At the school bookstore, Rylan bought two spiral notebooks and one folder and paid $6. 70. Olivia bought three spiral notebooks and five folders and paid $12. 85. Find the cost of each folder

The sound level of the vocalizations of a blue whale is approximately 140.28 decibels.

The question asks us to find the sound level in decibels (db) of the vocalizations of a blue whale, given its sound intensity of 106.8 watts/meter2.

The formula for sound level in decibels is:

L = 10log(i/I0)

Where L is the sound level in decibels, i is the sound intensity in watts/meter2, and I0 is the smallest sound intensity that can be heard by the human ear (approximately 1x10⁻¹² watts/meter2).

Plugging in the given values, we get:

L = 10log(106.8/1x10⁻¹² )

Simplifying this expression, we get:

L = 10log(1.068x10¹⁴)

L = 10(14.028)

L = 140.28

Therefore, the sound level of the vocalizations of a blue whale is approximately 140.28 decibels.

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The height of the new player is given as follows:

210 cm.

The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.

Considering the frequencies and the height of the new player of x, the sum of the observations is given as follows:

198 x 1 + 199 x 3 + 200 x 2 + 201 x 5 + 202 x 2 + x = 2604 + x.

The total number of players, considering the new player, is given as follows:

1 + 3 + 2 + 5 + 2 + 1 = 14.

The mean is of 201 cm, hence the height of the new player is obtained as follows:

(2604 + x)/14 = 201

2604 + x = 14 x 201

x = 14 x 201 - 2604

x = 210 cm.

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The actual measure of major arc JML is approximately 289.33°, which is closest to 287°.

We know that minor arc KL is supplementary to major arc JML. So,

m∠KL = 180° - m∠JML

Substituting the given values, we get:

8x - 6 = 180 - (25x - 13)

Solving for x, we get:

33x = 193

x = 193/33

Substituting this value of x in the expression for m∠JML, we get:

m∠JML = 25(193/33) - 13

m∠JML = 1468/3

m∠JML ≈ 489.33°

However, since lines KL and JM appear tangent, we know that minor arc KL and major arc JML share the same endpoint and thus are part of the same circle. So, the actual measure of major arc JML is:

m(arc JML) = 360° - m(arc KL)

We can find m(arc KL) by subtracting m∠KLM from 180°:

m(arc KL) = 180° - m∠KLM

m(arc KL) = 180° - (8(193/33) - 6)

m(arc KL) ≈ 70.67°

Substituting in the formula for m(arc JML), we get:

m(arc JML) = 360° - 70.67°

m(arc JML) ≈ 289.33°

Therefore, the actual measure of major arc JML is approximately 289.33°, which is closest to option (b) 287°.

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The equations that have the same value of x as 3/5 (30 x - 15) = 72 are C, D, and E.

To solve for x in 3/5 (30 x - 15) = 72, we can first simplify the left side by distributing the 3/5:

3/5 (30 x - 15) = 18 x - 9

Now we can solve for x by setting the right side equal to 72:

18 x - 9 = 72

Adding 9 to both sides:

18 x = 81

Dividing by 18:

x = 4.5

So we know that option E is one of the correct answers.

To check which of the other options have the same value of x, we can substitute x = 4.5 into each equation and see if it simplifies to 72:

A. 18 x - 15 = 72

18(4.5) - 15 = 72

81 - 15 = 72 (not equivalent)

B. 50 x - 25 = 72

50(4.5) - 25 = 200 - 25 = 175 (not equivalent)

C. 18 x - 9 = 72

18(4.5) - 9 = 72 (equivalent)

D. 3 (6 x - 3) = 72

3(6(4.5) - 3) = 3(24) = 72 (equivalent)

Therefore, the equations that have the same value of x as 3/5 (30 x - 15) = 72 are C, D, and E.

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Thus, the coordinates of ΔF'I'G' after a rotation of 90° clockwise about the origin. are - F'(4,-2), I'(4,-5) and G'(2,-3).

