The tree diagram shows the sample space of​ two-digit numbers that can be created using the digits 9,7,1, and 8.What is the probability of choosing a number from the sample space that contains both 7 and 8.

Answer:-2

Step-by-step explanation:

x(4)+16=8

The given shape is a quadrilateral and a trapezoid.

A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles.

A trapezoid is a quadrilateral with at least one pair of parallel sides.

In a trapezoid, the parallel sides are called the bases of the trapezoid.

The other two sides are called the legs of the trapezoid.

The given shape is not a parallelogram because only one pair of sides are parallel and other sides are not.

Hence, the given shape is a quadrilateral and a trapezoid.

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The equation for the trigonometric graph is:

y = 4 cos(x/2)

We have given is a graph with maximum point at 4 and minimum point at -4 with x intercepts are = -π, π, 3π, 5π, 7π, 9π, 11π.....

We need to identify the equation of the trigonometric graph.

To identify the equation of the given graph with maximum point at 4 and minimum point at -4, we can start by analyzing the characteristics of the cosine function.

The cosine function oscillates between 1 and -1 and has a period of 2π. The general form of the cosine function is:

y = A cos(Bx + C) + D

where A represents the amplitude, B represents the frequency (1/period), C represents the phase shift, and D represents the vertical shift.

Given that the maximum point is at 4 and the minimum point is at -4, we can determine the amplitude, A, to be 4.

y = 4 cos(Bx + C) + D

Next, let's consider the x-intercepts given as -π, π, 3π, 5π, 7π, 9π, 11π, and so on. The x-intercepts of the cosine function occur when the angle inside the cosine function, Bx + C, is equal to (2n + 1)π/2, where n is an integer.

We can start by analyzing the first x-intercept, which is -π. Setting Bx + C equal to -π, we have:

B(-π) + C = -π

Simplifying, we get:

B = -1 - C/π

Similarly, for the x-intercepts at π, 3π, 5π, 7π, 9π, 11π, and so on, we can write the following equations:

Bπ + C = π

B(3π) + C = π

B(5π) + C = π

B(7π) + C = π

B(9π) + C = π

B(11π) + C = π

Now, notice that these equations are satisfied when C is an odd multiple of π, and B = 1/2.

Therefore, we can conclude that:

C = (2n + 1)π

and

B = 1/2

Substituting these values into the equation, we have:

y = 4 cos((1/2)x + (2n + 1)π) + D

Finally, considering the vertical shift, we know that the maximum point is at 4 and the minimum point is at -4. Since the amplitude is 4, the vertical shift is the average of the maximum and minimum points, which is 0. Therefore, D = 0.

The final equation for the trigonometric graph is:

y = 4 cos(x/2)

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Answer(s):

[tex]\displaystyle y = 4sin\:(\frac{1}{2}x + \frac{\pi}{2}) \\ y = -4cos\:(\frac{1}{2}x \pm \pi) \\ y = 4cos\:\frac{1}{2}x[/tex]

Step-by-step explanation:

[tex]\displaystyle y = Asin(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{-\pi} \hookrightarrow \frac{-\frac{\pi}{2}}{\frac{1}{2}} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{4\pi} \hookrightarrow \frac{2}{\frac{1}{2}}\pi \\ Amplitude \hookrightarrow 4[/tex]

OR

[tex]\displaystyle y = Acos(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 0 \\ Horisontal\:[Phase]\:Shift \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{4\pi} \hookrightarrow \frac{2}{\frac{1}{2}}\pi \\ Amplitude \hookrightarrow 4[/tex]

