Find the maximum value of = √ subject to the cost constraint K + 4L = 16.
Estimate the change in the optimal value of Q if the cost constraint is changes to K +
4L = 17

Answer: Technically its π (pie) 4

Step-by-step explanation:

If tan (x)=1 , then sin x = cos x . We know this is true for x=π4 as a base case. p = nπ , where n is an integer. The first positive value x0 for which tanx=1 is, as stated before, π4 . yw.

By answering the presented question, we may conclude that Therefore, null hypothesis the test statistic is -1.942 and the pis 0.050.

A null hypothesis is a sort of statistical hypothesis that asserts that a certain set of observations has no statistical significance. Sample data is used to assess the feasibility of ideas. H0 represents what is sometimes referred to as "zero." The researchers make the premise that there may be a link between the parameters. The null hypothesis, on the other hand, claims that no such association exists. While it may not appear to be relevant, the null hypothesis is a crucial aspect of research.

p = 15/23 = 0.6522

z = (p' - p) / √(p * (1-p) / n)

z = (0.6522 - 0.78) / √(0.78 * (1 - 0.78) / 23) = -1.942

p= 2 * P(Z < -1.942) = 2 * 0.025 = 0.050

Therefore, the test statistic is -1.942 and the 0.050.

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A binomial experiment has given number of trails n and the given success probability n=2, p=0.2 then P(0) = 1.

A binomial experiment is an experiment where there have a fixed number of independent trials with only have two outcomes either success or failure.

A binomial experiment has given number of trails n and the given success probability p.

n=2, p=0.2

The probability of obtaining x successes in n independent trials of a binomial experiment, where the probability of success is p, is given by binomial probability distribution   P(x)= ₙ c ₓ p ˣ  (1-p)ˣ

Where x = 0, 1, 2, ---- , n

so, P(0)= ₂C₀ (0.2)⁰ (1-0.2)⁰

             = 1×1×1

             =1

Hence P(0)= 1

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Complete question:

Answer:

The reason why -2^2=-4 and (-2)^2=4 is due to the rules of mathematical operations and the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

According to the order of operations, when you have an expression with multiple operations, you must perform the operations in a specific order. Exponents come before multiplication and division, and before addition and subtraction.

So, when evaluating -2^2, you first evaluate the exponent, which is 2. However, the negative sign in front of the 2 is not part of the exponent, but rather an arithmetic operation. Therefore, the expression is equivalent to -(2^2), which is equal to -4.

On the other hand, when evaluating (-2)^2, the parentheses indicate that the negative sign is part of the base, so you must first evaluate the base, which is -2, and then apply the exponent of 2. This gives you (-2)^2 = 4.

Step-by-step explanation:

The minus in (-2)^2 is enclosed in parentheses, so it is also squared:

minus × minus = plus

And talking about this one: -2^2, the minus is not included in the parentheses, that's why we only square the number (2 in this case) and keep the minus in front of it

I hope this will help...

1. The 95% confident that the true population mean is 64.97 and 435.03 hours. 2. The 99% confident is between 0.407 and 0.500. 3. The 95% estimate of the mean is (45,424, 48,576). 4. Sample size is 91.27.

A fundamental idea in statistics known as the central limit theorem (CLT) argues that, under specific circumstances, the sampling distribution of the mean of a random sample from any population tends to resemble a normal distribution as the sample size rises.

1. For 95% confidence interval we have:

CI = x ± z*(σ/√n)

Substituting the values:

CI = 400 ± 1.96*(39/√30) = (364.97, 435.03)

2. For 99% confidence interval we have:

CI = p-cap ± z*(√(p - cap(1- p-cap)/n))

Substituting the values:

CI = 165/360 ± 2.58*(√((165/360)*(195/360)/360)) = (0.407, 0.500)

Hence, we can be 99% confident that the true population proportion of married men, 18 or older, is between 0.407 and 0.500.

