Prove each of the following statements using mathematical inductions. (a) Show that + - + · + 2 = 1 - 22 23 for all integer n ≥ 1. 27 272 (b) Show that 89 | (5³n – 6²n) for all integer n ≥ 0. +

Answer:

Okay bestie ‼️

Step-by-step explanation:

We can conclude that, at the 0.01 significance level, there is sufficient evidence to support the claim that the mean rate charged on credit cards is greater than 14%.

The test statistic is calculated as follows:

t = (x - μ) / (s / √n)

In this case, the sample mean is 15.66%, the sample standard deviation is 1.544%, and the sample size is 10. Plugging these values into the formula for the test statistic, we get:

t = (15.66 - 14) / (1.544 / √10)

= 3.4

The critical value is the value of the test statistic that separates the rejection region from the non-rejection region. The critical value for a two-tailed test with a significance level of 0.01 and 9 degrees of freedom (10 - 1 = 9) is 2.821.

Since the test statistic (3.4) is greater than the critical value (2.821), we reject the null hypothesis. This means that there is sufficient evidence to conclude that the mean rate charged is greater than 14%.

We can conclude that, at the 0.01 significance level, there is sufficient evidence to support the claim that the mean rate charged on credit cards is greater than 14%.

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P(A) = 6/36 = 1/6, P(B) = 21/36 = 7/12, P(A|B) = P(A∩B) / P(B) = 6/36 / 21/36 = 6/21 = 2/7.  The outcomes in both A and B are: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}. The number of outcomes in A and B = 6.

4.1 Sample Space and Events in Set Notation:

When a blue and red die are thrown simultaneously, the sample space, denoted by S, consists of all possible outcomes. Since each die has six faces numbered 1 to 6, there are 36 possible outcomes in total.

Sample Space (S): {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

Event A represents the outcomes where the values on both dice are the same:

A = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

Event B represents the outcomes where the value on the red die is greater than or equal to the value on the blue die:

B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (4,4), (4,5), (4,6), (5,5), (5,6), (6,6)}

4.2 Calculating Probabilities:

P(A): To find the probability of event A, we divide the number of favorable outcomes (6) by the total number of outcomes (36).

P(A) = 6/36 = 1/6

P(B): To find the probability of event B, we divide the number of favorable outcomes (21) by the total number of outcomes (36).

P(B) = 21/36 = 7/12

P(A|B): To find the conditional probability of event A given event B, we need to find the probability of A and B occurring together and divide it by the probability of event B.

The outcomes in both A and B are: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

The number of outcomes in A and B = 6.

P(A|B) = P(A∩B) / P(B) = 6/36 / 21/36 = 6/21 = 2/7

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Answer:

141.78 m

Step-by-step explanation:

Let d = distance of boat from the base of the lighthouse

Tan 10 = 25/d

d tan 10 = 25

d = 25/tan 10

d = 141.78 m

Correct question:

The experimental probability that Cindy will catch a fly ball is equal to 3/7. About what percent of the time will Cindy catch a fly ball?

Answer:

42.9%

Step-by-step explanation:

Given that:

Experimental probability of catching a fly is 3/7

This can be interpreted as : Out of 7 tries, Cindy caught a fly only 3 times

Expressing this as a percentage :

3/7 * 100%

0.4285714 * 100%

42.857%

= 42.9%

Hence, Cindy will catch a fly at about 42.9% of the time

Answer:

x = 6.9168 or x = 7

Step-by-step explanation:

Hope that helps :)

Answer:

13x degrees

Step-by-step explanation:

brainliest????!!!

Answer:

oh no homework not done

Step-by-step explanation:

mom:

The roots using the trigonometric formula is -2 + √3

The cubic formula is ax3 + bx2 + cx + d = 0. There is a wondering relation between the roots and the coefficients of a cubic polynomial.