You will need the above information to help you interpret the graph. First off, keep in mind that although this looks EXACTLY like the cosine graph, if you plan on writing your equation as a function of sine, then there WILL be a horisontal shift, meaning that a C-term will be involved. As you can see, the photograph on the right displays the trigonometric graph of [tex]\displaystyle y = 4sin\:\frac{1}{2}x,[/tex] in which you need to replase “cosine” with “sine”, then figure out the appropriate C-term that will make the graph horisontally shift and map onto the cosine graph [photograph on the left], accourding to the horisontal shift formula above. Also keep in mind that the −C gives you the OPPOCITE TERMS OF WHAT THEY REALLY ARE, so you must be careful with your calculations. So, between the two photographs, we can tell that the sine graph [photograph on the right] is shifted [tex]\displaystyle \pi\:units[/tex] to the right, which means that in order to match the cosine graph [photograph on the left], we need to shift the graph BACKWARD [tex]\displaystyle \pi\:units,[/tex] which means the C-term will be negative; so, by perfourming your calculations, you will arrive at [tex]\displaystyle \boxed{-\pi} = \frac{-\frac{\pi}{2}}{\frac{1}{2}}.[/tex] So, the sine equation of the cosine graph, accourding to the horisontal shift, is [tex]\displaystyle y = 4sin\:(\frac{1}{2}x + \frac{\pi}{2}).[/tex] Now, with all that being said, in this case, sinse you ONLY have a graph to wourk with, you MUST figure the period out by using wavelengths. So, looking at where the graph hits [tex]\displaystyle [-2\pi, -4],[/tex] from there to [tex]\displaystyle [-6\pi, -4],[/tex] they are obviously [tex]\displaystyle 4\pi\:units[/tex]apart, telling you that the period of the graph is [tex]\displaystyle 4\pi.[/tex] Now, the amplitude is obvious to figure out because it is the A-term, but of cource, if you want to be certain it is the amplitude, look at the graph to see how low and high each crest extends beyond the midline. The midline is the centre of your graph, also known as the vertical shift, which in this case the centre is at [tex]\displaystyle y = 0,[/tex] in which each crest is extended four units beyond the midline, hence, your amplitude. So, no matter how far the graph shifts vertically, the midline will ALWAYS follow.

I am delighted to assist you at any time.

As of December 31, Deloitte should recognize $15,000 as revenue for the advisory services completed.

To solve the given question, we need to determine the amount of revenue that Deloitte should recognize as of December 31, based on the percentage of services completed.

Here's how we can calculate it:

Calculate the total revenue for the contract:

Total revenue = $20,000

Determine the percentage of services completed:

Percentage of services completed = 75%

Calculate the revenue recognized as of December 31:

Revenue recognized = Percentage of services completed × Total revenue

= 75% × $20,000

= $15,000

Therefore, as of December 31, Deloitte should recognize $15,000 as revenue for the advisory services completed.

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Answer:

Because we are adding 2/5, we would be moving in the positive direction, which is to the right.

Answer:

See below

Step-by-step explanation:

Squaring a number means you multiply it by itself, and the exponent is 2, whereas multiplying a number by 2 means you're doubling it.

Answer:

Step-by-step explanation:

To find out how many people would have an IQ between 106 and 125, we need to calculate the area under the normal distribution curve between these two IQ values. We can do this by calculating the z-scores corresponding to these IQ values and then using a standard normal distribution table or a calculator.

First, let's calculate the z-score for an IQ of 106 using the formula:

z = (x - μ) / σ

where x is the IQ score (106), μ is the mean (100), and σ is the standard deviation (15).

z = (106 - 100) / 15 = 0.4

Next, let's calculate the z-score for an IQ of 125:

z = (x - μ) / σz = (125 - 100) / 15 = 1.67

Using a standard normal distribution table or a calculator, we can find the corresponding cumulative probabilities for these z-scores.

The cumulative probability for a z-score of 0.4 is approximately 0.6554.The cumulative probability for a z-score of 1.67 is approximately 0.9525.

To find the proportion of people with an IQ between 106 and 125, we subtract the cumulative probability corresponding to the lower z-score from the cumulative probability corresponding to the higher z-score:

0.9525 - 0.6554 = 0.2971

This means that approximately 29.71% of the population falls within the IQ range of 106 to 125.

To find out how many people out of the randomly selected 1450 would have an IQ in this range, we multiply the proportion by the sample size:

0.2971 * 1450 ≈ 431.15

Rounding to the nearest whole number, we find that approximately 431 people would have an IQ between 106 and 125 out of the randomly selected sample of 1450 individuals.

Answer:

To solve this question, we'll need to use the properties of normal distribution. Specifically, we'll use the z-scores, which tell us how many standard deviations away from the mean a given value is. The formula for calculating a z-score is:

Z = (X - μ) / σ

where:

- X is the value we're interested in,

- μ is the mean, and

- σ is the standard deviation.

The z-score corresponding to an IQ of 106 is (106 - 100) / 15 ≈ 0.40, and the z-score corresponding to an IQ of 125 is (125 - 100) / 15 ≈ 1.67.