A rotation is a turn made about a specific axis. Both clockwise and anticlockwise rotations are possible. Whereas the image is really the rotating image, the pre-image is the original item.

From the pre-image point, calculate the image. The listed pre-image point is (x , y). Change the x and y coordinates, then multiply this same previous y coordinate by -1 to get a 90 degree anticlockwise rotation. Use the guidelines mentioned below to calculate each rotation.

Clockwise :

Given :

F(2, 4), I(5, 4) and G(3, 2)

After 90 degree rotation: (x , y) ----> (y , -x)

F'(4,-2), I'(4,-5) and G'(2,-3).

Thus, the coordinates of ΔF'I'G' after a rotation of 90° clockwise about the origin. are - F'(4,-2), I'(4,-5) and G'(2,-3).

Graphs for the both triangles are obtained.

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Correct question:

ΔFIG has vertices at F(2, 4), I(5, 4) and G(3, 2). Graph ΔFIG and ΔF'I'G' after a rotation of 90° clockwise about the origin.​

John will be able to buy the 10 baseball bats for his team with some money left over is -$6.50 and David will be able to buy the 3 pairs of running shoes for his kids with some money left over is $12.30.

Based on the given information, we can calculate the total cost for each customer after factoring in the sales tax and discounts.

For John:
The cost of 10 baseball bats without any discounts or tax would be 10 * $25 = $250.
Since baseball bats are 5% off, John will get a discount of 0.05 * $250 = $12.50.
So, the cost of 10 baseball bats after discount is $250 - $12.50 = $237.50.
Adding 8% sales tax to this amount, the total cost for John will be $237.50 + (0.08 * $237.50) = $256.50.

For David:
The cost of 3 pairs of running shoes without any discounts or tax would be 3 * $50 = $150.
Since running shoes are 15% off, David will get a discount of 0.15 * $150 = $22.50.
So, the cost of 3 pairs of running shoes after discount is $150 - $22.50 = $127.50.
Adding 8% sales tax to this amount, the total cost for David will be $127.50 + (0.08 * $127.50) = $137.70.

Therefore, John will be able to buy the 10 baseball bats for his team with some money left over ($250 - $256.50 = -$6.50) and David will be able to buy the 3 pairs of running shoes for his kids with some money left over ($150 - $137.70 = $12.30).

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a) Where the information about triangle ABC is given above, the measure of angle B is 60°

Using the Cos Rules,

(15√3)² = 30² + 15² - 2(15 x 30) cos B

⇒ 657 = 1125 - 900 cosB

⇒ 2 Cos B = 1 Cos B = 1/2

So

B = Cos ⁻¹(1/2)

Hence,

B = 60°

B) Given the above,

Now, we can use the tangent function to find tan B:

tan B = sin B / cos B

  = sin (π/3) / cos (π /3)

  = (√3 / 2) / 0.5

tan B = √3

C ) the area of the rriangle is given as

s = (a + b + c) /2

Substituting

s = (15 + 15√ 3 + 30)/2

s = 30 + 15√3

Area = √ [ (30 + 15√3)(15√3) (15)(30  - 15 - 15√3))

Simplifying  we can say

Area = √(30 + 15√3  )( 15√3)(15)(15 - 5√3 )]

= √[3(10 + 5√3)(15)(3√3 - 1)]

  = √[2250 - 1125√3]

Hence,

Area ≈ 17.36in

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Answer:

366 times

Step-by-step explanation:

Using  net worth statement, Andre's net worth is $159,573.

Andre's net worth can be calculated by subtracting his total liabilities from his total assets.

Total assets = $155,000 (condominium) + $8,100 (stock portfolio) + $9,100 (car value) + $2,600 (bank account) = $174,800

Total liabilities = $3,300 (car loan) + $7,600 (student loans) + $4,327 (credit card balance) = $15,227

Net worth = Total assets - Total liabilities = $174,800 - $15,227 = $159,573

Therefore, Andre's net worth is $159,573.