The given function is

f(x) = x3 – 15x – 4

Using the Cardanos method we have

[tex]\sqrt[3]{2+11i} + \sqrt[3]{2-11i}[/tex]

Recall that the sum of the cubic root u of 2+11i with a cubic root u of 2-11i

Such that uv = -15/3 = 5

Now take u = 2+i and v = 2-i The indeed u³ = 2+11i, v³ = 2+11i and uv = 5

Therefore, 4(-u+v) is a root

But now take ω = -1/2 + √3/2i, Then ω² = -1/2 - √3i/2, ω = 1

and if you take u' = ωu, v' ω²v

u'' = ω²u, and v'' = ∈v

Then u' +v and u'' +v'' will be roots too

This means that -2±√3, v' + u' = -2 √3 and u'' + v'' = -2 +√3

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51in

Step-by-step explanation:

15x5=75

8x3=24

75-25=51

Answer:

hi

Step-by-step explanation:

the area of the little rectangle:

3 in × 8 in= 24 in

tge area of the shaded ragion:

5in × 15in =75in

75in - 24 in=51 in

hope it helps

have a nice day

Answer:

(2x+3)(3x+2)

Step-by-step explanation:

.....

......

..............

Answer:

(2x + 3)(3x + 2)

Step-by-step explanation:

[tex]6x^2+13x+6 \\ \\ = 6x^2+9x + 4x+6 \\ \\ = 3x(2x + 3) + 2(2x + 3) \\ \\ = (2x + 3)(3x + 2)[/tex]

Answer:

the bottom choice

Answer:

∠ W = 51°

Step-by-step explanation:

Since WX is the diameter , then

∠ WYX = 90° ( angle in a semicircle )

The sum of the 3 angles in a triangle = 180° , then

x + x + 12 + 90 = 180

2x + 102 = 180 ( subtract 102 from both sides )

2x = 78 ( divide both sides by 2 )

x = 39

Then

∠ W = x + 12 = 39 + 12 = 51°

Given information: The hourly number of emergency telephone calls coming in to a police command and control center has approximately a normal distribution with a mean of 130 and standard deviation of 25. One operator can deal with 24 calls in an hour.

a) The probability that 6 operators will be able to deal with all the calls that arise in an hour is 0.7642.

b) The number of operators should be 203 to ensure that there is sufficient capacity to meet 95% of demand.

c) (i) The probability that 12 operators will be able to deal with all the calls that arise in an hour is 0.7852.

(ii) The number of operators should be 336 to ensure that there is sufficient capacity to meet 95% of demand.

a) Probability that 6 operators will be able to deal with all the calls that arise in an hour.

Mean, µ = 130, Standard Deviation, σ = 25.

Operator can deal with in an hour, n = 24.

Let X = number of emergency calls coming in an hour.

The number of emergency telephone calls coming in to a police command and control center in an hour can be assumed to be Poisson with λ = 130.

Since each operator can handle 24 calls in an hour, therefore, the number of operators required to handle all the calls can be obtained as follows: [tex]$$\frac{X}{24}$$[/tex].

This can be converted to a Standard Normal Variable Z using the formula:[tex]$$Z=\frac{(\frac{X}{24}-\mu)}{\sigma}$$[/tex].

Probability that 6 operators will be able to deal with all the calls that arise in an hour can be calculated as follows:

[tex]$$\begin{aligned} \frac{X}{24} &\leq 6 \\ X &\leq 6 \times 24 \\ X &\leq 144 \end{aligned}$$[/tex]

Now, we need to find the probability of Z ≤ [tex]$$(\frac{144}{24}-130)/25=0.72$$[/tex].

Using normal distribution tables, we get P(Z ≤ 0.72) = 0.7642.

Hence, the probability that 6 operators will be able to deal with all the calls that arise in an hour is 0.7642.

b) To find the number of operators should there be to ensure that there is sufficient capacity to meet 95% of demand.

Let X = number of emergency calls coming in an hour.

The number of emergency telephone calls coming in to a police command and control center in an hour can be assumed to be Poisson with λ = 130.

Since each operator can handle 24 calls in an hour, therefore, the number of operators required to handle all the calls can be obtained as follows:[tex]$$\frac{X}{24}$$[/tex].

