LESSON 30 SESSION 1
➤ Complete problems 3-5.
3
A spinner has 5 equal-size sections numbered 1 through 5.
The spinner is spun one time.
a. Is it more likely that the spinner will land on an even number or
the number of getting
an odd number? Why?
an odd number
number, therefore greater than getting an even
Id number
b. How likely is it to spin a 1?
c. Why is it just as likely to spin a number greater than 3 as a number less than 3?
4 Use the spinner from problem 3.
a. What are the possible outcomes of spinning the spinner?
b. What are the possible outcomes for the event of spinning a prime number?
c. What are the possible outcomes for the event of spinning a factor of 4?
is likely that the Spinner will land on a
d. What are the possible outcomes for the event of spinning an even
number? An odd number?
5 Suppose you spin the spinner from problem 3 once. Give the possible
outcomes, if any, for each event.
Event
spinning a number less
than or equal to 2
spinning a factor of 6
spinning a 6
2
Outcomes Probability
unlikely
likely
impossible
3
5
4
Vocabulary
event
a set of one or more
outcomes of an
experiment.
outcome
one of the possible
results of a chance
experiment.
probability
a number between 0
and 1 that expresses
the likelihood of an
event occurring.

To calculate the probability that two people chosen at random were born during the same month of the year, we need to consider the total number of possible outcomes and the favorable outcomes.

There are 12 months in a year, so the total number of possible outcomes is 12 (one for each month).

Now, let's consider the favorable outcomes. To have two people born in the same month, we need to choose any one of the 12 months for the first person, and then the second person should also be born in the same month.

The probability that the second person is born in the same month as the first person is 1/12 since there is only one favorable outcome out of 12 possible outcomes.

Therefore, the probability that two people chosen at random were born during the same month of the year is 1/12 or approximately 0.0833 (rounded to four decimal places).

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There is insufficient evidence to conclude that the mean blood pressure for women is different from the mean blood pressure for men. To determine whether the mean blood pressure for women is equal to the mean blood pressure for men, a t-test is conducted using the sample data of 16 women (group 1) and their respective brothers' blood pressures (group 2).

The null hypothesis states that the means are equal, and the alternative hypothesis suggests they are different. The test statistic is calculated, critical points are identified, and the null hypothesis is either accepted or rejected based on the test results.

To carry out the t-test, we first calculate the test statistic. The formula for the test statistic (t) in this case is:

t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Given:

x1 = 119.4 (average blood pressure for women)

x2 = 121.2 (average blood pressure for men)

s2d = 25 (pooled sample variance)

n1 = n2 = 16 (sample sizes)

α = 0.05 (Type I error value)

Now, we can calculate the test statistic:

t = (119.4 - 121.2) / sqrt((25/16) + (25/16))

 = -1.8 / sqrt(3.125 + 3.125)

 = -1.8 / sqrt(6.25)

 = -1.8 / 2.5

 = -0.72

Next, we determine the relevant critical point(s) for the t-test. Since the sample size is small (n1 = n2 = 16), we refer to the t-distribution with degrees of freedom equal to n1 + n2 - 2 = 30 - 2 = 28. Using a significance level (α) of 0.05, the critical value for a two-tailed test is approximately ±2.048.

Since the absolute value of the test statistic (0.72) is less than the critical value (2.048), we fail to reject the null hypothesis.

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(i) (CUD-E) specified using set builder notation:

(CUD-E) = {x ∈ R | (x > 0 ∧ x ≤ 10) ∨ (x > 9 ∧ x ≤ 15) ∧ x ∉ {1, 2, 3}}

(ii) (CE) specified using interval notation and set operations concisely:

(CE) = (0, 10] ∩ {1, 2, 3} = {1, 2, 3}

(iii) (CDF) specified using the most concise notation:

(CDF) = (C ∩ D) ∩ F

(b) Proof using the element argument method:

Given: A and B are sets such that P(A) ≤ P(B).

To prove: A ≤ B.