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The equations represent four different parabolas with different shapes and orientations, but all of them have the same axis of symmetry, which is the y-axis (because there is no x term).

The collection of all points in a plane that are equally spaced from a fixed line and another fixed point in the plane that is not on the line is known as a parabola. The focus of the parabola is the fixed point (F) and the fixed point (D) is known as the directrix.

The equation of a parabola in standard form is:

y = a x² + b x + c

where "a" is the coefficient of the quadratic term (x^2), "b" is the coefficient of the linear term (x), and "c" is the constant term.

Looking at the given equations:

So, the equations represent four different parabolas with different shapes and orientations, but all of them have the same axis of symmetry, which is the y-axis (because there is no x term).

To graph each parabola, you can use the vertex form of the equation:

y = a (x - h)² + k

where (h, k) is the vertex of the parabola.

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The function in vertex form is f(x) = 4(x - 6)² - 26.

The function in vertex form is f(x) = 4(x - 6)² - 26.

To get to this form, the first step is to factor out the coefficient of x², which is 4:

f(x) = 4(x² - 12x) + 10

Next, complete the square by adding and subtracting (12/2)² = 36 inside the parenthesis:

f(x) = 4(x² - 12x + 36 - 36) + 10

Simplify the expression inside the parenthesis and combine like terms:

f(x) = 4((x - 6)² - 36) + 10

f(x) = 4(x - 6)² - 134

Therefore, the function in vertex form is f(x) = 4(x - 6)² - 26.

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The location of the point Kristin will add to represent the 13 bottles of water sold at 39 degrees Fahrenheit is the bottom left of the scatter plot.

A scatter plot represents the relationship between two variables. In this case, the temperature (independent variable) is plotted along the x-axis, while the number of bottles of water sold (dependent variable) is plotted along the y-axis. As the temperature increases, it is expected that more bottles of water would be sold.

The bottom left area of the scatter plot is where lower values of both temperature and the number of bottles sold would be found. Since 39 degrees Fahrenheit is relatively low and 13 bottles of water is a lower quantity, the point representing this data will be in the bottom left quadrant of the scatter plot.

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Complete question:

Kristin made a scatter plot to represent the relationship between the temperature and the number of bottles of water sold at her concession stand at a soccer tournament. Which is a description of the location of the point Kristin will add to represent the 13 bottles of water that were sold when the temperature was 39 degrees Fahrenheit? Select one:

O  Top left of the scatter plot

O  Bottom right of the scatter plot

O  Top right of the scatter plot

O  Bottom left of the scatter plotâ

The divergence of vector fields at all points where they are defined ar 4x - 2xcos(xz) for all points in R3.

The divergence of the given vector field F = (2x^2 - sin(xz)) i + 5j - (sin (xz)) k can be found using the formula for divergence:

div(F) = (∂Fx/∂x) + (∂Fy/∂y) + (∂Fz/∂z)

Here, Fx = (2x² - sin(xz)), Fy = 5, and Fz = -sin(xz). Taking the partial derivatives, we get:

∂Fx/∂x = 4x - zcos(xz)

∂Fy/∂y = 0

∂Fz/∂z = -xcos(xz)

Therefore, the divergence of F is:

div(F) = (∂Fx/∂x) + (∂Fy/∂y) + (∂Fz/∂z) = 4x - zcos(xz) - xcos(xz) = 4x - 2xcos(xz)

The divergence of F is defined for all points where F is defined, which is the entire 3-dimensional space. So, the divergence of F is 4x - 2xcos(xz) for all points in R3.

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Therefore, the height of the cone is approximately 4.23 centimeters.

Volume is a measure of the amount of space occupied by a three-dimensional object or region. It is the total amount of space enclosed by the boundaries of the object or region. Volume is usually measured in cubic units, such as cubic centimeters (cm³) or cubic meters (m³). Knowing the volume of an object can be useful for many purposes, such as determining how much material is needed to fill a container or how much space is needed to store a certain quantity of objects.