This can be converted to a Standard Normal Variable Z using the formula:[tex]$$Z=\frac{(\frac{X}{24}-\mu)}{\sigma}$$[/tex].

To ensure sufficient capacity to meet 95% of demand, we need to find the value of X such that: P(X ≤ x) = 0.95.

Using the Z table, we can find that the probability of Z ≤ 1.645 is 0.95.

Now, we can use the formula:

[tex]$$\frac{X}{24}-130/25=1.645$$[/tex]

[tex]$$X= 1.645\times 25\times 24+130$$[/tex]

[tex]$$X=202.63$$[/tex]

Therefore, the number of operators should be 203 to ensure that there is sufficient capacity to meet 95% of demand.

c) Two centers are combined and let X_1 and X_2 be the number of calls at centers 1 and 2, respectively.

Then the total number of calls, X = X_1 + X_2, follows a normal distribution with

mean = 130 + 130

mean = 260, and

standard deviation = sqrt(25^2 + 25^2)

= 35.36

i) Probability that 12 operators will be able to deal with all the calls that arise in an hour can be calculated as follows:

[tex]$$\begin{aligned} \frac{X}{24} &\leq 12 \\ X &\leq 12 \times 24 \\ X &\leq 288 \end{aligned}$$[/tex]

Now, we need to find the probability of Z ≤ [tex]$$(\frac{288}{24}-260)/35.36=0.789$$[/tex].

Using normal distribution tables, we get P(Z ≤ 0.789) = 0.7852.

Hence, the probability that 12 operators will be able to deal with all the calls that arise in an hour is 0.7852.

ii) To ensure sufficient capacity to meet 95% of demand, we need to find the value of X such that: P(X ≤ x) = 0.95.

Using the Z table, we can find that the probability of Z ≤ 1.645 is 0.95.

Now, we can use the formula:

[tex]$$\frac{X}{24}-260/35.36=1.645$$[/tex]

[tex]$$X= 1.645\times 35.36\times 24+260$$[/tex]

[tex]$$X=335.58$$[/tex]

Therefore, the number of operators should be 336 to ensure that there is sufficient capacity to meet 95% of demand.

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Answer:

8x+5005073

Step-by-step explanation:

Answer:

c. some of the toys are magical

Step-by-step explanation:

The detail in the excerpt that identifies it as fantasy is "Some of the toys are magical." The correct option is 3.

J.R.R. Tolkien's novel The Fellowship of the Ring was published in 1954. The Lord of the Rings is the first book in the epic fantasy series.

"Some of the toys are magical," says the excerpt, identifying it as fantasy. Magical toys imply the presence of magical elements in the story, which is a common feature of fantasy literature.

Other details, such as great gifts being given at a party and the children almost forgetting to eat, are not necessarily unique to fantasy and could be found in other genres.

The fact that the toys were ordered from the Mountain and Dale a year before.

Thus, the correct option is 3.

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The function f(x) = 5x^2 + 20x + 25 can be rewritten in vertex form as f(x) = 5(x + 2)^2 + 5, and its vertex is located at (-2, 5).

To rewrite the function f(x) = 5x^2 + 20x + 25 in vertex form, we complete the square. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.

Let's complete the square:

f(x) = 5(x^2 + 4x) + 25

To complete the square, we need to add and subtract (4/2)^2 = 4 to the expression inside the parentheses:

f(x) = 5(x^2 + 4x + 4 - 4) + 25

Rearranging the terms:

f(x) = 5((x^2 + 4x + 4) - 4) + 25

Now we can factor the perfect square inside the parentheses:

f(x) = 5((x + 2)^2 - 4) + 25

Expanding and simplifying further:

f(x) = 5(x + 2)^2 - 20 + 25

f(x) = 5(x + 2)^2 + 5

Therefore, the function f(x) = 5x^2 + 20x + 25 in vertex form is f(x) = 5(x + 2)^2 + 5. The vertex is given by the coordinates (-2, 5).