3. 95% estimate of the mean:

CI = x ± t*(s/√n)

Substituting the values:

CI = 47,000 ± 2.064*(1,050/√26) = (45,424, 48,576)

4. For sample size:

CI = 47,000 ± 2.064*(1,050/√26) = (45,424, 48,576)

Substitute the values:

n = (1.96*39 / 20)^2 = 91.27

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Step-by-step explanation:

if we extend the line VT to reach also the opposite side of the circle, we see that this line cuts the circle in half.

the whole arc angle of a half-circle is 360/2 = 180°.

now we can use the exterior angle rule :

the vertex angle (V) = 1/2 × difference of the angles of the intersected arcs.

the intersected arcs are

shorter = angle at T : the arc from U to the circle intersection with the line VT.

longer : the arc from U to the circle intersection with the extended line VT.

shorter + longer = 180°

longer = 180 - shorter

30 = 1/2 × ((180 - shorter) - shorter) =

= 1/2 × (180 - 2×shorter) = 90 - shorter

-60 = -shorter

shorter = 60°

the angle at T = 60°.

Answer:

  147.6 m²

Step-by-step explanation:

You want the surface area of a triangular pyramid with base sides of length 8 m and a slant height of 10 m. (The altitude of the base triangle is approximately 6.9 m.)

The surface area is the sum of the areas of the triangular faces.

  SA = 1/2(8 m)(6.9 m) + 3 × 1/2(8 m)(10 m)

  = 1/2(8 m)(6.9 m + 3×10 m) = (4)(36.9) m² = 147.6 m²

The surface area of the pyramid is 147.6 square meters.

Answer:

45.32

Step-by-step explanation:

La hipotenusa mide 137 cm.

Podemos utilizar el teorema de Pitágoras para resolver este problema, que establece que en un triángulo rectángulo, el cuadrado de la hipotenusa es igual a la suma de los cuadrados de los catetos.

Sea "a" la medida del otro cateto, entonces:

a² + 77² = (a+49)²

Desarrollando los cuadrados y simplificando, tenemos:

a² + 5929 = a²+ 98a + 2401

Restando a ambos lados a², obtenemos:

5929 = 98a + 2401

Restando 2401 a ambos lados, obtenemos:

3528 = 98a

Dividiendo por 98, obtenemos:

a = 36

Por lo tanto, el otro cateto mide 36 cm, y la hipotenusa se puede calcular utilizando el teorema de Pitágoras:

h² = a² + b²

h2 = 36² + 77²

h² = 12996 + 5929

h² = 18925

Tomando la raíz cuadrada en ambos lados, obtenemos:

h = 137

Por lo tanto, la hipotenusa mide 137 cm.

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the length of side AB of triangle ABC is 6 units.

How to find the length?

To find the length of side AB of triangle ABC, we can use the distance formula which is given by:

distance = sqrt((x2 - x1)² + (y2 - y1)²)

Where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment.

Using the distance formula, we can calculate the length of side AB as follows:

AB = sqrt((2 - (-4))²+ (3 - 3)²)

AB = sqrt((6)²+ (0)²)

AB = sqrt(36)

AB = 6

Therefore, the length of side AB of triangle ABC is 6 units.

We can also visualize the triangle on the graph and see that side AB is a horizontal line segment between the points (-4,3) and (2,3), which have the same y-coordinate. This makes the length of AB equal to the horizontal distance between these two points, which is simply 2 - (-4) = 6.

It is important to note that the distance formula can be applied to find the length of any line segment in the coordinate plane, as long as we know the coordinates of its endpoints.

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That sounds like a smart plan! Before investing his savings, Jose should do some research to determine the best investment strategy for his goals.

What is research?

He will need to consider factors such as his risk tolerance, the potential return on investment, and the length of time he has to invest. Once he has identified an investment strategy that aligns with his goals, he should start investing his savings.

Over time, as his investments generate returns, he can use these returns to fund his business venture. It is important for Jose to regularly monitor his investments and adjust his strategy as needed to ensure that he is on track to achieve his goals. With careful planning and a well-executed investment strategy, Jose can achieve his goal of retiring early and starting his own coaching classes.

What is an investment ?

An investment refers to the purchase of an asset with the goal of generating income, capital appreciation, or both. The asset could be anything of value, such as stocks, bonds, real estate, commodities, or mutual funds. Investors typically make an investment with the expectation of earning a return on their investment over a period of time.

The return could be in the form of interest, dividends, or capital gains, depending on the type of investment. Investors may choose to invest in different types of assets based on their investment goals, risk tolerance, and investment horizon. The process of investing involves conducting research, analyzing market trends, and making informed decisions based on the available information.

