,S = [(2,-3,0),(-1,1,0),(0,0,1)]b. Finding the coordinates of v in B if the coordinate vector of v in B1 is c. Let c be the coordinate vector of v with respect to B1. Then we know that v = c1e1 + c2e2 + c3e3. We are to find the coordinate vector of v with respect to B.We know that B is a basis for the vector space of lower triangular matrices with zero trace, so any vector in this space can be expressed uniquely as a linear combination of the vectors in B. Thus, we can write v as a linear combination of the vectors of B.v = a1x1 + a2x2 + a3x3We are to find the coefficients x1, x2 and x3. We do this by using the fact that the transition matrix S from B1 to B is such that v = Sc where c is the coordinate vector of v with respect to B1. Hence, v = Sc = [b1,b2,b3][c1,c2,c3] = (2c1 - c2) b1 - (c1 - c2) b2 + c3 b3Using the expressions for b1, b2 and b3 in terms of the standard basis vectors, we obtainv = (2c1 - c2)(2e1 - e2) - (c1 - c2)(-e1 + e3) + c3e3

Expanding this expression and comparing coefficients with the equation for v above yields(2c1 - c2)(2e1 - e2) - (c1 - c2)(-e1 + e3) + c3e3 = c1e1 + c2e2 + c3e3Therefore, we have the system of equations2(2c1 - c2) - (c1 - c2) = c11(2c1 - c2) + (c1 - c2) = c20 = c3Solving for x1, x2 and x3 yieldsx1 = c2/2, x2 = c1/2, and x3 = 0Therefore, the coordinate vector of v with respect to B is given by the vector( c2/2, c1/2, 0).c. Finding [v]B2 in 200 wordsWe are to find the coordinate vector of v with respect to B2. Since we already have the coordinate vector of v with respect to B1, we can use the transition matrix S from B1 to B2 to obtain this coordinate vector.Let c be the coordinate vector of v with respect to B1. Then, we know that v = c1e1 + c2e2 + c3e3. Since the coordinate vector of v with respect to B1 is c, we have the equationc = [c1,c2,c3]Using the transition matrix S from B1 to B2, we can write the coordinate vector of v with respect to B2 as[x1,x2,x3] = S[c1,c2,c3]Multiplying these matrices together yields the equation[x1,x2,x3] = [(2,-3,0),(-1,1,0),(0,0,1)][c1,c2,c3]

Expanding this equation gives the system of equations2c1 - c2 = x1-3c1 + c2 = x2c3 = x3Solving this system of equations for c1, c2 and c3 yieldsc1 = (x2 - x1)/4, c2 = (3x2 + x1)/4, and c3 = x3Therefore, the coordinate vector of v with respect to B2 is given by the vector((x2 - x1)/4, (3x2 + x1)/4, x3).

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Flipping a coin 3 times, the probability of getting 3 heads is 0.125.

The probability of getting 3 heads when a coin is flipped 3 times can be calculated using the binomial probability formula. The formula is:

P(X=k) = nCk × p^k × (1-p)^(n-k)

where n is the number of trials, k is the number of successes, p is the probability of success, and (1-p) is the probability of failure.

Here, n=3, k=3 and p=0.5 (since the coin has two sides and the probability of getting heads or tails is 0.5 each). Substituting the values, we get:

P(X=3) = 3C3 × 0.5³ × (1-0.5)⁽³⁻³⁾ = 1 × 0.125 × 1 = 0.125

Therefore, the probability of getting 3 heads as a decimal rounded to 3 decimal places is 0.125.

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For the transformations of translation, the following is correct -

A. by vector (7, 11).

What is translation?

A translation in mathematics does not turn a shape; instead, it moves it left, right, up, or down. They are congruent if the translated shapes (or the image) seem to be the same size as the original shapes. They have only changed their direction or directions.

The single transformation that represents the following sequence of transformations -

i) a translation vector by (2, -3) followed by

ii) a translation vector by (-5, 8)

is a translation by the vector (-3, 5).

To see why, we can combine the two translation vectors by adding them component-wise -

(2, -3) + (-5, 8) = (-3, 5)

This vector represents the total displacement of a point on the plane after both translations are applied.

So, a single translation by the vector (-3, 5) will have the same effect as applying both of the original translations in sequence.

Therefore, the correct statement is, by vector (7, 11).

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Answer: 0.96

Step-by-step explanation:

To round 0.956 to one decimal place, we need to look at the second decimal place (the hundredths place), which is 5. Since 5 is greater than or equal to 5, we need to round up the first decimal place (the tenths place), which is 9. Therefore, the rounded number to one decimal place is:

0.956 rounded to one decimal place = 0.96

W in terms of t can be represented as W = 7.5t + 10.