Here,

The formula for the volume of a cone is given by:

V = (1/3)πr²h

where V is the volume, r is the radius, and h is the height.

We are given that the volume of the cone is 3.0144 cubic centimeters, so we can plug this into the formula:

3.0144 = (1/3)πr²h

Next, we need to find the radius of the cone. Since we are not given the radius directly, we may need to use other information that is not given in the problem. If we assume that the cone is a right circular cone, then we can use the fact that the radius and height are proportional to find the radius:

r/h = 1/3

r = (1/3)h

We can substitute this expression for r into the volume formula:

3.0144 = (1/3)π((1/3)h)²h

Simplifying this equation:

3.0144 = (1/27)πh³

Multiplying both sides by 27/π:

h³ = 84.96

Taking the cube root of both sides:

h = 4.23 (rounded to the nearest hundredth)

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(a) Boris performed better on the aptitude test, since its z-score was higher.

(b) Callie performed better on the knowledge test.

(c) The z-score for the aptitude test was still higher than the z-score for the knowledge test, so Boris performed better on the aptitude test.

(a) To determine which test Boris performed better on, we need to compare his z-scores for the knowledge test and the aptitude test.

For the knowledge test, his z-score is calculated as:

z = (55 - 60) / 3 = -1.67

For the aptitude test, his z-score is:

z = (106 - 110) / 6.8 = -0.59

Since the z-score for the aptitude test is higher than the z-score for the knowledge test, Boris performed better on the aptitude test.

(b) To determine which test Callie performed better on, we need to compare her z-scores for the knowledge test and the aptitude test.

For the knowledge test, her z-score is:

z = (67 - 60) / 3 = 2.33

For the aptitude test, her z-score is:

z = (119 - 110) / 6.8 = 1.32

Since the z-score for the knowledge test is higher than the z-score for the aptitude test, Callie performed better on the knowledge test.

(c) Boris' z-score on the logic test (-0.43) is unrelated to his performance on the knowledge and aptitude tests, so it does not change the answer to which test he performed better on. The z-score for the aptitude test was still higher than the z-score for the knowledge test, so Boris performed better on the aptitude test.

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Answer:

-----------------------

F(x) is in the format of a quadratic function:

Hence it is a parabola.

The table represents a relation with two x-intercepts and two y-intercepts (points with the coordinate of 0).

We know that parabola can have maximum of one y-intercept, hence G(x) is not a parabola.

The matching answer choice is the first one.

The term f(4) < f(11) means that the temperature is lower at 4 A.M. than it is at 11 A.M., and this inequality can be used to make predictions about temperature changes over time.

The function f(t) represents the temperature at a specific time t. In this case, f(t) is the outside temperature (in degrees Fahrenheit) 7 hours after 2 A.M. So, we can think of f(4) as the temperature 4 hours after 2 A.M. and f(11) as the temperature 11 hours after 2 A.M.

Now, the inequality f(4) < f(11) means that the temperature 4 hours after 2 A.M. is less than the temperature 11 hours after 2 A.M. In other words, the temperature is lower at 4 A.M. than it is at 11 A.M. This might seem obvious, as we generally expect temperatures to rise as the day progresses and the sun comes up. However, this inequality is useful for making more specific predictions about temperature changes.

For example, if we know that f(4) < f(11), we can predict that the temperature will increase between 4 A.M. and 11 A.M. This might be important information if you're planning outdoor activities or need to dress appropriately for the day's weather.

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The measure of angle U in triangle UVW is approximately 28 degrees. This is found by using the inverse tangent function to solve for angle U given the lengths of two sides and the fact that angle W is a right angle.

