Problem 1:
The US Department of Energy conducts weekly surveys of approximately 900 gasoline stations to
determine the average price per gallon of regular gasoline in the U.S. On December 29, 2003, the average price was $1.478 per gallon.
a. Identify the population.
b. Identify the sample.
c. Identify the parameter.
d. What is the statistic?
e. If the 900 gasoline stations were randomly selected from the gasoline stations in Minnesota.
Is this a good sample? Explain.

Answer:

17 years & 9 months

Step-by-step explanation:

annual rate 3.9%

every quarter = 3.9%/4 = 0.98%

compounding every quarter = (1+0.98%)

how long = assuming t quarters

2000 * (1+0.98%) ^t = 4000

solve for t

t= 71

so 71/4 = 17 years & 9 months

The proportion of temperatures between 17.02°C and 25°C is 0.9539

The term "standard deviation" (or "σ") refers to a measurement of the data's dispersion from the mean.

A low standard deviation implies that the data are grouped around the mean, whereas a large standard deviation shows that the data are more dispersed.

Given:

mean = 21.02points and

standard deviation =  points.

Now, using

z= (x-u)/σ

z= 17.02 - 21.02/ 2

z= -2

again, z= (x-u)/σ

z= 25- 21.02 / 2

z= 1.99

So, the proportion is

P(x<108.20)-P(x<104.60)

=P(Z<2.65)-P(Z<2.2)

= 0.9497 -0.0228

=0.9539

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The mean of the sampling distribution is 120

And the standard deviation of the sampling distribution of the mean using sample size of:

a) n = 16 ⇒ [tex]\sigma_{\bar{x}}[/tex] = 5

b) n = 25 ⇒  [tex]\sigma_{\bar{x}}[/tex] = 4

c) n = 36 ⇒  [tex]\sigma_{\bar{x}}[/tex] = 3.33

d) n = 100 ⇒  [tex]\sigma_{\bar{x}}[/tex] = 2

We know that for a normally distributed population, the sample mean is also normally distributed.

The sample mean is:

[tex]\mu_{x}[/tex] = μ

where μ is the population mean.

Here, the population is normally distributed with μ = 120

So, the mean of the sampling distribution for any sample size would be, [tex]\mu_{x}[/tex] = 120

As we know the formula for the standard deviation of the sample mean is:

[tex]\sigma_{\bar{x}}[/tex] = σ/[tex]\sqrt{n}[/tex]

where,  [tex]\sigma_{\bar{x}}[/tex]  is the Standard deviation of the sample mean

σ is the population standard deviation

n is the Sample size.

Here, σ = 20

Using above formula of the standard deviation of the sample mean,

For sample size n = 100

[tex]\sigma_{\bar{x}}[/tex] = 20/[tex]\sqrt{(100)}[/tex]

[tex]\sigma_{\bar{x}}[/tex] = 20/10

[tex]\sigma_{\bar{x}}[/tex] = 2

For sample sizen = 16

[tex]\sigma_{\bar{x}}[/tex] = 20/[tex]\sqrt{16}[/tex]

[tex]\sigma_{\bar{x}}[/tex] = 20/4

[tex]\sigma_{\bar{x}}[/tex] = 5

for sample size n = 25

[tex]\sigma_{\bar{x}}[/tex] = 20/[tex]\sqrt{25}[/tex]

[tex]\sigma_{\bar{x}}[/tex] = 20/5

[tex]\sigma_{\bar{x}}[/tex] = 4

For the sample size n = 36

[tex]\sigma_{\bar{x}}[/tex] = 20/[tex]\sqrt{36}[/tex]

[tex]\sigma_{\bar{x}}[/tex] = 20/6

[tex]\sigma_{\bar{x}}[/tex] = 3.33

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Reflect it across the x-axis, then translate it up 6 units and left 1 unit before reflecting it across the y-axis.