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Complete question is: Jose is a middle-aged professor who wants to retire early and start his own coaching classes. He decides to invest his savings and use the returns to fund his business venture. Before investing his savings, Jose should do some research to determine the best investment strategy for his goals.

The total direct labor hours for each product to get the total direct labor hours required for the projected sales during the coming year.

To determine the contribution margin per direct-labor hour expended on each product, you will need to calculate the contribution margin for each product and divide it by the number of direct labor hours required to produce that product.
Contribution Margin = Sales Revenue - Variable Costs
Once you have calculated the contribution margin for each product, you can divide it by the number of direct labor hours required to produce that product to get the contribution margin per direct-labor hour.
To prepare a schedule showing the total direct labor hours that will be required to produce the projected sales during the coming year, you will need to estimate the number of units that will be sold for each product and the number of direct labor hours required to produce each unit.

Multiply the number of units by the number of direct labor hours per unit to get the total direct labor hours required for each product.

Contribution Margin = $40 - $20 - $5 - $3 = $12

Direct Labor Hours per unit = 2

Contribution Margin per Direct-Labor Hour = $12 / 2 = $6 MMK YU GHTJUIO7U

Contribution Margin = $60 - $25 - $10 - $5 = $20

Direct Labor Hours per unit = 4

Contribution Margin per Direct-Labor Hour = $20 / 4 = $5

Contribution Margin = $80 - $30 - $15 - $10 = $25

Direct Labor Hours per unit = 5

Contribution Margin per Direct-Labor Hour = $25 / 5 = $5

To prepare a schedule showing the total direct labor hours that will be required to produce the projected sales during the coming year, we need to estimate the number of units that will be sold for each product and the number of direct labor hours required to produce each unit.

Assuming that the projected sales for the coming year are as follows:

Product A: 10,000 units

Product B: 8,000 units

Product C: 5,000 units

And the number of direct labor hours required to produce each unit are as follows:

Product A: 2 hours per unit

Product B: 4 hours per unitF

Product C: 5 hours per unit

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For the givn steam and leaf plot, the mean= 11.61 and the median = 7.

What is mean?

In mathematics, the mean is a measure of central tendency of a set of numbers. It is also known as the average.

To find the mean of the data represented by the stem and leaf plot, we need to first determine the actual values represented by the plot.

The stem and leaf plot indicates that we have the following values:

For stem 1: 0, 0, 2, 8

For stem 3: 1, 7, 9

For stem 5: 0, 5, 7, 7, 7

For stem 11: 3, 4, 6

For stem 22: 3

For stem 23: 5

For stem 31: 0, 2, 9

To calculate the mean, we add up all these values and divide by the total number of values.

There are a total of 18 values, so:

(1x4 + 3x3 + 5x5 + 11x3 + 22x1 + 23x1 + 31x3) / 18

= (4 + 9 + 25 + 33 + 22 + 23 + 93) / 18

= 209 / 18

= 11.61 (rounded to two decimal places)

Therefore, the mean of the data represented by the stem and leaf plot is approximately 11.61.

To find the median of the data represented by the stem and leaf plot, we need to first determine the actual values represented by the plot.

The stem and leaf plot indicates that we have the following values:

For stem 1: 0, 0, 2, 8

For stem 3: 1, 7, 9

For stem 5: 0, 5, 7, 7, 7

For stem 11: 3, 4, 6

For stem 22: 3

For stem 23: 5

For stem 31: 0, 2, 9

We can list all the values in order from smallest to largest:

0, 0, 2, 3, 3, 4, 5, 5, 6, 7, 7, 7, 8, 9, 11, 29, 31

There are a total of 17 values. Since there is an odd number of values, the median is the middle value.

The middle value is the ninth value in the ordered list, which is 7.

Therefore, the median of the data represented by the stem and leaf plot is 7.

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To construct a confidence interval for the proportion of American adults who viewed China as either "unfriendly" or "an enemy," we can use the following formula:

CI = p ± z*(√(p(1-p)/n))

where:

CI is the confidence interval

p is the sample proportion (0.29 + 0.09 = 0.38)

z is the critical value for the desired confidence level (95% confidence level corresponds to z = 1.96)

n is the sample size (2,047)

Plugging in the values, we get:

CI = 0.38 ± 1.96*(√(0.38*(1-0.38)/2047))

CI = 0.38 ± 0.018

CI = (0.362, 0.398)

Therefore, we can say with 95% confidence that the true proportion of American adults who view China as either "unfriendly" or "an enemy" is between 0.362 and 0.398.

