A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.80 with a sample standard deviation of 1.57. Assume the underlying population is normally distributed. Find the 95% Confidence Interval for the true population mean for the amount of soda served.

a. (12.42, 14.18)

b. (12.32, 14.29)

c. (12.50, 14.10)

d. None of the above

e. Impossible to determine

Answer:

eventually the gender ratio of population in this society will be 50% male and 50% female.

Step-by-step explanation:

For practical purposes we will think that every couple is healthy enough to give birth as much children needed until giving birth a girl.

As the problem states, "each couple continue to have more children until they get a girl and once they have a girl they will stop having more children". Then, every couple will have one and only one girl.

We will denote for P(Bₙ) the probability of a couple to have exactly n boys.

Observe that statement 1 implies that:

[tex]P(B_{n-1})=(0.5)^{n}[/tex].

Then, the average number of boys per couple is given by

[tex]\sum^{\infty}_{n=1}(n-1)P(B_{n-1})=\sum^{\infty}_{n=1}(n-1)(\frac{1}{2} )^n=\sum^{\infty}_{n=2}n(\frac{1}{2} )^n=\\\\=\sum^{\infty}_{m=2}\sum^{\infty}_{n=m}(\frac{1}{2} )^n=\sum^{\infty}_{m=2}(\frac{1}{2} )^{m-1}=\sum^{\infty}_{m=1}(\frac{1}{2} )^{m}=1.\\[/tex]

This means that in average every couple has a boy and a girl. Then eventually the gender ratio of population in this society will be 50% male and 50% female.

The average rate of change of a function is found with the formula (f(b) - f(a)) / (b - a). When applying this formula to the function p(x) = 6x + 7 over the interval [2, 2 + h], we find that the average rate of change is 6.

In Mathematics, the average rate of change of a function on the interval [a, b] is given by the formula (f(b) - f(a)) / (b - a). In this case, our function is p(x) = 6x + 7, and the interval is [2, 2 + h]. So, we can plug these values into the formula to get an expression for the average rate of change.

First, calculate p(2 + h) and p(2). Here they are:

Substitute these expressions into the average rate of change formula:

(P(2 + h) - P(2)) / (2 + h - 2) = (19 + 6h - 19) / (h) = 6h / h = 6.

So, the average rate of change of the function p(x) = 6x + 7 on the interval [2, 2 + h] is 6.

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Answer:

  $607.75

Step-by-step explanation:

As a first approximation, compound interest will be slightly higher than simple interest for a relatively short time period. Here simple interest at 5% for 4 years will add 4×5% = 20% to the value, adding about $100 to the initial $500 value. That is, we expect the value in 4 years to be slightly more than $600.

The appropriate answer choice is $607.75.

_____

The actual amount can be calculated using the multiplier 1.05 for each of the 4 years, or 1.05^4 ≈ 1.21550625 for the entire period. Then the predicted item value is ...

  $500 × 1.21550625 = $607.753125 ≈ $607.75

Answer:

Answer: $607.75

Step-by-step explanation:

Answer:

 $607.75

Step-by-step explanation:

As a first approximation, compound interest will be slightly higher than simple interest for a relatively short time period. Here simple interest at 5% for 4 years will add 4×5% = 20% to the value, adding about $100 to the initial $500 value. That is, we expect the value in 4 years to be slightly more than $600.

The appropriate answer choice is $607.75.

_____

The actual amount can be calculated using the multiplier 1.05 for each of the 4 years, or 1.05^4 ≈ 1.21550625 for the entire period. Then the predicted item value is ...

 $500 × 1.21550625 = $607.753125 ≈ $607.75

Answer:

Suppose that you initially have $100 to spend on books or movie tickets.

The books start off costing $25 each and the movie tickets start off costing $10 each.

a. Your budget increases from $100 to $150 while the prices stay the same.

Increase

b. Your budget remains $100, the price of books remains $25, but the price of movie tickets rises to $20.

Decrease

c. Your budget remains $100, the price of movie tickets remains $10, but the price of a book falls to $15.

Increase

answer is x is less than or equal to -13

hope this helps:-)

Answer:

x  ≤ -13.