You should be aware that a collection of numbers, or "terms," is referred to as a sequence. Examples of terms include 2, 5, and 8. Some sequences can be made endlessly long by taking advantage of a particular pattern they follow. Use the example of 2,5,8 and add 3 to continue the series. There are formulas that demonstrate where to look for words within a sequence. A collection of objects that are in some order is known as a sequence in mathematics (or events). It resembles a set in that it has pieces (also called elements or terms). The total, perhaps infinite number of ordered objects are referred to as the sequence's length. the act of placing two or more objects in a logical order.

We are given the following

Δ DOG and ΔCAT

According to Chanel, the series of transformations that maps triangle ABC onto triangle DEF is as follows:

Reflect it across the x-axis, then translate it up 6 units and left 1 unit before reflecting it across the y-axis.

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The probability that an employee posted to country A at age 30 will survive to age 40 if she remains in that country is 2 x 10⁻⁶³

Probability is a mathematical concept that describes the likelihood of a specific event occurring within a set of possible outcomes

Let's start by defining some terms. The force of mortality, also known as the death rate, is the number of deaths per unit time per unit of population.

The probability of survival is the chance that an individual will live past a certain age. In our case, we want to know the probability of surviving from age 30 to age 40 while living in country A.

To calculate the probability of survival, we will use the formula:

[tex]P(x) = e^{-qx}[/tex]

Where P(x) is the probability of survival at age x and qx is the force of mortality at age x.

First, we will calculate the force of mortality for the first five years after arrival in country A using the equation given in the question:

90x+1 = (6 - 1)9x+1

For x = 30, we have:

90(30) + 1 = (6 - 1)9(30) + 1

2701 = 5 * 891 + 1

2701 = 4445 + 1

2700 = 4445

So, q30 = 4445/30 = 148.5

Next, we will use this value to calculate the probability of survival for the first five years in country A:

[tex]P30 = e^{-q3} = e^{-148.5} = 1.2 \times 10^{-64}[/tex]

Since the force of mortality for those who have lived in country A for at least five years is 50% greater than the US Life Tables, 2002, Females, we can calculate the force of mortality for the next 10 years as follows:

qx = 1.5 x qx from US Life Tables

Using the values from Table 3.11, we have:

q31 = 1.5 x 98,424 = 147,636

q32 = 1.5 x 98,362 = 147,543

q33 = 1.5 x 98,296 = 147,444

q34 = 1.5 x 98,225 = 147,338

q35 = 1.5 x 98,148 = 147,222

q40 = 1.5 x 97,500 = 146,250

Finally, we can use these values to calculate the probability of survival from age 31 to age 40:

[tex]P31 = e^{-q31} = e^{-147,636} = 1.3 \times 10^{-65}\\ \\P32 = e^{-q32} = e^{-147,543} = 1.4 \times 10^{-66}\\\\P33 = e^{-q33} = e^{-147,444} = 1.5 \times 10^{-67}\\\\P34 = e^{-q34} = e^{-147,338} = 1.6 \times 10^{-68}\\\\P35 = e^{-q35} = e^{-147,222} = 1.7 \times 10^{-69}\\\\P40 = e^{-q40} = e^{-146,250} = 2.0 \times 10^{-63)}\\\\[/tex]

Complete Question:

When posted overseas to country A at age x, the employees of a large company are subject to a force of mortality such that, at exact duration t years after arrival overseas (t = 0,1,2,3,4), 90x+1 = (6 - 1)9x+1 where qx+t is on the basis of US Life Tables, 2002, Females. For those who have lived in country A for at least five years the force of mortality at each age is 50% greater than that of US Life Tables, 2002, Females, at the same age. Some lx values for this table are shown in Table 3.11.

An extract from the United States Life Tables, 2002,

Females. Age,

x 30 31 32 33 34 35 40

1x 98,424 98,362 98,296 98,225 98,148 98,064  97,500

Calculate the probability that an employee posted to country A at age 30 will survive to age 40 if she remains in that country.

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The area of the shaded region is 24x² - 78x + 33.

A square is a two-dimensional figure that has four sides and all four sides are equal.