Proof:

1. Let x be an arbitrary element in A.

2. Since x is in A, by definition, x is a subset of A. Hence, x ⊆ A.

3. Since x ⊆ A and A ≤ B, by the definition of ≤, x ⊆ B.

4. Therefore, x is a subset of B. Hence, x ∈ P(B), where P(B) is the power set of B.

5. Since x ∈ P(B), by definition, x is a subset of B. Hence, x ⊆ B.

6. Since x is an arbitrary element in A and x ⊆ B, by definition, A ≤ B.

7. Therefore, if P(A) ≤ P(B), then A ≤ B.

In this proof, we used the fact that if x is an element of A, then x is a subset of A. Also, if x is a subset of A and A ≤ B, then x is a subset of B. These properties are based on the definitions of subsets and the order relation between sets.

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The experimental probability that Sue will hit the bullseye on her next toss is 2/7.

We have,

The proportion of outcomes where a specific event occurs in all trials, not in a hypothetical sample space but in a real experiment, is known as the empirical probability, relative frequency, or experimental probability of an event.

Here, we have

Given: Sue is playing darts. So far, she has hit the bullseye 4 times and missed the bullseye 10 times.

We have to find the experimental probability that Sue will hit the bullseye on her next toss.

experimental probability = 4/(4+10) = 4/14 = 2/7

The next toss = P = 2/7

Hence,  the experimental probability that Sue will hit the bullseye on her next toss is 2/7.

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complete question:

Sue is playing darts. So far, she has hit the bullseye 4 times and missed the bullseye 10 times. What is the experimental probability that Sue will hit the bullseye on her next toss?

The area of pentagon ABCDE is 36 times the area of pentagon PQRST.

Any five-sided polygon or 5-gon is referred to as a pentagon. The area of pentagon ABCDE is 36 times the area of pentagon PQRST.

We have,

Any five-sided polygon or 5-gon is referred to as a pentagon. A basic pentagon's interior angles add up to 540°. A pentagon might be straightforward or self-intersecting.

We know the formula for the area of a pentagon, therefore, the area of the pentagon PQRST can be written as,

A = 1/4 * √5(5+25)*a²

Given that the side of the side length of pentagon ABCDE is 6 times the side length of pentagon PQRST, therefore, the area of the pentagon ABCDE can be written as,

ABCDE = 1/4 * √5(5+25)* 6a²

ABCDE = 36 * A

Hence, The area of pentagon ABCDE is 36 times the area of pentagon PQRST.

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complete question:

Pentagon ABCDE is similar to pentagon PQRST. If the side length of pentagon ABCDE is 6 times the side length of pentagon PQRST, which

statement is true?

A.

The area of pentagon ABCDE IS 6 times the area of pentagon PQRST.

B.

The area of pentagon ABCDE is 12 times the area of pentagon PQRST.

C.

The area of pentagon ABCDE is 36 times the area of pentagon PQRST.

D.

The area of pentagon ABCDE IS 216 times the area of pentagon PQRST.​

Answer: y=x²+5x−24

Step-by-step explanation:

In the context of the linear search algorithm, the time complexity in the worst-case scenario remains the same, whether the list is sorted or unsorted.

The linear search algorithm has a time complexity of O(n) in the worst case, which means that it takes n steps to search through a list of n elements.
Step-by-step explanation:
1. Start at the first element of the list.
2. Compare the current element with the target value.
3. If the current element is equal to the target value, return the index of the current element.
4. If the current element is not equal to the target value, move on to the next element.
5. Repeat steps 2-4 until you reach the end of the list or find the target value.
In the worst-case scenario, the target value is either at the end of the list or not present in the list. In both sorted and unsorted lists, the algorithm has to traverse the entire list to determine the result. Therefore, there is no asymptotic difference in the worst-case scenario between sorted and unsorted lists when using the linear search algorithm.

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By the Mean Value Theorem, there exists a value c in the interval [1,4] such that f'(c) is equal to the average rate of change of f on the interval [1,4], which is (f(4) - f(1))/(4-1).

We can start by computing f(4) and f(1):

f(4) = 5cos(2(4^2)) + ln(4+1) - 3 = -0.841 + 1.609 - 3 = -1.232

f(1) = 5cos(2(1^2)) + ln(1+1) - 3 = 2.531 - 0.693 - 3 = -1.162

Then, we can compute the average rate of change:

(f(4) - f(1))/(4-1) = (-1.232 - (-1.162))/3 = -0.023

To satisfy the conclusion of the Mean Value Theorem, we need to find a value c in the interval [1,4] such that f'(c) = -0.023. From the given expression for f'(x), we can see that there is no value of c that satisfies this equation, since f'(x) can never be negative. Therefore, there is no value of c that satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4].