To find the measure of ∠U in ΔUVW, we can use trigonometry. We know that sin(∠U) = opposite/hypotenuse, which is equal to UW/VW. Therefore, we can plug in the given values and solve for sin(∠U)

sin(∠U) = UW/VW = 2.2/4.7 = 0.4681

Next, we can use the inverse sine function (sin⁻¹) to find the measure of ∠U

∠U = sin⁻¹(0.4681) = 28.34 degrees (rounded to the nearest degree)

Therefore, the measure of ∠U in ΔUVW is approximately 28 degrees.

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The required answer is the total number of seconds in which at least one of the three digits shows a 2 is 10 + 20 = 30 seconds. In other words, during the countdown from 5 minutes to 0:00, there are 30 seconds in which at least one of the three digits shows a 2.

To determine the number of seconds in which at least one of the three digits on a digital timer shows a 2 while counting down from 5 minutes (5:00) to 0:00, we need to consider the various possibilities.

Step 1: Determine the total number of seconds in 5 minutes.

There are 60 seconds in a minute, so 5 minutes would be equal to 5 * 60 = 300 seconds.

Step 2: Consider each second from 0 to 300 and check if any of the three digits (hundreds, tens, or ones) contains the digit 2.

To simplify the calculation, we can focus on the ones digit for the first 60 seconds (from 0:00 to 0:59). In this range, the ones digit contains the digit 2 ten times (2, 12, 22, 32, 42, 52, 62, 72, 82, 92). So, in the first minute, there are 10 seconds in which the ones digit shows a 2.

For the remaining 240 seconds (from 1:00 to 4:59), we need to consider both the tens and ones digits. In each minute within this range, the tens digit can have a digit 2 for all ten seconds (20, 21, 22, ..., 29). Additionally, the ones digit can have a digit 2 for ten seconds in each minute. So, in the remaining 240 seconds, there are 10 * 2 = 20 seconds in which at least one of the tens or ones digits shows a 2.

Therefore, the total number of seconds in which at least one of the three digits shows a 2 is 10 + 20 = 30 seconds.

Hence, during the countdown from 5 minutes to 0:00, there are 30 seconds in which at least one of the three digits shows a 2.

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She requires 55 consecutive successful throws to get a success rate of 60%

The number of successful throws she made is 8 out of 50

Now to calculate the percentage we will use the formula

[tex]\frac{8}{50} X 100 = 16[/tex]

According to the problem, she needs to make her success percentage 60% and for that, we need to estimate the number of consecutive successful throws. Hence the new equation will be

[tex]\frac{8+x}{50+x} X 100 = 60[/tex]

[tex]or, \frac{8+x}{50+x} = 0.6[/tex]

This is the equation concerned.

Now,to solve this, we will take the denominator on the other side will give us

[tex]or, 8+x = 0.6(50+x)[/tex]

[tex]or, 8+x = 30+ 0.6x[/tex]

[tex]or, x - 0.6x = 30 -8[/tex]

[tex]or, 0.4x= 22[/tex]

or, x = 55

Hence we see that she requires 55 consecutive successful throws to get a success rate of 60%

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Complete Question

A basketball player made 8 out of 50 free throws she attempted which is 16%. She wants to know how many consecutive free throws more in a row she would have to make to raise her overall successful baskets divided by attempts or percent of successful free throws to 60%.

(a) Write an equation to represent this situation.

(b) Solve the equation. How many consecutive free throws would she have to make to raise her percent to 60%?