Step-by-step explanation:

2(x + 1 ) - 3(x + 5) ≥ 0      Distribute the 2 and -3 over the parentheses:

2x + 2 - 3x - 15   ≥  0

-x  ≥  15 - 2

-x  ≥  13

x  ≤  -13      Note: when dividing by a negative the inequality sign is flipped .

Answer: 33%

Step-by-step explanation:

Given : The temperature dropped from 75 degrees to 50 degrees.

Decrease in temperature ( in degrees) = 75-50=25

The formula to find the percent decrease :-

[tex]\dfrac{\text{Decrease in temperature}}{\text{Initial temperature}}\times100\\\\=\dfrac{25}{75}\times100\\\\=33.3333333333\approx33\%\ \ \text{[Rounded to the nearest whole percent.]}[/tex]

Hence, the percent decrease in the temperature = 33%

Answer:

If the mean of the sample standard deviations is equal to the population standard​ deviation, then the sample standard deviations target the population standard deviation and are called an unbiased estimator.  

If the mean of the sample standard deviations is not equal to the population standard​ deviation, then the sample standard deviations target the population standard deviation and are called a biased estimator.

A biased estimator regularly underestimates or overestimates the parameter.  

Final answer:

Sample standard deviations do not target the population standard deviation but serve as estimators. The accuracy of the estimation depends on the sample size.

Explanation:

The sample standard deviations do not target the exact value of the population standard deviation. Instead, they serve as estimators of the population standard deviation. In general, sample standard deviations can be good estimators of population standard deviations, but the accuracy of the estimation depends on the sample size. When the sample size is large, the sample standard deviation tends to be close to the population standard deviation. However, when the sample size is small, the sample standard deviation may not accurately estimate the population standard deviation.

Answer: The shortest living man at that time had the height that was more​ extreme.

Step-by-step explanation:

We will z scores to solve this exercise. The formula we need is:

 [tex]z=\frac{x-\mu}{\sigma}[/tex]

Where [tex]x[/tex] is the raw score, [tex]\mu[/tex] is the mean and [tex]\sigma[/tex] is the standard deviation.

We know at that time heights of men had a mean of 170.53 centimeters and a standard deviation of 5.91 centimeters, then:

[tex]\u=170.53\\\\\sigma=5.91[/tex]

Knowing that the tallest living man at that time had a height of 230 centimeters, we get:

[tex]z=\frac{230-170.53}{5.91}\approx10.07[/tex]

And knowing that the shortest living man at that time had a height of 91.3 centimeters, we get:

[tex]z=\frac{91.3-170.53}{5.91}\approx-13.40[/tex]

Based on this, we can conclude that the shortest living man at that time had the height that was more​ extreme.

Answer: 3

Step-by-step explanation:

Given : Pooled variance : [tex]\sigma^2=18[/tex]

Sample sizes of each sample = [tex]n_1=n_2=4[/tex]

We know that the standard error for the sample mean difference is given by :-

[tex]S.E.=\sqrt{\sigma^2(\dfrac{1}{n_1}+\dfrac{1}{n_2})}\\\\=\sqrt{(18)(\dfrac{1}{4}+\dfrac{1}{4})}\\\\=\sqrt{(18)(\dfrac{1}{2})}=\sqrt{9}=3[/tex]

Hence, the estimated standard error for the sample mean difference =3

Answer:

The answer to your question is: m∠LOM = 44°

Step-by-step explanation:

Data

m< LOM = 3x +38

m< MON= 9x+28

m< LOM = ?

Process

They are complementary angles

                    m∠LOM + m∠MON = 90°

                   3x + 38 + 9x + 28 = 90°

                   12x + 66 = 90°

                   12x = 90 - 66

                   12x = 24

                   x = 24/12

                   x = 2

m∠LOM = 3(2) + 38

             = 6 + 38

            = 44°

m∠ MON = 9(2) + 28

m∠MON = 18 + 28

m∠MON = 46°

Answer:

  5 1/2

Step-by-step explanation:

Your calculator can give you the correct answer.