When a baby weighs 10 pounds at birth and 40 pounds four years later, we must apply the linear equation to represent W in terms of t.

y = mx + b,

where

m indictaes the slope of the line while

b is the y-intercept.

The formula to find the slope of a line is as follows:

Slope (m) = (y2 - y1) over (x2 - x1)

Given that a baby weighs 10 pounds at birth, and four years later, the child's weight is 40 pounds, we can determine the slope of the line as:

Slope (m) = (40 - 10) over (4 - 0) = 30 / 4 = 7.5

Thus, the linear equation relating childhood weight W (in pounds) to age t (in years) is:

W = 7.5t + 10

Therefore, we can express W in terms of t as W = 7.5t + 10.

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Three numbers have a product of 115 and a sum of 3(√115), which is the smallest possible sum.

In mathematics, a positive number is any number that is greater than zero. This includes all numbers that are written without a minus sign or are explicitly denoted as positive, such as 1, 2, 3, 4, 5, and so on

To find three positive numbers whose product is 115 and whose sum is as small as possible, we can use the AM-GM inequality. In other words, if we have three positive numbers x, y, and z, then:

(x + y + z)/3 ≥ (xyz)^(1/3)

If we rearrange this inequality, we get:

x + y + z ≥ 3(√(xyz))

Now, let's apply this inequality to the given problem. We want to find three positive numbers x, y, and z whose product is 115 and whose sum is as small as possible. Therefore, we want to minimize x + y + z while still satisfying the condition xyz = 115.

Using the AM-GM inequality, we have:

x + y + z ≥ 3(√(xyz)) = 3(√115) ≈ 16.75

Therefore, the sum of the three numbers is at least 16.75. To find three numbers that achieve this minimum sum, we can use trial and error or solve the system of equations:

xyz = 115

x + y + z = 3(√115)

One solution to this system is:

x = √(115/3)

y = √(115/3)

z = 3(√(115/3)) / 5

These three numbers have a product of 115 and a sum of 3(√115), which is the smallest possible sum.

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The complete question is Find three positive numbers whose product is 115.

Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

To place two bench press machines in a small area, you will need a space of approximately 10 feet by 10 feet. Let's discuss it in detail below. Here are the dimensions of the bench press machine, which can help determine the amount of space required to fit two bench press machines in a small area:

The length of the bench press machine is between 48 inches and 54 inches.

The width of the bench press machine is between 28 inches and 32 inches.

The height of the bench press machine is between 48 inches and 56 inches.

Based on the above dimensions of the bench press machine, two machines can be placed in a small area of 10 feet by 10 feet. However, for safe use, the following guidelines should be followed:

There should be at least 6 feet of distance between the two bench press machines. There should be at least 3 feet of clearance in the front of the bench press machine to allow for safe movement during exercises. There should be at least 2 feet of clearance behind the bench press machine to allow for a safe exit in case of an emergency.

Thus, to place two bench press machines in a small area, you will need a space of approximately 10 feet by 10 feet.

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The procedure of surveying 33 people to determine which statistics courses they have taken does not result in a binomial distribution because it does not meet the condition of having only two possible outcomes for each trial.

The outcome of each trial is not a success or failure, but rather a specific course that the individual has taken. Binomial distribution requires a fixed number of independent trials with only two possible outcomes (success or failure) for each trial.

It does not satisfy the four conditions required for a binomial distribution as follows:

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Match the following with the given options:

1. An angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon - Central angle of a regular polygon2. The common center of its inscribed and circumscribed circles - Center of a regular polygon3. The distance from the center of the polygon to a vertex - Radius of a regular polygon4. The perpendicular distance from the center of the polygon to a side - Apothem of a regular polygonA polygon is a two-dimensional object that has three or more straight sides and angles. A regular polygon is a polygon with equal sides and equal angles.

A regular polygon is a shape that has all its sides the same length and all of its angles the same degree. Below are the matched answers for the given options.1. An angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon - Central angle of a regular polygonThe central angle of a regular polygon is defined as the angle whose vertex is the center of the polygon and whose sides contain consecutive vertices of the polygon.