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The solutions of the quadratic equation x² - 7x = -12 are x = 3 or x = 4

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

x² - 7x=-12

Express properly

So, we have

x² - 7x = -12

Add 12 to both sides

x² - 7x + 12 = 0

When factored, we have

(x - 3)(x - 4) = 0

Using the zero product property , we have

x - 3 = 0 or x - 4 = 0

Evaluate

x = 3 or x = 4

Hence, the solutions of the quadratic equation are x = 3 or x = 4

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The value of angle x in the kite is 243 degrees.

The diagram above is a kite. The sum of angle in a kite is 360 degrees. Therefore, a kite kites have two sets of equivalent adjacent sides and one set of congruent opposite angles.

Therefore,

360 - 318 + 84 + 2y = 360

where

y are the inner congruent angles

Therefore,

42 + 84 + 2y = 360

2y = 360 - 126

2y = 234

divide both sides of the equation by 2

y = 234  /2

y = 117 degrees

Therefore,

x = 360 - 117

x = 243 degrees

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Answer:

51

Step-by-step explanation:

Sum of interior angles in a triangle is 180.

x + 90 + 39 = 180

x + 129 = 180

x = 180 - 129

x = 51

Answer:

51 degrees

Step-by-step explanation:

We are asked to find angle x.

To find angle x, we have to write an equation to find it.

We know that one angle is 39 degrees, and the other is 90 degrees because of the small box.  We also know that all 3 angles in a triangle have to add up to 180.

Here's the equation for this:

180=39+90+x

simplify

180=129+x

subtract 129 from both sides

51=x

So, angle x is 51 degrees.

Hope this helps! :)

If the universal set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} then (A ∪ B) ∩ C = {1, 3, 5}, A' U B = {0, 2, 4, 6, 7, 8, 9} and  (A ∩ C)' = {0, 2, 4, 6, 7, 8, 9}.

The universal set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

To find (A ∪ B) ∩ C, we first need to find A ∪ B and then find the intersection with C.

A ∪ B is the set of all elements that are in A or B, so:

A ∪ B = {1, 2, 3, 4, 5, 6, 8}

Now we need to find the intersection of A ∪ B and C:

(A ∪ B) ∩ C = {1, 3, 5}

Therefore, (A ∪ B) ∩ C = {1, 3, 5}.

b. A' is the complement of A, which means it is the set of all elements in U that are not in A.

A' = {0, 6, 7, 8, 9}

A' U B is the set of all elements that are in A' or B, so:

A' U B = {0, 2, 4, 6, 7, 8, 9}

Therefore, A' U B = {0, 2, 4, 6, 7, 8, 9}.

c. A ∩ C is the set of all elements that are in both A and C:

A ∩ C = {1, 3, 5}

(A ∩ C)' is the complement of A ∩ C, which means it is the set of all elements in U that are not in A ∩ C:

(A ∩ C)' = {0, 2, 4, 6, 7, 8, 9}

Therefore, (A ∩ C)' = {0, 2, 4, 6, 7, 8, 9}.

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Answer:

113.1 cm³

Step-by-step explanation:

diameter = 2 X radius

Volume of sphere = (4/3) X π X r ³

= (4/3) π (3)³

= 36π

= 113.1 cm³ to nearest tenth

The correct answer is: Nicole’s rate is 45 mi/h, and Kim’s rate is 40 mi/h.

Nicole and Kim are driving towards each other at a combined speed of 170 miles in 2 hours, so their average speed is 85 miles per hour. Let's assume that Kim's speed is x miles per hour, then Nicole's speed is x+5 miles per hour.

So, the equation we get from their combined speed is:

x + (x+5) = 85

Simplifying the equation, we get:

2x + 5 = 85

2x = 80

x = 40

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The value of the measure of Arc CDE is,

⇒ Arc CDE = 128 degree

Since, An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.

Here, A circle is shown in figure.

And, By circle we have;

The measure of Arc CDE is,

⇒ Arc CDE = 128 degree

Thus, The value of the measure of Arc CDE is,

⇒ Arc CDE = 128 degree

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The probability that the average height of 3 young women is at most 63 inches is approximately 12.38%. Here option A is the correct answer.

To calculate the probability that the average height of 3 young women is at most 63 inches, we need to use the central limit theorem, which states that the distribution of the sample means of a sufficiently large sample from any population with a finite mean and variance will be approximately normally distributed.