Answer: 20%

(Please give me brainliest if you can tomorrow, hopefully this helps)

Explanation:

(So if it's even)

There are 5 possible outcomes and 2 evens on the spinner, so 2 favorable outcomes. That gives us a fraction of 2/5.

Flipping a coin, either heads or tails, it'd be 1/2.

Since it's a compound event we would multiply 2/5 and 1/2, getting:

[tex]\frac{2}{5} *\frac{1}{2} = \frac{2}{10}[/tex]

To change that into a percentage, we divide the numerator by the denominator and multiply by 100 which would give us:

.2 * 100 = 20%

Answer: To find the points where the two skaters might collide, we need to solve the system of equations:

y = -2x + 14 (Equation 1)

y = -2x^2 + 14x (Equation 2)

We can substitute Equation 1 into Equation 2 to eliminate y and get:

-2x + 14 = -2x^2 + 14x

Simplifying, we get:

2x^2 - 16x + 14 = 0

Dividing both sides by 2, we get:

x^2 - 8x + 7 = 0

This quadratic equation factors as:

(x - 1)(x - 7) = 0

So the possible values of x where the skaters might collide are x = 1 and x = 7.

To find the corresponding y-values, we can plug each value of x into either Equation 1 or Equation 2. For x = 1:

y = -2(1) + 14 = 12 (using Equation 1)

y = -2(1)^2 + 14(1) = 12 (using Equation 2)

So the skaters might collide at the point (1, 12).

For x = 7:

y = -2(7) + 14 = 0 (using Equation 1)

y = -2(7)^2 + 14(7) = 0 (using Equation 2)

So the skaters might collide at the point (7, 0).

Therefore, the two skaters might collide at points (1, 12) and (7, 0).

Step-by-step explanation:

To solve the integral [tex]\rm\int_0^7 e^x \sin(x) dx\\[/tex], we can use integration by parts. Let u = sin(x) and dv = [tex]\rm e^x[/tex] dx, then we have:

[tex]\begin{align} \rm\int \rm e^x \sin(x) dx &= \rm-e^x \cos(x) + \rm\int e^x \cos(x) dx \\&= \rm -e^x \cos(x) + e^x \sin(x) - \int e^x \sin(x) dx\end{align}[/tex]

Rearranging, we get:

[tex] \begin{align}2 \rm \int e^x \sin(x) dx &= \rm e^x (\sin(x) - \cos(x)) \bigg|^7_0 \\& \rm= e^7 (\sin(7) - \cos(7)) - 1\end{align}[/tex]

Dividing both sides by 2, we get:

[tex] \rm\int_0^7 e^x \sin(x) dx = \frac{e^7 (\sin(7) - \cos(7)) - 1}{2} \\ [/tex]

Therefore, the value of the integral is

[tex] \rm \boxed{ \rm\frac{e^7 (\sin(7) - \cos(7)) - 1}{2}}[/tex]

The indicated derivatives in the question can be found to be :

To find the derivative of e^(x^3) + log_3(pi) with respect to x, we apply the rules of differentiation to each term:

Using the chain rule, we have:

d/dx (e^(x^3)) = e^(x^3) x (3x^2)

The derivative of log_3(pi) is 0, because log_3(pi) is a constant and the derivative of a constant is 0.

So, the derivative of the entire expression is:

d/dx (e^(x^3) + log_3(pi)) = e^(x^3) x 3x^2 + 0 = 3x^2 x e^(x^3)

To find the derivative of (x^(1/6))^ln(x) with respect to x, we first simplify the expression by using the power rule of logarithms:

(x^(1/6))^ln(x) = x^(ln(x)/6)

Now, we find the derivative of x^(ln(x)/6) using the chain rule:

d/dx (x^(ln(x)/6)) = (ln(x)/6) × x^((ln(x)/6) - 1) × d/dx (x)

The derivative of x is 1, so:

d/dx ((x^(1/6))^ln(x)) = (ln(x)/6) × x^((ln(x)/6) - 1) × 1 = (ln(x)/6) × x^((ln(x)/6) - 1)

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Answer:

4.67

Step-by-step explanation:

The probability of choosing all diet drinks is then: P(all diet drinks) = 24 / 70 ≈ 0.343 or about 34.3%. P(first two are diet drinks and second two are regular) = 288 / 70 ≈ 4.114 or about 4.1%

It is expressed as a number between 0 and 1, where 0 represents an impossible event (i.e., it will never happen), and 1 represents a certain event (i.e., it will always happen).