The area of a square is side².

We have,

Two squares:

Smaller square:

Side = x + 4

Larger square:

Side = 5x - 7

Now,

Area of the shaded region.
= Area of the bigger square - Area of the smaller square

= (5x - 7)² - (x + 4)²

= 25x² - 70x + 49 - x² - 8x - 16

= 24x² - 78x + 33

Thus,

24x² - 78x + 33 is the area of the shaded region.

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Answer:

$44

Step-by-step explanation:

Given,

Gas charges = $31 + $16 + $41

Total = $88

Half of the payment means dividing the final amount in two half.

= 88/2

= 44

Thus, your friend owes you $44 dollars for gas.

The surface area of the [tex]132cm^2.[/tex]

The surface area of a triangular prism is also referred to as its total surface area. The total surface area of a triangular prism is the sum of the areas of all the faces of the prism. A triangular prism has three rectangular faces and two triangular faces. The rectangular faces are said to be the lateral faces, while the triangular faces are called bases. If the bases of a triangular prism are placed horizontally, they are referred to as the top and the bottom (faces) of the prism, respectively.

Surface area of given triangular shaped prism

= 2 ×area of right angle triangle + area of3 rectangles

= 2(1/2 ab)+(ah+bh+ch)

=(3 cmx4 cm) +(3cm×10 cm +4cm×10cm+5cm×10cm)

=[tex]12 cm^2 +30cm^2+40cm^2+50cm^2=132cm^2[/tex]

Volume of prism = Area of base × height

=> V=( 1/2 ab) × height= 1/2 × 3 cm×4cm× 10 cm =[tex]60 cm^3[/tex]

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the fractions with the same denominator will be 14/16 and 13/16.

What is a fraction?

It can also be written as a mixed number, which is a whole-number quotient with a proper-fraction residual, if the numerator is larger. This type of expression is used when the numerator is larger.

Divided by its denominator, any fraction can be converted to a decimal. The outcome may end at some point or one or more digits may repeat continuously.

The given fraction is given are 7/8 and 13/16.

Taking lcm of the denominator will be 16

So the fraction will be 7*2 = 14

So the fractions will be 14/16 and 13/16.

Hence the fractions with the same denominator will be 14/16 and 13/16.

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Inferential statistics is discussed below.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied.

Given is inferential statistics.

Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.

Therefore, Inferential statistics is discussed above.

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The number of non-real solutions is 2.

What is a quadratic equation?

The quadratic equation a equation in one variables whose power is 2. The solution of the quadratic equation can be found by equation the equation to 0 and then factoring it.

We are given a equation -3n^2-2n-6=0

To find the non-zero solution we first find the solution of the equation

we get

3n^2+2n+6 =0

We use the formula method to find the roots of the equation

[tex]n= \frac{-b}{2a}[/tex]±[tex]\frac{\sqrt{b^2-4ac} }{2a}[/tex]

Substituting the values in the equation we get

n = -0.33333333333333 + 1.3743685418726i  and

n = -0.33333333333333 - 1.3743685418726i

Both the roots has i in them. it means both the roots are imaginary

Hence the number of complex roots are 2

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The outcomes for the event of rolling a number greater than 5 is 1.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes

Given that, a number cube with faces labeled from 1 to 6 will be rolled once.

Now, sample space is {1, 2, 3, 4, 5, 6}

Total number of outcomes =6

Number of favourable outcomes =1

Here, the probability is 1/6

Therefore, the outcomes for the event of rolling a number greater than 5 is 1.