Next, we need to determine the proportion of individuals in a normal distribution that falls between these z-scores. This is found by looking up these z-scores in a standard normal distribution table, or using a software function that provides the cumulative distribution function of the standard normal distribution.

The approximate values are:

- The cumulative probability associated with a z-score of 0.40 is about 0.6554.

- The cumulative probability associated with a z-score of 1.67 is about 0.9525.

The proportion of individuals with IQs between 106 and 125 is the difference between these probabilities, or about 0.9525 - 0.6554 = 0.2971.

Now, to find the number of people out of 1450 with IQs between 106 and 125, we simply multiply this proportion by 1450 and round to the nearest whole number:

0.2971 * 1450 ≈ 431

So, we expect about 431 people out of the randomly selected 1450 to have an IQ between 106 and 125.

The center of mass of the system is approximately (3.7, 2.6).

The center of mass of a system of objects is the point where all the weight of the system appears to be concentrated. It can be defined as the average location of the weighted parts of the system.

The center of mass of a system is dependent on the mass of the objects in the system and their positions.

Let's determine the center of mass of the system with masses of 2, 3, and 5 at the points (-1, 2), (1, 1), and (3, 3), respectively. Let's name the masses m1, m2, and m3, respectively, and the coordinates (x1, y1), (x2, y2), and (x3, y3).

The x-component of the center of mass is given by the formula:

x= (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)

The y-component of the center of mass is given by the formula:

y= (m1y1 + m2y2 + m3y3) / (m1 + m2 + m3)

By using the given values, let's calculate the x and y components of the center of mass:

x = (2 x -1 + 3 x 1 + 5 x 3) / (2 + 3 + 5) = 37/10 ≈ 3.7y

= (2 x 2 + 3 x 1 + 5 x 3) / (2 + 3 + 5)

= 26/10 = 2.6

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Answer:

See below

Step-by-step explanation:

Since the spinner has the numbers 1, 2, and 3 on it, and we want to find the probability of spinning a number less than 2, there is only one possible outcome that satisfies this condition, which is spinning a 1. Therefore, the probability of spinning a number less than 2 is:

P(less than 2) = P(1) = 1/3

So the probability of spinning a number less than 2 is 1/3.

There are 14 Geometry problems in Vardan's homework assignment.

The number of geometry problems in Vardan's homework assignment, we need to calculate 58 1/3 percent of the total number of problems.

First, let's convert 58 1/3 percent to a decimal by dividing it by 100:

58 1/3 percent = 58.33/100 = 0.5833

Next, we multiply the decimal by the total number of problems:

Number of geometry problems = 0.5833 * 24

To calculate this, we can multiply 0.5833 by 24:

Number of geometry problems = 0.5833 * 24 = 14

Therefore, there are 14 geometry problems in Vardan's homework assignment.

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Answer:

theta = n pi and theta = (2n + 1)^pi/2, Where n is an integer.

Step-by-step explanation:

cos(2theta) + (sin(2theta) = 0

2 cos(theta) sin(theta) + 2 sin(theta) cos(theta) = 0

2 sin(theta) cos(theta)(1 + 1) = 0

4 sin(theta) cos(theta) = 0

For sin(theta) = 0.theta = n pi,  Where n is an integer.

For cos(theta) = 0.theta  = (2n + 1)^pi / 2, Where n is an integer

The solution for the original equation is:

theta = n pi, and  theta = (2n + 1)^pi / 2, Where n is an integer.

Hope this helps!

Answer:

[tex]\ln(14k)=\ln(14)+\ln(k)[/tex]

Step-by-step explanation:

Recall the following property of logarithmic function:

[tex]\ln(ab)=\ln(a)+\ln(b)[/tex]

By using the property above, we expand the given logarithmic expression as follows:

[tex]\ln(14k)=\ln(14)+\ln(k)[/tex]

Answer:

To expand the logarithm as much as possible, we need to use the properties of logarithms that allow us to rewrite the expression as a sum, difference, or product of logs. One such property is the product rule, which states that ln(ab) = ln(a) + ln(b) for any positive numbers a and b. Using this rule, we can rewrite ln(14k) as ln(14) + ln(k). This is the most simplified form of the expression, as we cannot further expand ln(14) or ln(k) using the properties of logarithms. Therefore, the final answer is:

ln(14k) = ln(14) + ln(k)

Note that we enclosed the arguments of the logarithm functions in parentheses and included a multiplication sign between 14 and k, as instructed.