There are a total of 8 cans of soft drinks, and the child chooses 4 cans to arrange in a row. The total number of ways to choose 4 cans out of 8 is:

C(8,4) = 8! / (4! * 4!) = 70

(a) To find the probability that all four cans are diet drinks, we need to count the number of ways to choose 4 cans that are all diet drinks. Since there is one diet drink for each brand, there are 4 choices for the first can, 3 choices for the second can, 2 choices for the third can, and 1 choice for the fourth can. Therefore, the number of ways to choose 4 cans that are all diet drinks is:

4 * 3 * 2 * 1 = 24

The probability of choosing all diet drinks is then:

P(all diet drinks) = 24 / 70 ≈ 0.343 or about 34.3%

(b) To find the probability that the first two cans are diet drinks and the second two cans are regular drinks, we can count the number of ways to choose 2 diet drinks out of 4 and the number of ways to choose 2 regular drinks out of 4, and then multiply these two numbers together. There are:

C(4,2) = 4! / (2! * 2!) = 6

ways to choose 2 diet drinks out of 4, and also 6 ways to choose 2 regular drinks out of 4. Once we have chosen the 2 diet drinks and 2 regular drinks, there are 4! ways to arrange them in a row. Therefore, the total number of arrangements that have the first two cans as diet drinks and the second two cans as regular drinks is:

6 * 6 * 4! = 288

The probability of this arrangement is then:

P(first two are diet drinks and second two are regular) = 288 / 70 ≈ 4.114 or about 4.1%

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Answer:

(-1.29, 8.29) or (9.29,-2.29)

Step-by-step explanation:

Find what y equals in terms of x for both equations.

Set those two equations equal to each other (Because y=y)

Solve for x (you may need to use the quadratic formula for this step)

Plug the value(s) of x into the original equation to find y.

The graph of y = sec(x) will have vertical asymptotes at these three values of x.

Vertical asymptotes are vertical lines on a graph where a function approaches either positive or negative infinity as the independent variable approaches a certain value.

The secant function, y = sec(x), is defined as the reciprocal of the cosine function:

sec(x) = 1/cos(x)

The cosine function has vertical asymptotes at x = (2n + 1)π/2 for any integer n, since cosine is undefined at these points (division by zero). Therefore, the secant function will have vertical asymptotes at the same points, since dividing 1 by a very small number (close to zero) results in a very large number (tending towards infinity).

So, the values of x in the interval -2π ≤ x ≤ 2π that will cause the secant function to have vertical asymptotes are:

x = (2n + 1)π/2, where n is an integer

For n = -2, we have x = -5π/2, which is outside the interval, so we can ignore it.

For n = -1, we have x = -π/2

For n = 0, we have x = π/2

For n = 1, we have x = 3π/2

For n = 2, we have x = 5π/2, which is also outside the interval.

Therefore, the values of x in the interval -2π ≤ x ≤ 2π that will cause the secant function to have vertical asymptotes are:

x = -π/2, π/2, 3π/2

So, the graph of y = sec(x) will have vertical asymptotes at these three values of x.

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Using division, we can find that Marcus will take 2 hours or 120 minutes to read the book.

Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. By dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers.

Marcus is reading a book with 80 pages.

He reads first 20 pages in 30 mins.

So, to read 1 page time taken

= 30/20

= 3/2

= 1.5 mins

Now to read 80 pages, time taken will be:

80 × 1.5

= 120 mins.

Therefore, Marcus will take 2 hours or 120 minutes to read the book.

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We can conclude that m∠MNP has a measure of 0 degrees in parallelogram MNPQ.

A parallelogram is a four-sided geometric shape in which opposite sides are parallel and congruent (equal in length) to each other. It is a special case of a quadrilateral, which is any four-sided polygon.