The angle measures (360 / n) degrees, where n is the number of sides of the polygon.2. The common center of its inscribed and circumscribed circles - Center of a regular polygonThe center of a regular polygon is the point at which its axes of symmetry intersect. The center is also the common center of the circumscribed and inscribed circles.3. The distance from the center of the polygon to a vertex - Radius of a regular polygonThe radius of a regular polygon is defined as the distance from the center of the polygon to any vertex. It is also the distance from the center of the polygon to the midpoint of a side

.4. The perpendicular distance from the center of the polygon to a side - Apothem of a regular polygonThe apothem of a regular polygon is defined as the perpendicular distance from the center of the polygon to a side. It is also the radius of the inscribed circle.

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Based on the full rank transformation matrix t, it is true that the eigenvalues of ay and a from part (a) are the same.

We have to show that the eigenvalues of ay and a from part (a) are the same.

Let A be a full rank transformation matrix.

A is a full rank matrix.

Therefore, the rank of A is equal to the number of rows and columns of A.

Let λ be an eigenvalue of Ay.

Then, Ay - λy = 0, where y is the eigenvector corresponding to λ.

Now let us take a look at the eigenvalues of A(y).

A(y) is a full rank matrix since A is a full rank matrix.

This means that det(A(y)) ≠ 0.

We know that λ is an eigenvalue of Ay if and only if λ is a root of the characteristic polynomial det(Ay - λy) = 0.

Using the fact that A(y) is full rank, we have that det(Ay - λy) = det(A(y)A⁻¹yAy - λy) = det(A(y)(A⁻¹yAy - λI)y) = det(A(y))det(A⁻¹yAy - λI).

Since det(A(y)) ≠ 0, we have that λ is an eigenvalue of Ay if and only if λ is a root of the characteristic polynomial det(A⁻¹yAy - λI) = 0.

However, det(A⁻¹yAy - λI) is just the characteristic polynomial of A(y) and therefore has the same roots as the characteristic polynomial of A(y).

Thus, the eigenvalues of Ay and A(y) are the same.

Thus, the statement above is true.

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Answer: your answer should be 12.48

have a good day

sorry I didn't read that well

for your calculator you should type in 0.13 divided by 96

I appreciate you reading my communication, and I will be back. I sincerely hope you'll think about taking part in our upcoming fundraising gathering .

An electronic mail, or email, is a digital communication that is sent and received online. It is a popular method of communication that is utilised for both personal and business reasons. Sending and receiving information, communicating, marketing, and staying in contact with friends and family are just a few of the many uses for emails.

given

A Charity Fundraising Event That Was Effective

Hello [Name of Friend],

I pray you are well and reading this email. I'm writing to let you know about some exciting developments regarding the fundraising gathering I took part in last weekend in our neighborhood.

The goal of this charity event was to raise funds for the underprivileged children who are battling to get access to basic education and healthcare facilities. A nearby nonprofit group that works to better the lives of underprivileged kids arranged it.

I recommend that you participate in comparable fundraising events in the future. You'll not only be helping a worthy cause, However, you will also have the opportunity to join a group that is dedicated to changing the world. I'm confident you'll find it to be a very rewarding experience.

I appreciate you reading my communication, and I will be back. I sincerely hope you'll think about taking part in our upcoming fundraising gathering .

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The Statement 1: She is 95% confident that the mean systolic blood pressure for students in the sample is between 108.6 and 120.3 mm Hg. This statement is accurate and conveys the meaning of a 95% confidence interval.

The confidence interval is the range of values within which a statistical parameter is estimated to be accurate with a certain level of confidence.

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. Suppose that a student is working on a statistics project using data on systolic blood pressure collected from a random sample of 100 students from her college.

Statement 2: She is 95% confident that mean systolic blood pressure for students is between 108.6 and 120.3 mm Hg. This statement is inaccurate because the confidence interval obtained from the sample cannot be generalized to the entire population of students.

The correct statement should mention that the confidence interval is only for the random sample.Statement 2 is the incorrect statement.

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Answer:

The cost per apple is $1.45

Step-by-step explanation:

I will help with the solution, as I do not know the standard for the picture/estimate.

$37.70 / 26 is simply the cost per apple.

The cost per apple is $1.45

Hope this helps!

Answer:

(1,3)

Step-by-step explanation:

Given system of equations is :-

We can solve this by using substitution method by substituting the value of y from equation (1) into equation (2) as ,

2x + 1 = x + 2

Subtract x on both sides,

2x - x + 1 = 2

Simplify,

x = 2 - 1

x = 1

Substitute this value of x into equation (2) as ,

y = 1 + 2

y = 1 + 2

y = 3

Hence the required answer is (1,3) .

and we are done!

Answer: 118

Step-by-step explanation: 38 is the highest value, which is the fifth number in the pattern, and we need to find out what the 15th number is. We simply need to add (8x10) to 38.