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The cost of four points is:4 x $1,370 = $5,480Thus, the four points will cost the homebuyers $5,480.

Points can help lower mortgage rates on fixed-rate loans. The concept of points, which are basically prepaid interest, is a little complicated.

Each point is worth one percent of the loan amount, and paying points can lower your interest rate by a certain amount, typically about one-eighth to one-quarter of a percentage point.

The cost of points in the given scenario can be found using the following steps:

The loan amount to purchase a condominium is $137,000. The homebuyers obtained a 15-year fixed-rate loan with a rate of 5.05%.

If the homebuyers opt for four points, their loan rate will decrease to 4.81%.

To figure out how much the points will cost the homebuyers, we must first determine the cost of one point. Since one point is equal to 1% of the loan amount, one point on a $137,000 loan is:1% of $137,000 = $1,370

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Answer:

2.5

Step-by-step explanation:

The explanation is attached below.

Answer:

195

Step-by-step explanation:

a = 3/2

According to the formula tn= a + (n-1)d

81/2= 3/2 + (13 - 1)d

81/2= 3/2 + 12d

81/3 = 12d

Therefore 27/12 = d

Sn= n/2 [2a + (n-1)d]

[tex]S_{13}[/tex] = 13/2 [2(3/2) + (13-1)(27/12)]

     = 13/2 (3 + 27)

     = 39/2 + 351/2

     = 390/2

     = 195

The total surface area of the cylindrical brass pipe, including the internal surface area, is approximately 34183.08 cm².

The formula for the surface area of a cylinder includes the curved surface area (CSA) and the two circular base areas.

1. Outer Surface Area:

The curved surface area (CSA) of a cylinder is given by the formula: CSA = 2πrh, where r is the radius and h is the height.

Given:

Outer diameter = 10 cm

Outer radius (R) = 10 cm / 2 = 5 cm

Height (h) = 600 cm

Outer CSA = 2π(5)(600) = 6000π cm²

The circular base areas can be calculated using the formula: Base area = πr², where r is the radius.

Outer base area = π(5)² = 25π cm²

Therefore, the total outer surface area is the sum of the curved surface area and the two circular base areas:

Total outer surface area = Outer CSA + 2 * Outer base area = 6000π + 2 * 25π = 6050π cm²

2. Inner Surface Area:

The inner radius (r) is given as 4 cm. We can use the same formulas to calculate the inner surface area.

Inner CSA = 2π(4)(600) = 4800π cm²

Inner base area = π(4)² = 16π cm²

Total inner surface area = Inner CSA + 2 * Inner base area = 4800π + 2 * 16π = 4832π cm²

Total surface area = Total outer surface area + Total inner surface area = 6050π + 4832π = 10882π cm²

To find the numerical value, we can use the approximation π ≈ 3.14:

Total surface area ≈ 10882 * 3.14 = 34183.08 cm²

Therefore, the total surface area of the cylindrical brass pipe, including the internal surface area, is approximately 34183.08 cm².

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Answer:

B) [tex]A=0.5(f\cdot g)(x)[/tex]

Step-by-step explanation:

[tex]\displaystyle A=\frac{1}{2}bh\\\\A=\frac{1}{2}(2x)(x^2+3)\\\\A=\frac{1}{2}g(x)f(x)\\\\A=\frac{1}{2}(f\cdot g)(x)[/tex]

Answer:

Option 2 is the correct answer.

Step-by-step explanation:

The equation that can be used to represent the area of the triangle is:

A(x) = 0.5 * (f ⋅ g)(x)

Let's break it down step by step:

Step 1: Understanding the equation components

f(x) represents the height of the triangle, given as x² + 3.

g(x) represents the base of the triangle, given as 2x.

(f ⋅ g)(x) represents the product of f(x) and g(x), or the multiplication of their respective values at a specific x.

Step 2: Evaluating the equation

To find the area of the triangle when x = 2, we substitute x = 2 into the equation.

A(2) = 0.5 * (f ⋅ g)(2)

Now, let's substitute the functions f(x) and g(x) with their corresponding values at x = 2.

f(2) = 2² + 3 = 4 + 3 = 7

g(2) = 2(2) = 4

Substituting these values into the equation:

A(2) = 0.5 * (7 ⋅ 4)

= 0.5 * 28

= 14

Therefore, when x = 2, the area of the triangle is 14.