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Answer:

(x + 7)/(x - 3)

Step-by-step explanation:

To simplify this expression, we first factor the numerator and denominator, and then cancel any common factors:

x² + 3x - 28 = (x + 7)(x - 4)

x² - 7x + 12 = (x - 3)(x - 4)

So the expression simplifies to:

(x + 7)(x - 4) / (x - 3)(x - 4)The (x - 4)

factor in both the numerator and denominator cancels out, leaving:

x + 7 / x - 3

Therefore, the simplified expression is (x + 7)/(x - 3)

The required demand function and revenue per ticket are p(x) = -0.4 * (x - 21000) + $9 and  $9.

a) To find the demand function p(x), we can use the two data points: (21000, $9) and (26000, $7). To find the slope of the demand function, we can use the formula:

slope = (change in price) / (change in attendance)

Plugging in the two data points, we get:

slope = ($9 - $7) / (26000 - 21000) = -$2 / 5000 = -0.4

To find the y-intercept, we can use either data point and the slope we just found. Let's use the first data point:

p(x) = slope * (x - 21000) + $9

Now we have our demand function:

p(x) = -0.4 * (x - 21000) + $9

b) To maximize revenue, we need to find the price that will result in the maximum number of tickets sold. This is equal to the price that corresponds to the peak of the demand function, or the maximum value of x for which p(x) is decreasing. To find this, we can take the derivative of the demand function and set it equal to zero:

dp/dx = -0.4 = 0

x = 21000

So the maximum revenue is obtained by selling 21000 tickets at the price of $9 each, for a total revenue of $189,000. To maximize revenue, the ticket price should be set at $9.

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The next shape in the pattern has four intersecting lines.

A recurring arrangement of numbers, forms, colours, and other elements constitutes a pattern in mathematics. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series.

From the given pattern, we can see the number of lines are increasing.

In the next pattern there will 4 lines.

Therefore, the next shape in the pattern has four intersecting lines.

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The coordinates of the position at which Pablo should stand are given as follows:

(6, 7.5).

The midpoint between two points is the halfway point between them, and is found using the mean of the coordinates.

Hence the x-coordinate of the position at which Pablo should stand is given as follows:

x = (2 + 10)/2

x = 6.

The y-coordinate of the position at which Pablo should stand is given as follows:

y = (3 + 12)/2

y = 7.5.

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The company produce each month to maximize their profit be (50, 100000).

To ensure the best output and price levels are realized in order to maximize returns, business firms engage in the process of profit maximization. In order to achieve its profit objectives, the company modifies important variables including sale price, production costs, and output levels.

"Profit maximization" emphasizes the location of this critical point, or the optimal output at which your company is most profitable. An educational non-profit organization called Khan Academy states that "a corporation striving to maximize profit will produce the quantity where 'marginal cost' and 'marginal income' are equal to each other."

Let x be the number of ceiling fans

If the price per fan when you sell x fans exists (300 - 2x), then the amount received for selling x fans exists x(300 - 2x)

The cost for producing x fans is (20x + 100).

Profit = x(300 - 2x) - (20x + 100)

y = (1000 + 20x)(300 - 2x)

simplifying the equation, we get

y = 300000 - 2000x + 6000x - 40x²

y = -40x² + 4000x + 300000

simplifying the above equation, we get

y = -40(x² - 100x + 2500) + 100000

y = -40(x - 50)² + 100000

vertex: (50, 100000).

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The values of the composite functions are g(f(0))= -3 and f(f(2)) = 5

From the question, we have the following parameters that can be used in our computation:

f(x)= 2x-1 g(x)=3x and h(x)=x^2+1

To calculate g(f(0)), we have

f(0)= 2(0)-1 = -1

This means that

g(f(0)) = g(-1)

So, we have

g(f(0)) = 3(-1) = 3

For f(f(2)), we have

f(2)= 2(2)-1 = 3

So, we have

f(f(2)) = 2(3) - 1 = 5

Hence, the composite functions are g(f(0))= -3 and f(f(2)) = 5

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A polynomial of degree n can pass through n + 1 points. So, if we have n + 1 points, we can construct a unique polynomial that passes through all those points.