Answer:

[tex]20,000 \text{ kWh}[/tex]

Step-by-step explanation:

We can convert 68 million British Thermal Units (BTUs) to kilowatt-hours (kWh) using the given conversion ratio:

[tex]\dfrac{1 \text{ kWh}}{3400 \text{ BTUs}}[/tex]

Multiplying by the ratio:

[tex]68,000,000 \text{ BTUs} \cdot \dfrac{1 \text{ kWh}}{3,400 \text{ BTUs}}[/tex]

↓ canceling the BTU units

[tex]68,000,000\cdot \dfrac{1 \text{ kWh}}{3,400}[/tex]

↓ executing multiplication

[tex]\dfrac{68,000,000}{3,400} \text{ kWh}[/tex]

↓ rewriting as a decimal

[tex]\boxed{20,000 \text{ kWh}}[/tex]

Its translation is 5×3-3=12

The balance when $3000 is invested at 4% rate for 20 years and compounded annually, semi-annually, quarterly, monthly, daily, and continuously are $6,372.76, $6,454.81, $6,506.71, $6,535.94, $6,546.49, and $6,549.18 respectively.

Compounding Frequency Balance after 20 Years

Annually $6,372.76

Semi-annually $6,454.81

Quarterly $6,506.71

Monthly $6,535.94

Daily $6,546.49

Continuous $6,549.18

Using the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the balance after t years, P is the principal (amount invested), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

For p = $3000, r = 4%, t = 20 years, and the different compounding frequencies, we get:

Annually: A = $3000(1 + 0.04/1)^(1*20) = $6,372.76

Semi-annually: A = $3000(1 + 0.04/2)^(2*20) = $6,454.81

Quarterly: A = $3000(1 + 0.04/4)^(4*20) = $6,506.71

Monthly: A = $3000(1 + 0.04/12)^(12*20) = $6,535.94

Daily: A = $3000(1 + 0.04/365)^(365*20) = $6,546.49

Continuous: A = $3000e^(0.0420) = $6,549.18 (where e is the constant 2.71828...)

Therefore, the balance a when $3000 is invested at 4% rate for 20 years and compounded n times per year (where n is the different frequencies given) are as mentioned above.

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The general expression would be A_shaded = πr² - (1/2)r²θ(π/180).

To find an expression in terms of π that represents the area of the shaded part of circle R, we need to :
1. Identify the radius (r) of circle R.
2. Determine the area of the entire circle using the formula A_circle = πr².
3. Identify the angle measure (θ) in degrees of the sector corresponding to the shaded part.
4. Convert the angle measure to radians by multiplying by (π/180).

5. Calculate the area of the sector using the formula A_sector = (1/2)r²θ.
6. Subtract the area of the sector from the area of the entire circle to find the area of the shaded part: A_shaded = A_circle - A_sector.
By following these steps, you will obtain an expression in terms of π that represents the area of the shaded part of circle R. However, the general expression would be A_shaded = πr² - (1/2)r²θ(π/180).

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Therefore, Jamison made $96.30 in tips on Friday night.

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, left-hand side (LHS) and right-hand side (RHS), connected by an equal sign (=). The LHS and RHS can contain numbers, variables, operators, and functions, and the equal sign indicates that the value of the expression on the LHS is equal to the value of the expression on the RHS.

Here,

Jamison made $42.80 in tips on Thursday night.

To find out how much he made in tips on Friday night, we need to multiply the Thursday amount by 2 1/4, which is the same as multiplying it by 9/4.

$42.80 x 9/4 = $96.30

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We know that both np and nq are greater than or equal to 10, it is appropriate to use a normal distribution to model the sampling distribution of p^ in this case.

(a) To calculate the mean and standard deviation of the sampling distribution of p^, we'll use the formulas:

Mean (μ) = p
Standard deviation (σ) = √(p*q/n)

where p is the proportion of individuals with allergies (0.12), q is the proportion of individuals without allergies (1 - p = 0.88), and n is the sample size (100).

Mean (μ) = 0.12
Standard deviation (σ) = √(0.12 × 0.88 / 100) = 0.02887

(b) The standard deviation of 0.02887 in this context represents the variability in the proportions of individuals with allergies that we would expect to observe in different random samples of 100 adult patients each. It quantifies the dispersion of p^ around the true population proportion.