Step 3: Interpretation and Conclusion

The equation A(x) = 0.5 * (f ⋅ g)(x) correctly represents the area of the triangle. It takes into account the height function f(x) = x² + 3 and the base function g(x) = 2x, multiplied together and multiplied by 0.5 (or divided by 2), which is a common factor in the formula for the area of a triangle.

Hence, the equation A(x) = 0.5 * (f ⋅ g)(x) can be used to represent the area of the triangle for any given value of x.

Answer:

The lengths are equal so the triangle is equilateral

Step-by-step explanation:

We can write the points as follows,

(2,0), (-1,[tex]\sqrt{3}[/tex]) (-1,-[tex]\sqrt{3}[/tex])

now if it is an equilateral triangle, all side lengths must be equal

first we compute the sides(vectors)

(2-(-1),-[tex]\sqrt{3}[/tex]) = (3,-[tex]\sqrt{3}[/tex]) = side 1

(2-(-1),[tex]\sqrt{3}[/tex]) = (3,[tex]\sqrt{3}[/tex]) = side 2

(-1+1,[tex]\sqrt{3}[/tex]+[tex]\sqrt{3}[/tex]) = (0,2[tex]\sqrt{3}[/tex]) = side 3

now we compute the lengths of the sides using pythagoras theorem

(3)^2 + (-[tex]\sqrt{3}[/tex])^2 = (length of side 1)^2 = 9 + 3 = 12

similarly, (3)^2 + ([tex]\sqrt{3}[/tex])^2 = 12 = Length of side 2 squared

and,( 2[tex]\sqrt{3}[/tex])^2 = length of side 3 squared = 12

since the squares are equal, so the lengths must also be equal

so the triangle is equilateral

this is just a quick addition to the superb posting by "hamza0100" above

well, indeed, in the argand or imaginary plane, for those values above we have the coordinates of A(2 , 0) , B(-1 √3) and C(-1 , -√3), let' use the distance formula for those fellows

[tex]~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ A(\stackrel{x_1}{2}~,~\stackrel{y_1}{0})\qquad B(\stackrel{x_2}{-1}~,~\stackrel{y_2}{\sqrt{3}}) ~\hfill AB=\sqrt{(~~ -1- 2~~)^2 + (~~ \sqrt{3}- 0~~)^2} \\\\\\ ~\hfill AB=\sqrt{( -3)^2 + ( \sqrt{3})^2} \implies \boxed{AB=\sqrt{ 12 }}[/tex]

[tex]B(\stackrel{x_1}{-1}~,~\stackrel{y_1}{\sqrt{3}})\qquad C(\stackrel{x_2}{-1}~,~\stackrel{y_2}{-\sqrt{3}}) \\\\\\ BC=\sqrt{(~~ -1- (-1)~~)^2 + (~~ -\sqrt{3}- \sqrt{3}~~)^2} \\\\\\ ~\hfill BC=\sqrt{( 0)^2 + ( -2\sqrt{3})^2} \implies \boxed{BC=\sqrt{ 12 }}[/tex]

[tex]C(\stackrel{x_1}{-1}~,~\stackrel{y_1}{-\sqrt{3}})\qquad A(\stackrel{x_2}{2}~,~\stackrel{y_2}{0}) ~\hfill CA=\sqrt{(~~ 2- (-1)~~)^2 + (~~ 0- (-\sqrt{3})~~)^2} \\\\\\ ~\hfill CA=\sqrt{( 3)^2 + (-\sqrt{3})^2} \implies \boxed{CA=\sqrt{ 12 }} \\\\[-0.35em] ~\dotfill\\\\ AB=BC=CA=\sqrt{12}\implies 2\sqrt{3}\hspace{5em}\qquad equilateral\textit{\LARGE \checkmark}[/tex]

Step-by-step explanation:

therefore, the correct option should be A.

Answer: 2

Step-by-step explanation: 10 divided by 5 equals 2

The probability of writing the first 7 digits of your phone number is 1/604,800.

Given data ,

To determine the probability of writing the first 7 digits of your phone number, we need to consider the total number of possible outcomes and the favorable outcomes.

Total number of possible outcomes:

When picking 7 digits without replacement from the set of 10 digits (0-9), the total number of possible outcomes is given by 10P7 (permutations of 10 objects taken 7 at a time):

10P7 = 10! / (10 - 7)! = 10! / 3! = 10 * 9 * 8 * 7 * 6 * 5 * 4 = 604,800.