Answer: 118

Step-by-step explanation:

you just keep going up at the rate of 8

Answer:

1,000

Step-by-step explanation:20*50

The complete table for the amount balance, A when P dollars is invested at rate r for t years and compounded n times per year is present in above figure 2.

The compound interest formula is written as A = P( 1 + r/n)ⁿᵗ

where, A--> total Amount of money after t years

P --> Principal

r --> Annual rate of interest (as a decimal)

t --> Number of years:

n--> number of times interest is compounded per year

Here, principle, P = $1300, rate of interest, r = 8.5% = 0.085 , time periods, t = 11 years. We have to complete the above table for compound interest.

Case 1: n = 1

Substitute the known values in above formula, A = 1300( 1 + 0.085/1)¹¹

= 1300( 1.085)¹¹

= 3,189.12

Case 2: n = 2

A = 1300( 1 + 0.085/2)²²

= 1300( 2.085/2)²²

= 1300( 1.0425)²²

= 3,248.01

I'll let you work out the cases where n = 4, 12 and 365 since all you need to do is place those in for n as done in the 1st 2 cases. For the Compounded continuously case, the formula becomes,

[tex] A = Pe^{rt}[/tex]

Where: A-> Total amount of money after t years

P --> Principal Amount

e --> Natural log constant:

r = Annual rate of interest (as a decimal)

Case: Continuous: e = 2.71828 (approx), r = 0.085

A = 1300( 2.71828)⁰·⁰⁸⁵⁽¹¹⁾

= 1300(e)⁰·⁹³⁵ = 3,311.34

Hence, required value is $3,311.3775.

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Complete question :

The above table completes the question.

Complete the table by finding the balance A when P dollars is invested at rate r for t years and compounded n times per year. (Round your answers to the nearest cent. ) P = $1300, r = 8. 5%, t

= 11 years

The car will never have a value of £0 because the formula for depreciation approaches but never reaches zero.

Percent is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin per centum, which means "per hundred". Percentages are often used to represent rates, ratios, or proportions of a quantity. For example, if a student scores 90 out of 100 on a test, their score can be expressed as 90%, or 0.9 as a decimal. Similarly, if a store offers a discount of 20% on a product, it means that the customer can purchase the product for 80% of its original price, or 0.8 as a decimal. Percentages are commonly used in finance, science, and everyday life to represent various types of information, such as interest rates, probabilities, taxes, and discounts.

Here,

1. Amount after 1 year:

The amount after 1 year can be calculated using the formula for compound interest:

A = P(1 + r/n)ⁿˣ

Where A is the amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

Substituting the given values, we get:

A = 1000(1 + 0.05/1)¹*¹= £1050

Therefore, the total amount in the bank account after 1 year is £1050.

2. Amount after 2 years:

Using the same formula, we get:

A = 1000(1 + 0.05/1)¹*² = £1102.50

Therefore, the total amount in the bank account after 2 years is £1102.50.

3. Amount after 3 years:

Using the same formula, we get:

A = 1000(1 + 0.05/1)¹*³ = £1157.63

Therefore, the total amount in the bank account after 3 years is £1157.63.

4. Amount after 6 years:

Using the same formula, we get:

A = 1000(1 + 0.05/1)¹*⁶ = £1345.99

Therefore, the total amount in the bank account after 6 years is £1345.99.

5. The amount of interest earned after 6 years is not equal to double the amount of interest earned after 3 years because the interest earned in each year is added to the principal amount, so the interest earned in subsequent years is based on a larger principal amount. This leads to exponential growth in the amount of interest earned over time.

6. The value of the car after n years can be calculated using the formula:

V = P(1 - r/100)ⁿ

where V is the value of the car after n years, P is the initial value of the car, r is the rate of depreciation per year, and n is the number of years.

Using this formula, we can calculate the value of the car after:

1 year: V = 1000(1 - 0.05)¹ = £950

2 years: V = 1000(1 - 0.05)² = £902.50

3 years: V = 1000(1 - 0.05)³ = £857.38

6 years: V = 1000(1 - 0.05)⁶ = £735.03

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Step-by-step explanation:

it would mean that she made 53 batches of soap and 4 batches of lotion.

now, is it a solution ?

then both inequalities must be true with these values.

5×53 + 15×4 <= 325

265 + 60 <= 325

325 <= 325 correct

20×53 + 35×4 <= 1200

remember, 1 hour = 60 minutes.

1060 + 140 <= 1200

1200 <= 1200 correct

so, (53, 4) is the intersection point of both limit lines. and it is as such an extreme point and optimum.