(c) To determine if it's appropriate to use a normal distribution to model the sampling distribution of p^, we can check if both np and nq are greater than or equal to 10:

np = 100 × 0.12 = 12
nq = 100 × 0.88 = 88

Since both np and nq are greater than or equal to 10, it is appropriate to use a normal distribution to model the sampling distribution of p^ in this case.

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The paper initial area of the paper is 90 square inches and needs to be folded at least 5 times before its area becomes less than 1 square inch.

a. When n=0, the expression for the area of the paper after folding becomes:

a = 90•(1/3)⁰ = 90•1 = 90 square inches

This means that the initial area of the paper before any folding is 90 square inches.

b. To find out how many folds are needed before the area is less than 1 square inch, we can set the expression for the area of the paper after folding to be less than 1 and solve for n:

a = 90•(1/3)ⁿ < 1

(1/3)ⁿ < 1/90

Taking the logarithm of both sides with base 1/3, we get:

n > log(1/90)/log(1/3)

n > 4.8

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The soccer ball will land approximately 1 seconds after it was kicked upward. This is found by setting the height equation to 0 and solving for t using the quadratic formula.

To solve for the time the soccer ball lands, we need to find the time when h = 0. We can use the given equation

h = -4.9t² + vt + s

where v = 4.9 m/s (since it's kicked upward) and s = 0 (since it starts at ground level).

Substituting those values, we get

0 = -4.9t² + 4.9t

Factoring out 4.9t, we get

0 = 4.9t(-t + 1)

So, either t = 0 or -t + 1 = 0

Since time cannot be negative, we discard the second solution and solve for t

-t + 1 = 0

t = 1

Therefore, the soccer ball lands after approximately 1 second.

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To find the probability that the point lies in the shaded region of the circle, we need to compare the area of the shaded region to the total area of the circle. Let's say the radius of the circle is r.

The area of the shaded region can be found by subtracting the area of the unshaded region from the total area of the circle. The unshaded region is a square with side length equal to the radius of the circle. Therefore, its area is r^2. The total area of the circle is πr^2. So the area of the shaded region is:
πr^2 - r^2 = r^2(π - 1)

Now, if we choose a point at random from the circle, any point has an equal chance of being chosen. So the probability of choosing a point in the shaded region is equal to the area of the shaded region divided by the total area of the circle:

P(shaded) = (r^2(π - 1))/πr^2 = π - 1

Therefore, the probability of choosing a point in the shaded region of the circle is π - 1.

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measure of arc JG = 160 degrees

Main Concept: Intersecting chords

Chords are line segments with ends points that are both on the edge of the circle.  Intersecting chords are a pair of chords on the same circle that intersect.

In an extreme example, the chords may intersect at one of the end points, making the intersecting chords an inscribed angle.

Because Intersecting chords intersect, if the line segments are extended into lines, the lines form two pairs of vertical angles.  Vertical angles are congruent.  Given one vertical angle pair, they will contain two arcs (in the extreme case, the arc will have a measure of zero).

The measure of each of the vertical angles is the average of the two contained arcs.

This problem

For this problem, FG and HJ are chords of the same circle, and they intersect.

If we call the intersection P, angle GPJ is given with a measure of 100 degrees.

Angle GPJ and Angle FPJ form a vertical angle pair, so they are congruent, because vertical angles are congruent.

The measure of each of the vertical angles is the average of the two contained arcs.

The two arcs that this vertical angle pair contain are the arc JG and arc FH.

The measure of arc FH is given as 40 degrees.

Substitute these known quantities into the equation describing the relationship between one of the vertical angles and the contained arcs.

[tex]m \angle GPJ=\frac{1}{2}(m ~\text{arc}JG + m ~\text{arc}FH)[/tex]

[tex](100^o)=\frac{1}{2}(m ~\text{arc}JG + (40^o))[/tex]

Multiply both sides by 2...

[tex]200^o=m ~\text{arc}JG + 40^o[/tex]

Subtract 40 degrees from both sides...

[tex]160^o=m ~\text{arc}JG[/tex]