Favorable outcomes:

To have written the first 7 digits of your phone number, you need to pick those specific digits in the correct order. Since the phone number has no repeated digits, there is only one specific order that satisfies this condition.

Therefore, the number of favorable outcomes is 1.

Probability:

The probability is given by the ratio of favorable outcomes to total possible outcomes:

Probability = Favorable outcomes / Total possible outcomes

= 1 / 604,800.

Hence, the probability is 1/604,800.

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Answer:

To find the value of the condominium in 2020, we can work backward from the given information. Since we know that the value appreciates by 15% each year, we can calculate the value for the previous year using the formula:

Value in previous year = Value in current year / (1 + appreciation rate)

Let's calculate the value of the condominium in 2020:

Value in 2021 = $899,300 / (1 + 0.15) ≈ $781,521.74

Value in 2020 = Value in 2021 / (1 + 0.15) ≈ $781,521.74 / (1 + 0.15) ≈ $678,309.34

Therefore, the value of the condominium in 2020 is approximately $678,309.34.

Step-by-step explanation:

Answer: 8C3

Step-by-step explanation: You need to use Combinations for this. Out of 8, you need to select 3, so answer is 8C3.

Multiply three consecutive digits backwards starting from 8, and divide by 3 factorial

(8*7*6)/(3*2*1)

=56

(a) The null hypothesis is that the mean birth weight of babies born at full term is 7.2 pounds. The alternative hypothesis is that the mean birth weight of babies born at full term is greater than 7.2 pounds.

(b) If the scientist decides to reject the null hypothesis, she might be making a Type I error.

(c) A Type II error occurs when the null hypothesis is false, but the scientist fails to reject it.

a A Type I error occurs when the null hypothesis is true, but the scientist rejects it. In this case, the null hypothesis is that the mean birth weight of babies born at full term is 7.2 pounds. If the scientist rejects this hypothesis, she is saying that she believes that the mean birth weight is greater than 7.2 pounds. However, if the null hypothesis is true, then the mean birth weight is actually 7.2 pounds, and the scientist has made a mistake.

b In this case, the scientist would fail to reject the null hypothesis and conclude that the mean birth weight of babies born at full term is 7.2 pounds. However, the true mean birth weight is 7.7 pounds, so the scientist would be making a Type II error.

c In the context of a Type II error, suppose the null hypothesis is false, meaning there is indeed a significant difference or relationship. However, due to various factors such as insufficient sample size, low statistical power, or other limitations, the scientist fails to reject the null hypothesis. Consequently, they accept the null hypothesis even though it is false, leading to a Type II error.

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The fish tank has a total volume of 288 inch³. As a result, Jessica's new fish tank has a capacity of 288 inch³ for water.

The volume of a rectangular prism can be calculated using the formula:

V = l x b x h..........(i)

where,

V ⇒ Volume

l  ⇒ length

b ⇒ width

h ⇒ height

From the question, we are given the values,

l = 8 inches

b = 4 inches

h = 9 inches

Putting these values in equation (i), we get,

V = 8 x 4 x 9

⇒ V = 288 in³

Therefore, the fish tank has a total volume of 288 inch³. As a result, Jessica's new fish tank has a capacity of 288 inch³ for water.

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Whwn the line of sight from the park ranger station, P, to the lifeguard chair, L, the length of PL is 0.76 miles.

It should be noted that since the line of sight from the park ranger station, P, to the lifeguard chair, L, on the beach of a lake is perpendicular to the path joining the campground, C, and the first aid station, F, then the path from the park ranger station to the lifeguard chair is the hypotenuse of a right triangle with legs of 0.35 miles and 0.65 miles.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Therefore, the length of PL is equal to:

= ✓(0.35² + 0.65²)

= 0.76 miles.

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The measure of angle 1 and angle 4 are equal, that is 62°, by vertical angle theorem.

Given that, m∠1=62° and lines t and l intersect.

From the given figure,

m∠1 and m∠4 are vertically opposite angles.

Vertical angles theorem or vertically opposite angles theorem states that two opposite vertical angles formed when two lines intersect each other are always equal (congruent) to each other.

Here, m∠1 =m∠4=62°

Hence, it is proved.

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