___

[tex]\sqrt[3]{343}+\dfrac{3}{4}\sqrt[3]{-8}=7+\dfrac{3}{4}(-2)=7-\dfrac{3}{2}=5\frac{1}{2}[/tex]

Answer:

  your selection is correct

Step-by-step explanation:

You can divide by 9 to get ...

  | x+4 | ≥ 6

This resolves to two inequalities:

These are disjoint intervals, so the solution set is the union of them:

  x ≤ -10 or 2 ≤ x

Answer:

Step-by-step explanation: A is correct

Answer:

  A)  The functions are not inverses of each other.

Step-by-step explanation:

[tex]f(g(x))=\sqrt{(x^2+3)-3}=\sqrt{x^2}=|x|\ne x[/tex]

The result of f(g(x)) is not always x, so the functions are not inverses of each other.

In general, a quadratic (or any even-degree polynomial) such as g(x) cannot have an inverse function because it does not pass the horizontal line test.

Answer: 11

Step-by-step explanation:

We know that the degree of freedom for a t-distribution is given by :-

[tex]df=n-1[/tex], where n is the sample size.

Given : A repeated-measures experiment and a matched-subjects experiment each produce a t statistic with df = 10.

Then, the number of  individuals participated in each study = [tex]df+1=10+1=11[/tex]

Hence, the number of  individuals participated in each study =11.

Final answer:

The limit of f(x)/g(x) as x approaches 3 is negative infinity, since f(x) approaches 150 and g(x) approaches 0 with g(x) < 0.

Explanation:

We are given that as x approaches 3, f(x) approaches 150, and g(x) approaches 0 while being less than zero. The question is to determine the limit of f(x)/g(x) as x approaches 3.

To find this limit, we should consider the behavior of both f(x) and g(x) as x approaches 3.

Since f(x) approaches a finite number and g(x) approaches 0, the limit of the quotient could potentially be infinity or negative infinity, depending on the sign of g(x).

Since g(x) is less than 0 as x approaches 3, the quotient f(x)/g(x) will approach negative infinity.

Hence, the limit limx→3 f(x)/g(x) = -∞.

Answer:

72 alligators, 128 turtles

Step-by-step explanation:

When you put the words into the form of an equation, with a being alligators, you get

3a-16=200

So you have to do 'letters left numbers right'. This gives you

3a=216

Now you have to divide. 216 divided by 3 equals 72. So there are 72 alligators. 200 minus 72 equals 128, so there are 128 turtles

Answer:

The number of alligators and number of turtles are 54 and 146 respectively.

Step-by-step explanation:

Given :

The number of turtles is 16 less than 3 times the number of alligators in the lake

There are 200 reptiles total.

To Find :  Find the numbers of alligators and turtles .

Solution:

Let the number of alligators be x

So, The number of turtles is 16 less than 3 times the number of alligators

Number of turtles = 3x-16

Now we are given that there are 200 reptiles in total .

[tex]x+3x-16=200[/tex]

[tex]4x-16=200[/tex]

[tex]4x=216[/tex]

[tex]x=54[/tex]

Number of alligators = 54

Number of turtles = 3(54)-16 = 146

Hence the number of alligators and number of turtles are 54 and 146 respectively .

Answer:

The answer to your question is: Perimeter = 108 in

Step-by-step explanation:

Data

Length (l) = 5 width (w)

A = 405 in²

Perimeter = ?

Formula

Area = l x w

Perimeter = 2w + 2l

Process

                      405 = 5w x w

                       405 = 5w²

                     405/5 = w²

                       w = √81

                       w = 9 in

                       l = 5(9) = 45 in

Perimeter = 2(9) + 2(45)

                 = 18 + 90

                 = 108 in

Answer: First, suppose that A is a knight. then when he says "i am a knave" he would be lying, so you have a logical failure because knights can't lie.

if A is a knave and says "I am a Knave, but B isn’t" then he would be telling a truth in the first part.

now you have two paths to tink itm as A said a truth, he can't be a knave. but if you consider the whole sentence can be splitted in two sentences.

I am a Knave ----- wealready know that will be a truth.

but B isn’t----- and now, as the first sentence is true, this must be false, so the sum of both sentences is false.

so A is a knave and B is a knave.