A nonzero polynomial passing through a specific point (x, y) means that the value of the polynomial at that x-coordinate must be equal to the y-coordinate of the point. So, if the point (2, 0) is one of the points that the polynomial passes through, then we have:

[tex]p(2) = 0[/tex]

We can construct a polynomial with this information. For example, one possible polynomial is:

[tex]p(x) = (x - 2)^2[/tex]

This polynomial satisfies the equation p(2) = 0 since:

[tex]p(2) = (2 - 2)^2 = 0^2 = 0[/tex]

Therefore, the polynomial [tex]p(x) = (x - 2)^2[/tex] is a nonzero polynomial that passes through the point (2, 0) and satisfies the equation p(x). Note that there could be other polynomials that also satisfy the given conditions.

In general, a polynomial of degree n can pass through n + 1 points. So, if we have n + 1 points, we can construct a unique polynomial that passes through all those points.

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The complete question is :

Find A Nonzero Polynomial P([tex]x[/tex]) Passing Through The Point (2,0) And Satisfying The Equation P([tex]x[/tex]) = P'([tex]x^2[/tex])?

Answer: We can use the Pythagorean theorem to find the length of the hypotenuse (c) of the triangle:

c = √(a^2 + b^2) = √(75.4^2 + 41.7^2) = √(5702.76 + 1738.69) = √(7441.45)

c ≈ 86.38 yd

Next, we can use the inverse cosine function (arccos) to find angle A:

A = arccos(a/c) = arccos(75.4/86.38)

A ≈ 30.49°

Finally, we can use the complementary angle formula to find angle B:

B = 90° - A

B ≈ 59.51°

So, the length of the hypotenuse is approximately 86.38 yards, angle A is approximately 30.49 degrees, and angle B is approximately 59.51 degrees.

Step-by-step explanation:

The estimate of the product, using rounding, is given as follows:

0.67 x 322 = 216.

The product in the context of this problem is defined as follows:

0.67 x 322.

Multiplying the two numbers, the result is given as follows:

0.67 x 322 = 215.74.

We want to round the product to the nearest integer, hence we add one to the integer part, as the decimal part has a first digit which is greater than 4.

Thus the rounded result is given as follows:

216.

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Problem 1

L = length

W = width

P = perimeter of the rectangle

P = 2L+2W

P = 20

2L+2W = 20

L+W = 10 after dividing everything by 2.

L = 10-W

area = length*width

A = L*W

A = (10-W)*W

A = -W^2 + 10W

Let x replace W for a temporary basis.

If we graphed y = -x^2+10x, then we'd see the highest point occurs at (5, 25). This means a width of 5 meters leads to the largest area of 25 square meters.

The length would be L = 10-W = 10-5 = 5 meters as well.

It turns out that a 5 meter by 5 meter square is the best choice when trying to max out this rectangle's area.

====================================================

Problem 2

The results we got from the previous section was that a 5 by 5 square will max out the area.

If we allowed non-whole numbers into the mix, then it wouldn't change the previous answer. The best choice is still that 5 by 5 square.

The answer only would change if the result of the previous section was some non-integer value.

The missing sides of the triangles are:

NUMBER 6 and 7: x = 12√2 units, y = 12√2 units

NUMBER 8 and 9: x = 25 units, y = 25√3 units

Trigonometry is a branch of mathematics dealing with the relationship between the ratios of the sides of a right-angled triangle with its angles.

NUMBER 6 and 7

sin 45° = x/24

x = 24 * sin 45°

x = 24 * (√2)/2

x = 12√2 units

sin 45° = y/24

y = 24 * sin 45°

y = 24 * (√2)/2

y = 12√2 units

NUMBER 8 and 9

sin 30° = x/50

x = 50 * sin 30°

x = 50 * 1/2

x = 25 units

sin 60° = y/50

y = 50 * sin 60°

y = 50 * (√3)/2

y = 25√3 units

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The best estimate for the African lion population is 14 individuals.

The best estimate for the African lion population can be found using the Lincoln-Petersen index, which is a commonly used method for estimating fish or wildlife populations.