In the given question,

Since opposite angles in a parallelogram are equal, we know that:

m∠MNP = m∠QMN

We also know that the sum of the interior angles of a triangle is 180 degrees. Therefore, we can find the measure of angle MQN as follows:

m∠MQN = 180 - m∠PQM = 180 - 135 = 45 degrees

Now, we can use the fact that the opposite angles in a parallelogram are equal to find the measure of angle QMN:

m∠QMN = m∠PQM = 135 degrees

Finally, we can use the fact that the angles in a triangle add up to 180 degrees to find the measure of angle MNP:

m∠MNP = 180 - m∠MQN - m∠QMN

= 180 - 45 - 135

= 180 - 180

= 0 degrees

Therefore, we can conclude that m∠MNP has a measure of 0 degrees. This means that the sides MN and PQ are parallel and do not intersect, so angle MNP does not exist.

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The probability of exactly 3 out of 10 vehicles exceeding the speed limit on US 131 is 0.250.

This problem can be solved using the binomial distribution. We are given that the probability of any given vehicle exceeding the speed limit is p = 0.6, and we are interested in the probability of 3 out of 10 vehicles exceeding the speed limit, i.e., X = 3.

The probability mass function of the binomial distribution is given by:

P(X = k) = (n choose k) ×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 (n choose k) is the binomial coefficient, which is equal to n!/(k! (n-k)!).

Substituting the given values, we get:

P(X = 3) = (10 choose 3) × 0.6³ ×0.4⁷

P(X = 3) = 0.250

Therefore, the probability of exactly 3 out of 10 vehicles exceeding the speed limit on US 131 is 0.250, rounded to three decimal places.

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To determine which graph has the larger range, we need to understand what is meant by the term “range” in the context of graphs. the answer is: O The first graph has the larger range.

In statistics, the range is defined as the difference between the maximum and minimum values in a dataset. It provides a measure of how spread out the data is, and can be used to identify the extent of variability within a dataset.

Looking at the first graph, we can see that the range of race times is from approximately 10 minutes to 25 minutes. Therefore, the range of this dataset is [tex]25 - 10 = 15[/tex] minutes.

Looking at the second graph, we can see that the range of race times is from approximately 8 minutes to 22 minutes. Therefore, the range of this dataset is [tex]22 - 8 = 14[/tex] minutes.

Since the range of the first graph is  [tex]15[/tex] minutes and the range of the second graph is  [tex]14[/tex] minutes, we can conclude that the first graph has the larger range.

Therefore, the answer is: O. The first graph has the larger range.

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According to the given question we can conclude that statement I and statement II are correct but not statement III.

Interest is the cost incurred when lending and borrowing a specific amount of money from a financial standpoint. It is crucial to note that interest is calculated as a percentage.

given,

Statement I is correct because the inflation rate averaged 1.3%, and Maria's savings account earned interest at a rate of 0.725%, which is lower than the inflation rate. This means that the buying power of her gift decreased by approximately 1.3% - 0.725% = 0.575% or 0.6% (rounded).

Statement II is also correct because Maria's savings account earned interest on the deposit of $2,000 for one year at a rate of 0.725%. The interest earned can be calculated as:

Interest = Principal x Rate x Time

Interest = $2,000 x 0.00725 x 1 year

Interest = $14.50

However, statement III is not correct. Since Maria deposited the entire gift of $2,000 into the savings account and did not make any withdrawals during the year, the balance in the account at the end of 2016 would be the sum of the initial deposit and the interest earned, which is $2,000 + $14.50 = $2,014.50. Therefore, the correct option is:

Statement III is incorrect, however Statements I as well as II are true.

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Answer:

Let the height be represented as x

width = 14 + x

Length of diagonal = 34 inches.

Taking one diagonal of the rectangle to form a triangle.

using the Pythagoras theorem.

34² = (14 + x)² + x²

1156 = 196 + 28x + x² + x²

2x² + 28x - 960 = 0

divide through by 2

x² + 14x - 480 = 0

x² + 30x - 16x - 480 = 0

x(x + 30) - 16 (x + 30) = 0

(x - 16) = 0 or (x + 30) = 0

x = 16 or x = - 30

since the length can't be negative

therefore 16 inches is correct