Final answer:

After analyzing the statements provided by A and the silence of B, the logical deduction reveals that both A and B are Knights, given the inherent contradictions in A's statement if he or B were Knaves.

Explanation:

The Island of Knights and Knaves presents a classic example of logical deduction. Given that A states he is a Knave but also says 'B isn’t a Knave', we can infer A's nature through contradiction. If A were a Knave, he would not tell the truth about himself or B, creating a paradox since a Knave can't tell the truth. If A is a Knight, his statement is also impossible since Knights cannot lie. Therefore, A must be a Knight, making the first part of his statement a lie (which is not possible for a Knight), but the second part true: B is not a Knave. Consequently, for the statement to uphold the rules of the island, B must be a Knight as well, which is consistent with the silent B offering no statements that could be lies.

Answer:

x = 24

Step-by-step explanation:

Given that y varies directly with x the the equation relating them is

y = kx ← k is the constant of variation

To find k use the condition y = 10 when x = 8, then

k = [tex]\frac{y}{x}[/tex] = [tex]\frac{10}{8}[/tex] = 1.25, thus

y = 1.25x ← equation of variation

When y = 30, then

30 = 1.25x ( divide both sides by 1.25 )

x = 24

Answer:

Step-by-step explanation:

$3x6=18 the six represents the apples and the 3 is the cost of each apple, 18 is the cost.

$2x9=$18 the nine is the oranges and the two is the money spent on each one. The total would be $36 in total for all the fruit.

So (9x2)+(3x6)=$36

Answer with Step-by-step explanation:

Susan needs to buy apples and oranges to make fruit salad.

She needs 15 fruits in all.

Let m represent the number of apples.

Number of oranges= 15-m

Apples cost $3 per piece, and oranges cost $2 per piece.

Amount spent= $ (3m+2(15-m))

                       = $ (3m + 2×15 - 2m)

                       = $ (m+30)

If she bought 6 apples.

i.e. m=6

Amount spent =$ (6+30)

                        = $ 36

Hence,

Expression that represents the amount Susan spent on the fruits is:

m+30

The amount Susan spent if she bought 6 apples is:

$ 36

Answer:

Step-by-step explanation:

g(x)= -2 x^2 + 3 x - 5

g(-2) = -2 . (-2)^2 + 3. (-2) - 5 = -2 . 4 - 6 - 5 = - 8 - 6 - 5 = - 19

g(0) = -2 . (0)^2 + 3 . 0 - 5 = -2 . 0 + 0 - 5 = 0 + 0 - 5 = - 5

g(3) = -2 . (3)^2 + 3 . (3) - 5 = -2 . 9 + 9 - 5 = -18 + 9 - 5 = - 14

The value of g(x) is the input values are  –2, 0, and 3 are -19, -5 and -14

Given the following function

g(x) =  –2x² + 3x – 5

For the input value of -2

g(-2) =  –2(-2)² + 3(-2) – 5

g(-2) = -8 - 6 - 5

g(-2) =-19

If the value of x is 0

g(0) =  –2(0)² + 3(0)– 5

g(0) = -5

If the vaue of x is 3

g(3) =  –2x² + 3x – 5

g(3) = -2(3)² + 3(3)– 5

g(3) =-18 + 9 - 5

g(3) = -14

Hence the value of g(x) is the input values are  –2, 0, and 3 are -19, -5 and -14

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Answer:

2

Step-by-step explanation:

You have to find a number that added with -6 equals -4, and multiplied equals -12.

Just have to do the opperations and that is all!

[tex](x-6)(x+2)=x^2+2x-6x-12=x^2-4x-12[/tex]

Hope you like it!

Answer:

Option C is the answer.

Step-by-step explanation:

Simon is factorizing the polynomial x² - 4x - 12 = (x - 6)( x + ......)

We will factorize the left hand side of the given expression

x² - 4x - 12

= x² - 6x + 2x - 12  

Now we will break 12 into the factors so that sum of the factors should equal to 4

{ 6 × 2 = 12 and 6 - 2 = 4]

= x(x - 6) + 2(x - 6)

= (x + 2)(x - 6)

Therefore, the blank space should be replaced by 2.