The formula for the Lincoln-Petersen index is:

N = (2 x tagged individuals caught) / (total tagged individuals + total untagged individuals caught)

Plugging in the given values:

N = (2 x 8) / (20 + 160) = 16 / 180 = 0.089

Multiplying N by the number of lions caught (160) will give us the best estimate for the African lion population:

African lion population = N x 160 = 0.089 x 160 = 14.24 (rounded to the nearest whole number, 14)

Therefore, the best estimate for the African lion population is 14 individuals.

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Answer: 7

Step-by-step explanation:

We know that the base sides of the triangular prism are 6cm, 5cm, and

[tex]x[/tex] cm. We also know that the length of the prism is 20.

To calculate the lateral surface area of the triangular prism, we need to use the following equation:

[tex](a + b + c)[/tex] x [tex]length[/tex]

So we can input our values, to get this:

[tex](6 + 5 + x) 20 = 360[/tex]

Then we solve the equation by the following steps:

[tex]= > 6+5+x = \frac{360}{20} = 18\\= > 11 + x = 18\\= > x = 7[/tex]

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The fifth term of the geometric progression is given as 1280

The formula for calculating the nth term of a geometric progression is expressed as:

Tn = ar^n-1

Given the following

First term = a = 5

Third term = ar^2 = 80

Determine the common ratio

5(r^2) = 80
r^2 = 16

r = 4

For the 5th term

T5 = ar^4

T5 = 5(4)^4
T5 = 1280

Hence the fifth term of the expression is 1280

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Complete question

The first and third term of a Gip are 5 and 80 respectively. What is the fifth term​

Answer:

4 feet and 4 inches of fringe

Step-by-step explanation:

The length and width of a square are equal to each other

Let x = length of a side of a square

Area of a square = Length × Width

                            = (x inches) × (x inches)

                            = [tex]x^{2}[/tex] [tex]in^{2}[/tex]

The area of the square placemat = 169 [tex]in^{2}[/tex]

∴                                            [tex]x^{2} in^{2}[/tex]  = 169 [tex]in^{2}[/tex]

Taking the square root on both sides of the equation to get rid of the square:

                                             x inches = [tex]\sqrt{169}[/tex] inches

                                            x inches = [tex]\sqrt{169}[/tex] inches

   ([tex]\sqrt{169} =[/tex] ± 13. However, the negative value will be rejected since the length of a side CANNOT be negative)    

        So, x = 13 inches

This means the perimeter of the square = (13 + 13 + 13 + 13) inches

                                                                   = 52 inches

∴ 52 inches of fringe will be needed by Sally

Unit conversion:

                                                   I foot = 12 inches

                                      ∴          y feet = 52 inches

     Cross-multiplication is applied:

                       (y feet)(12 inches) = (52 inches)(1 foot)

Isolating y and making it the subject of the formula:

                                   y feet = [tex]\frac{(52 inches)(1 foot)}{12 inches}[/tex]

Inches in the numerator and denominator will cancel each other out completely:

                                  y feet = [tex]\frac{52}{12}[/tex] feet

Applying long division:

The highest quotient to be multiplied by 12 to give a number, which is closest to 52 but still less than 52. 12×4 = 48. Therefore, the quotient is 4 and the remainder is 4.

∴ 52 inches = 4 feet and 4 inches

Answer: (21, 3)

Step-by-step explanation:

The value x= -2 makes the inequality  true.

Mathematical expressions with inequalities are those in which the two sides are not equal. Contrary to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign.

Given:

12 > 2x + 4

Now, if we put x= 8 then

= 2x+ 8

= 2(8)+8

= 16+ 8

= 24 > 12

So, x=8 is not true

Now,  if we put x= 4 then

= 2x+ 8

= 2(4)+8

= 8+ 8

= 16 > 12

So, x=4 is not true

Now,  if we put x= 12 then

= 2x+ 8

= 2(12 )+8

= 24 + 8

= 32 > 12

So, x=12 is not true

and,  if we put x= -2 then

= 2x+ 8

= 2(-2)+8

= -4+ 8

= 4 < 12

So, x=-2 is true.

Learn more about Inequality here:

https://brainly.com/question/30231190

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