Assuming the heights of young women follow a normal distribution, with a mean of μ and a standard deviation of σ, we can calculate the probability using the standard normal distribution table or a statistical software package.

First, we need to calculate the mean and standard deviation of the sample mean. The mean of the sample mean is equal to the population mean, μ, which we assume to be 65 inches. The standard deviation of the sample mean is equal to the population standard deviation divided by the square root of the sample size, which is 3 in this case. Assuming a standard deviation of 3 inches, the standard deviation of the sample mean is 3 / sqrt(3) = 1.73 inches.

Next, we need to calculate the z-score for a sample mean of 63 inches:

z = (63 - 65) / 1.73 = -1.16

Using the standard normal distribution table, we can find the probability that a z-score is less than or equal to -1.16. The probability is 0.1238, or approximately 12.38%.

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Complete question:

If we select 3 young women at random, what is the probability that their average height is shorter than / at most 63 inches (that is, they are at most 63 inches tall, on average)?

A - 12.38

B - 13.38

C - 15.48

D - 17.40

Answer:

In a two-tailed or nondirectional test, the alternative hypothesis claims its parameters don't equal the null hypothesis value. This means the two-tailed directional test states there are differences present that are greater than and less than the null value.

Step-by-step explanation:

have a nice day.

Models should be produced of the function C(x) = 10 + 2 x is 329.4 .

Cost of manufacturing is

C(x) = 10 + 2 x

Sale price in soles of each model is

P(x) = 20 - [tex]\frac{6x^{2} }{800}[/tex]

U(x) is the utility function

U(x) = x P(x) - C(x)

U(x) = x (20 - [tex]\frac{6x^{2} }{800}[/tex]  ) - (10 +2x)

U(x) = 20x - [tex]\frac{6x^{3} }{800}[/tex]   - 10 - 2x

U(x) = 18x - [tex]\frac{6x^{3} }{800}[/tex] - 10

U'(x) = 18 - 18x²/800

For maximum model U'(x) = 0

18 - 18x²/800 = 0

18x²/800 = 18

x² = 800

x = √800

x = 20√2

U(x) = 18(20√2 ) - [tex]\frac{6(20\sqrt{2} )^{2} }{800}[/tex] - 10

U(x) = 329.5

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The question is in Spanish question in English :

The manufacturing costs of models are modeled to the following function. C(x) = 10 + 2x. The manufacturer estimates that the sale price in soles of each model is given by: P(x) = 20- 6x2/800 How many models should be produced?

Answer:

Step-by-step explanation:

a= 3(x+11)

B=9(x+8)

C=5(x+5)

D=3(3x+2)

Answer:

4.2

Step-by-step explanation:

5x+3-8-9=7

5x=7-3+8+9

5x=21

X =4.2

The correct 4-bit combination for the decimal number 9 is 1001. To explain it in a long answer, we need to understand binary representation. In binary, each digit can either be 0 or 1, and the value of the digit depends on its position.

The rightmost digit represents the value 2^0 (which is 1), the next digit to the left represents the value 2^1 (which is 2), the next represents 2^2 (which is 4), and so on. To convert decimal number 9 to binary, we can start by finding the highest power of 2 that is less than or equal to 9, which is 2^3 (which is 8). We can subtract 8 from 9, and the remainder is 1. This means the leftmost digit in the binary representation is 1.

We repeat the same process with the remainder, which is 1, and find the highest power of 2 that is less than or equal to 1, which is 2^0 (which is 1). We subtract 1 from 1, and the remainder is 0. This means the rightmost digit in the binary representation is 0. Thus, the binary representation of decimal number 9 is 1001, which is the correct 4-bit combination.

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We can expect approximately 9 rats to die after being injected with cancerous cells based on probability.

To answer your question, we will use the given probability and the number of rats injected to find the expected number of rats that would die.

1. The probability that a rat injected with cancerous cells will live is 0.8.
2. Therefore, the probability that a rat will die is 1 - 0.8 = 0.2 (since the sum of probabilities of all possible outcomes should be equal to 1).
3. We have 45 rats injected with cancerous cells.
4. To find the expected number of rats that would die, multiply the total number of rats by the probability of a rat dying: 45 rats * 0.2 = 9 rats.

So, we can expect approximately 9 rats to die after being injected with cancerous cells.