Option C is the answer.

Answer:

x = 2

Step-by-step explanation:

Since the parabola is opening vertically up then the equation of symmetry is vertical and of the form x = c

The axis of symmetry passes through the vertex (2, 0), thus

equation of axis of symmetry is x = 2

To determine the number of different eight-note melodies that alternate between black and white keys within a single octave, you calculate the permutations starting with either type of key and add them together, resulting in 141,120 possible melodies.

The question asks: How many different eight-note melodies within a single octave can be written if the black keys and white keys need to alternate? To solve this, we need to understand the structure of a piano octave, which consists of seven white keys and five black keys. Since melodies must alternate between black and white keys, starting with a white key will always result in a pattern of white-black-white-black, and so on, until eight notes are reached. Conversely, starting with a black key follows a black-white pattern.

If we start with a white key, we have 7 options for the first note. The next note (a black key) gives us 5 options. This alternating pattern continues, decreasing the number of options by 1 for each type of key used, until we have selected all eight notes. Mathematically, this calculates as 7 × 5 × 6 × 4 × 5 × 3 × 4 × 2. Similarly, starting with a black key would result in a calculation of 5 × 7 × 4 × 6 × 3 × 5 × 2 × 4.

However, since an eight-note melody can start with either a white or a black key, we calculate both scenarios and add them together for the total amount of possible melodies. The sum of the series for both starting options gives us 141,120 possible eight-note melodies that alternate between black and white keys within a single octave.

To find the percentage rate of growth, calculate the percentage change in enrollment from one year to the next using the graph. Perform these calculations for each pair of consecutive years to determine the overall percentage rate of growth.

To determine the percentage rate of growth, we need to analyze the graph showing the enrollment growth of the school. Exponential growth is represented by a curve that increases more and more steeply over time. To find the rate of growth, we can calculate the percentage change in enrollment from one year to the next.

For example, if the enrollment was 100 in Year 1 and 200 in Year 2, the percentage change would be (200-100)/100 * 100 = 100%. This means the enrollment doubled from Year 1 to Year 2.

By performing similar calculations for each pair of consecutive years, we can find the percentage rate of growth over the entire period represented by the graph.

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The percentage rate of growth for the school's enrollment is approximately 24% per year.

To determine the percentage rate of growth, we need to find the common ratio and the common difference in the exponential function that represents the school's enrollment. From the given graph, we can see that the enrollment doubles approximately every three years. This means that the common ratio is 2. The initial enrollment is 500, which is represented by the value "a" in the exponential function. The final enrollment, which is approximately 4000, is represented by "an". Using these values, we can write the exponential function as follows:

[tex]an = a * 2^(n-1)[/tex]

Substituting the initial enrollment and final enrollment in this equation, we get:

[tex]4000 = 500 * 2^(n-1)[/tex]

Dividing both sides by 500 and simplifying, we get:

[tex]8 = 2^(n-1)[/tex]

Taking the logarithm of both sides with base 2, we get:

(n-1) = log2(8)

(n-1) = 3

Adding 1 to both sides, we get:

n = 4

This means that it takes approximately four years for the school's enrollment to double. To find the percentage rate of growth per year, we need to find the common difference in the exponential function. The common difference is calculated as follows:

Common difference =[tex]ln(y2 / y1) / (x2 - x1)[/tex]

Here, x1 and x2 are two consecutive years, and y1 and y2 are their corresponding enrollments. Using this formula, we can calculate the common difference as follows:

Common difference = ln(4000 / 3200) / (7 - 4) = 0.263975 (approximately 24%) per year. This means that every year, the school's enrollment grows by approximately 24%.

Answer:

V = 9

Step-by-step explanation:

You can see it in the picture.

The side length of the square is found using the diagonal, which is the distance between the two parabolas. The area of each square cross section is then integrated from 0 to 3 to find the solid's volume, which is 9 cubic units.