The correct question should be :

What is the vertex form of the function f(x) = 5x^2 + 20x + 25? Identify the coordinates of the vertex.

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Answer:

poison

Step by Step Explanation

When you reflect a shape, you flip over an axis or line, the answer is flip.

It is defined as the change in coordinates and the shape of the geometrical body. It is also referred to as a two-dimensional transformation. In the geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, and glide translation.

As we know the reflection will change the orientation not the shape or size after reflection we will get mirror image of the body.

When you reflect a shape, you flip over an axis or line.

Thus, when you reflect a shape, you flip over an axis or line the answer is flip.

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Step-by-step explanation:

As the required line is perpendicular to y = (1/2)x + 5, product of slope of y = (1/2)x + 5 and required line should be - 1.

=> (1/2)m = - 1

=> m = - 2

Thus, required line is

=> y - (-8) = (-2)(x - 3)

=> y + 8 = -2(x - 3)

Answer:

(a−5)(a−7)

Step-by-step explanation:

factor a2−12a+35

a2−12a+35

The middle number is -12 and the last number is 35.

Factoring means we want something like

(a+_)(a+_)

Which numbers go in the blanks?

We need two numbers that...

Add together to get -12

Multiply together to get 35

Can you think of the two numbers?

Try -5 and -7:

-5+-7 = -12

-5*-7 = 35

Fill in the blanks in

(a+_)(a+_)

with -5 and -7 to get...

(a-5)(a-7)

Answer:

hi

Step-by-step explanation:

Answer:

True

Step-by-step explanation:

it does add up to more then 60

It is clear that First-married adults had more positive perceptions of marriage than singles or remarried adults.

Attitudes towards relationships are often studied to determine how they affect people's perception of them. A survey was given to unmarried, currently married, and formerly married adults as part of a broader study of attitudes toward relationships. In this study, it was discovered that first-married adults had more positive attitudes toward marriage than single or remarried adults. The statistical values from the study are provided below:First married adults had more positive perceptions of marriage than singles or remarried adults, F(2, 39) = 5.34, p = 0.042.F stands for F-test, which is a statistical test used to compare whether the means of two or more groups differ from each other significantly. Here, the F-test indicated that there was a statistically significant difference in the attitudes of first-married adults, unmarried adults, and remarried adults towards marriage. Additionally, the p-value is 0.042, which indicates that there is a statistically significant difference between the groups' attitudes towards marriage.

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The sentence given "First married adults had more positive perceptions of marriage than singles or remarried adults, F(2, 39) = 5.34, p = 042" is a claim made in the larger study that was conducted investigating attitudes towards relationships.

The F(2, 39) = 5.34 indicates that the claim is statistically significant and the p-value is less than 0.05, which is the generally accepted level of significance, indicating that the findings are not due to chance.

The terms "part" and "positive" are related to the study but do not specifically apply to this claim. The claim made in this sentence is that first-married adults had more positive perceptions of marriage than singles or remarried adults. The F(2, 39) = 5.34 indicates that the claim is statistically significant. F-statistic is the ratio of between-group variance to within-group variance. Here, the between-group variance is the variance among the perceptions of different types of adults (i.e., first-married, singles, remarried) and the within-group variance is the variance within each group. Since the F-value is statistically significant, we can reject the null hypothesis and accept that there are differences in perceptions of different types of adults. The p-value is the probability of finding such results by chance. Here, the p-value is less than 0.05, which is the generally accepted level of significance, indicating that the findings are not due to chance.

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Answer:

[tex]x=-12[/tex]

Step-by-step explanation:

[tex]3x+5=2x-7[/tex]

Subtract 2x from each side,

[tex]x+5=-7[/tex]

Subtract 5 from each side to find x,

[tex]x=-12[/tex]

Answer:

The deepest that Amy dove below the ocean's surface is 5 feet.

Step-by-step explanation:

Let the deepest that Amy dove below the ocean's surface be x.

7x = 35

x = 5 ft