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Therefore, if the price of the product is increased by 2 units, the demand will decrease by 6 units.

To determine the change in demand when the price is increased by 2 units, we substitute the new price into the demand equation and compare it to the original demand.

Given:

ŷ = 120 - 3x

Let's assume the original price is denoted by x, and the new price is x + 2.

Original demand:

y = ŷ

= 120 - 3x

New demand:

y' = ŷ'

= 120 - 3(x + 2)

= 120 - 3x - 6

= 114 - 3x

Comparing the original demand (y = 120 - 3x) with the new demand (y' = 114 - 3x), we can see that the demand decreases by 6 units when the price is increased by 2 units.

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The probabilities of hitting the next bat are  Mitchell = 5/12 and Travis= 9/20

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

Mitchell hits 5 out of 12 times

Travis hits 9 out of 20 times

The probabilities of hitting the next bat is calculated as

P(Hit) = Hit/Total number of times

using the above as a guide, we have the following:

P(Mitchell Hit) = 5/12

P(Travis Hit) = 9/20

Hence, the probabilities of hitting the next bat are 5/12 and 9/20

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Answer:

Avg BC= 10.873 m

Step-by-step explanation:

See a picture

The system of first-order equations that is equivalent to the given second-order differential equation is dy/dt = z, dz/dt = w, and dw/dt = (-3z - 2w - 6y)/t².

To write the given second-order differential equation as a system of first-order equations, we need to introduce new variables.

Let z = dy/dt. Then, we can rewrite the given equation as

d³y/dt³ = dz/dt

d²y/dt² = dz/dt = z

Substituting these expressions into the original equation, we get

(t²) (d³y/dt³) + 3(d²y/dt²) + 2(dy/dt) + 6y = t² (dz/dt) + 3z + 2(dy/dt) + 6y

Simplifying and grouping the terms, we obtain

d/dt [y, z, w] = [z, w, (-3z - 2w - 6y)/t²]

where w = dt/dt = 1.

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[tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )[/tex] can be expressed in polar form as[tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )=\underline{\frac{\sqrt{30} }{3} } \ cis\ (\underline{\frac{4\pi}{3}})[/tex] . Therefore the values to be dragged in the box are [tex]\frac{\sqrt{30} }{3}[/tex]  and [tex]\frac{4\pi}{3}[/tex].

We have to express in polar form, polar form of complex number:

[tex]r(cos\theta+isin\theta) \rightarrow rcis\theta[/tex]

where, r = modulus of complex number

[tex]\theta[/tex] = argument of complex number

The division of two complex number, [tex]z=[/tex] [tex]r_{1} cis \theta_{1}[/tex] and [tex]x=[/tex] [tex]r_{2} cis \theta_{2}[/tex]

[tex]\frac{z}{x} = \frac{r_{1} }{r_{2} }\ cis(\theta_{1}- \theta_{2})[/tex]

Similarly, let a = [tex]6\sqrt{5} \ cis(\frac{11\pi}{6})[/tex]

                    b = [tex]3\sqrt{6} \ cis(\frac{pi}{2})[/tex]

[tex]\frac{a}{b}= \frac{6\sqrt{5} }{3\sqrt{6} } \ cis (\frac{11\pi }{6}- \frac{\pi}{2} )[/tex]

 [tex]= \frac{{\sqrt{2}}\times\sqrt{2}\times\sqrt{5} }{\sqrt{2} \times\sqrt{3} } \ cis(\frac{11\pi-3\pi}{6} )[/tex]

 [tex]=\sqrt{\frac{10}{3} }\ cis\ \frac{8\pi}{6}[/tex]

[tex]\frac{a}{b}= \sqrt{\frac{10}{3} } \ cis\ \frac{4\pi}{3}[/tex]

It can also be written as, [tex]\frac{a}{b}= \frac{\sqrt{30} }{3} \ cis\ \frac{4\pi}{3}[/tex]

⇒ [tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )= \frac{\sqrt{30} }{3} \ cis\ \frac{4\pi}{3}[/tex]

Comparing it with the question we get:

[tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )=\underline{\frac{\sqrt{30} }{3} } \ cis\ (\underline{\frac{4\pi}{3}})[/tex]

Therefore, the first blank is [tex]\frac{\sqrt{30} }{3}[/tex] and the second blank is [tex]\frac{4\pi}{3}[/tex].

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