For these types of volume problems, you'll need to integrate. However, first you have to find the area of the square formed by the diagonals. The distance between the parabolas y=-√x and y=√x forms the square's diagonal. This distance, or length of the diagonal, can be obtained by adding the y-values of the two parabolas which gives 2√x. Given the diagonal, the side length of the square (s) can be obtained from the diagonal using Pythagoras theorem: s=diagonal/√2 => s=2√x/√2 => s=√2* √x => s=√2x. The area of the square is the side length squared, A=s² => A= 2x. Now, integrate the area function from 0 to 3 to get the volume of the solid: Volume= ∫ from 0 to 3 [2x dx] = [x²] from 0 to 3 = 9 - 0 = 9 cubic units.

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Answer:

The bus fair is 50% increases.

Step-by-step explanation:

The percentage is the proportion of the relation to the whole.

Percentage increase is calculate as ratio of difference of original and new value to the original value. i.e.

[tex]\frac{ Percentage\ increase\ \ =\ \ new\ value - original \ value}{original\ \ value}[/tex]

∴ [tex]\frac{ Percentage\ increase\ \ =\ \ 2.25 - 1.50 }{1.50}[/tex]

⇒ Percentage increase = 50%

By using percentage, the result obtained is-

Percentage increase in bus fare in Chicago  = 50%

What is percentage?

Suppose there is a number and the number has to be expressed as a fraction of 100. The fraction is called percentage.

For example 2% means [tex]\frac{2}{100}[/tex]. Here 2 is expressed as a fraction of 100.

Here,

Bus fare in the year 1995  in Chicago= $1.50

Bus fare in the year 2008 in Chicago = $2.25

Increase in bus fare = $(2.25 - 1.50) = $0.75

Percentage increase in bus fare in Chicago = [tex]\frac{0.75}{1.50}\times 100[/tex]

                                                                         = 50%

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Step-by-step explanation:

a) Compute the probability that 2 or fewer will withdraw

First we need to determine, given 2 students from the 20. Which is the probability of those 2 to withdraw and all others to complete the course. This is given by:

[tex](0.3)^2(0.7)^{18}[/tex].

Then, we must multiply this quantity by

[tex]{20\choose2}=\frac{20!}{18!2!}=\frac{20\times19}{2}=190,[/tex]

which is the number of ways to choose 2 students from the total of 20. Therefore:

Following an analogous process we can determine that:

Finally, the probability that 2 or fewer students will withdraw is

[tex]190(0.3)^2(0.7)^{18}+20(0.3)(0.7)^{19}+(0.7)^{20}=(0.7)^{18}(190(0.3)^2+20(0.3)(0.7)+(0.7)^2)\approx0.0355[/tex]

b) Compute the probability that exactly 4 will withdraw.

Following the process explained in a), the probability that 4 student withdraw is given by

[tex]{20\choose4}(0.3)^4(0.7)^{16}=\frac{20\times19\times18\times17}{4\times3\times2} (0.3)^4(0.7)^{16}=4845(0.3)^4(0.7)^{16}\approx 0.1304.[/tex]

c) Compute the probability that more than 3 will withdraw

First we will compute the probability that exactly 3 students withdraw, which is given by

[tex]{20\choose3}(0.3)^3(0.7)^{17}=\frac{20\times19\times18}{3\times2} (0.3)^3(0.7)^{17}=1140(0.3)^3(0.7)^{17}\approx 0.0716.[/tex]

Then, using a) we have that the probability that 3 or fewer students withdraw is 0.0355+0.0716=0.1071. Therefore the probability that more than 3 will withdraw is 1-0.1071=0.8929

d) Compute the expected number of withdrawals.

As stated in the problem, 30% of the students withdraw, then, the expected number of withdrawals is the 30% of 20 which is 6.

This problem involves using the binomial distribution to compute probabilities of student withdrawal and the expected number of withdrawals. Probability values are computed using the binomial probability formula and the expected number of students withdrawing from the course is given by n*p.

To solve the probability and expected value questions, we need to use the binomial distribution since the event (a student withdraws or not) is independent and repeated a fixed number of times.

1. The probability that 2 or fewer will withdraw is [tex]P(X < =2) = P(0)+P(1)+P(2) where P(x) = C(n, x) * (p^x) * (q^(n-x)). For n=20, p=0.3, q=0.7.[/tex]

2. The probability that exactly 4 withdraw is given by the binomial probability formula P(X=4). Again, use the same values of n, p, and q.

3. The probability that more than 3 will withdraw is P(X > 3) which is 1 - P(X<= 3). Compute P(X<=3) similar to the first situation and subtract it from 1.

4. The expected number of withdrawals, or the expectation of a binomial distribution, is given by n*p.

Calculations using these formulas will give you the desired probabilities to the accuracy you require.

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Answer:

The lengths of the shelves are 3.5 feet , 4.5 feet , 6 feet

Step-by-step explanation:

* Lets explain how to solve the problem

- There are 3 shelves

- You have  one piece of lumber 14 feet long

- Your plane is:

# The top shelf is 1 foot shorter than the middle shelf

# The Bottom shelf a foot shorter than twice the length of the top shelf

* Assume that the length of the middle shelf is x feet

∵ The length of the middle shelf = x

∵ The top shelf is shorter by 1 foot

∴ The length of the top shelf = x - 1

∵ The length of the bottom shelf is 1 less than twice the length of

   the top shelf

- That means multiply the length of the top shelf by 2 and subtract

 1 from the product

∵ The length of the top shelf is x - 1

∴ The length of the bottom shelf = 2(x - 1) - 1

- Simplify it by multiplying the bracket by 2 and add like terms

∴ The length of the bottom shelf = 2x - 2 - 1

∴ The length of the bottom shelf = 2x - 3

* The sum of the lengths of the 3 shelves equal the length of lumber

∵ The length of the lumber is 14 feet

∵ The length of the 3 shelves are x - 1 , x , 2x - 3

∴ x - 1 + x + 2x - 3 = 14

- Add like terms in the left hand sides

∴ 4x - 4 = 14

- Add 4 for both sides

∴ 4x = 18

- Divide both by 4

∴ x = 4.5

- Lets find the length of each shelf

∵ The length of the top shelf is x - 1

∴ The length of the top shelf = 4.5 - 1 = 3.5 feet

∵ The length of the middle shelf is x

∴ The length of the middle shelf = 4.5 feet

∵ The length of the bottom shelf is 2x - 3

∴ The length of the bottom shelf = 2(4.5) - 3 = 9 - 3 = 6 feet

* The lengths of the shelves are 3.5 feet , 4.5 feet , 6 feet

Final answer:

To find the length of each shelf, a system of equations is created based on the conditions given. Solving this system shows the middle shelf to be 4.5 feet, the top shelf to be 3.5 feet, and the bottom shelf to be 6 feet long.

Explanation:

The question involves using a piece of lumber that is 14 feet long to make three shelves with specific relative lengths. We can let the length of the middle shelf be x feet. Therefore, the top shelf will be x - 1 feet long, and the bottom shelf will be 2(x - 1) - 1 feet long, which simplifies to 2x - 3 feet. Adding together the lengths of the three shelves gives us the total length of the lumber:

So: x + (x - 1) + (2x - 3) = 14.

Solving the equation:

Therefore, the middle shelf is 4.5 feet long, the top shelf is 3.5 feet (4.5 - 1), and the bottom shelf is 6 feet (2(3.5) - 1).

Answer:

Complete the square on x by adding​ 49 to both sides.

Complete the square on y by adding​ 81 to both sides.

Step-by-step explanation:

We have been given an equation [tex](x^2+14x)+(y^2+18y)=5[/tex]. We are asked to complete the squares for both x and y.

We know to complete a square, we add the half the square of coefficient of x or y term.

Upon looking at our given equation, we can see that coefficient of x is 14 and coefficient of y is 18.

[tex](\frac{14}{2})^2=7^2=49[/tex]

[tex](\frac{18}{2})^2=9^2=81[/tex]

Now, we will add 49 to complete the x term square and 81 to complete y term square on both sides of our given equation as:

[tex](x^2+14x+49)+(y^2+18y+81)=5+49+81[/tex]

Applying the perfect square formula [tex]a^2+2ab+b^2=(a+b)^2[/tex], we will get:

[tex](x+7)^2+(y+9)^2=135[/tex]

Therefore, We can complete the square on x by adding​ 49 to both sides and the square on y by adding​ 81 to both sides.