Use the Ratio Test to determine whether the series is convergent or divergent. Σ n=1 (-1)^n + 1 n^6 6^n/n!Identify an_____Evaluate the following limit. lim n--> [infinity] |an + 1|/|an|

Answer:

x = 7.15

Step-by-step explanation:

Start off by solving for the missing angle:

180° - 51° - 90° = 39°

Now knowing this angle, we can use the Law of Sines to solve for x.

sin(90°)/x= sin(39°)/4.5

Isolate x.

x = sin(90°)*4.5/sin(39°) ≈ 7.15

P.S. This is just my way of solving for x, be open-minded to other ways to solve for x.

The result of the equation 2.34 x 10²⁴ + 1.92 x 10²³ is 2.532 x 10²⁴.

To solve this given equation,

One first needs to take the common exponent out in both numbers

i.e. we need to take common from 2.34 x 10²⁴ and 1.92 x 10²³ which comes out to be 10²³

Therefore, using the distributive property of multiplication that states ax + bx = x (a+b)

we have, 2.34 x 10²⁴ + 1.92 x 10²³ = 10²³ (2.34 x 10 + 1.92)

= 10²³ (23.4 + 1.92)

=10²³ x 25.32

We convert this into proper decimal notation, and we get

=2.532 x 10²⁴

Therefore, we get 2.532 x 10²⁴ as the result of the given equation.

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Given the angles, the value of x in the triangle is 8

From the question, we have the following parameters

Angles (5x + 1)°, (4x - 5) and (14x)

By the theorem of adding angles in a triangle, we have

5x + 1 + 4x - 5 + 14x = 180

When evaluated, we have

23x = 184

Divide through the equation by 23

x = 8

Hence, the value of x is 8

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The probability that the student will get at least three questions correct is 0.1719

To solve this problem, we can use the binomial distribution formula, which gives the probability of getting exactly k successes in n independent Bernoulli trials, where each trial has a probability p of success:[tex]P(k successes) = (n choose k) p^k (1-p)^{n-k}[/tex]

In this case, n = 10 (the number of questions), p = 0.5 (the probability of getting a correct answer by guessing), and we want to find the probability of getting at least three questions correct. This means we need to add up the probabilities of getting exactly 3, 4, 5, ..., 10 questions correct.

[tex]=P(at least 3 correct) = P(3 correct) + P(4 correct) + ... + P(10 correct)[/tex]

[tex]= (10 choose 3) (0.5^3) (0.5^7 )+ (10 choose 4) (0.5^4) (0.5^6) + ... + (10 choose 10) (0.5^{10} )(0.5^0)[/tex]

[tex]= 0.1719[/tex]

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The statement "Investing $20,000 for 10 years at 12.25% compounded monthly will have a better yield than investing $20,000 at 8.5% compounded quarterly for 30 years." is false.

What is Compound Interest?

Compound interest is a type of interest that takes into account both the principal and the interest accrued during the previous period. This differs from simple interest, where only the principal is considered in the calculation of interest for each period. In mathematics, compound interest is often abbreviated as C.I.

To compare the yield of two investments with different interest rates, we need to calculate the future value of each investment at the end of its respective term.

For the first investment -

Principal = $20,000

To calculate the future value of an investment with monthly compounding, we need to determine the interest rate per month by dividing the annual interest rate by 12. For example, an annual interest rate of 12.25% would correspond to a monthly interest rate of 1.0208%.

Next, we need to determine the number of compounding periods, which is equal to the number of years multiplied by the number of compounding periods per year. In this case, a term of 10 years would correspond to 120 monthly compounding periods.

Once we have determined the interest rate and the number of compounding periods, we can use the formula for future value of a monthly compounded investment to calculate the value of the investment at the end of the term. It is important to use a reliable formula to ensure accurate calculations and to compare the yields of different investments effectively.

[tex]FV = $20,000 \times (1 + 0.010208)^{120} = $68,398.61[/tex]

For the second investment -

Principal = $20,000

To calculate the future value of an investment with quarterly compounding, we first need to determine the interest rate per quarter by dividing the annual interest rate by 4. For instance, an annual interest rate of 8.5% would correspond to a quarterly interest rate of 2.125%.

The number of compounding periods can be calculated by multiplying the number of years by the number of compounding periods per year. In this case, a term of 30 years would correspond to 120 quarterly compounding periods.

Using the formula for future value of a quarterly compounded investment, we can determine the value of the investment at the end of the term. It is essential to use an accurate formula to ensure that the calculations are reliable and that the yields of different investments can be compared effectively.

[tex]FV = $20,000 \times (1 + 0.02125)^{120} = $89,432.63[/tex]

Comparing the yields of different investments involves determining which investment will produce the higher future value at the end of the term. In this case, the two investments being compared are investing $20,000 for 10 years at 12.25% compounded monthly and investing $20,000 at 8.5% compounded quarterly for 30 years.

After calculating the future values of both investments using the appropriate formulas, it is found that the investment with the better yield is the one with the higher future value, which is the investment at 8.5% compounded quarterly for 30 years. Therefore, the statement "Investing $20,000 for 10 years at 12.25% compounded monthly will have a better yield than investing $20,000 at 8.5% compounded quarterly for 30 years" is false.

It is crucial to accurately compare the yields of different investments to make informed financial decisions and ensure the best returns on investments. Using reliable formulas and techniques can help ensure accurate calculations and better investment outcomes.

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The characteristic of independence or lack of redundancy has been violated in creating the index of "religious fundamentalism" by including an indicator of political conservatism.

Indexes and scales used in research are typically designed to measure specific constructs or concepts. One important characteristic of indexes and scales is independence or lack of redundancy, which means that each indicator or item included in the index should contribute unique and distinct information to the measurement of the construct. Including indicators that are redundant or overlapping violates this characteristic.

In this case, including an indicator of political conservatism in the index of "religious fundamentalism" may violate the characteristic of independence or lack of redundancy. This is because political conservatism and religious fundamentalism are distinct concepts, although they may be related or correlated in some cases. By including an indicator of political conservatism in the index of religious fundamentalism, the researcher may be overlapping or duplicating some of the measurement of the construct of religious fundamentalism with the measurement of political conservatism.

Therefore, including an indicator of political conservatism in the index of religious fundamentalism violates the characteristic of independence or lack of redundancy in index construction.

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Answer:

A 400 sq in

Step-by-step explanation:

10^2*3.14=314, I just rounded to the closest answer sorry if it's wrong.

Answer:

Response

The variable we are trying to predict.

Synonyms

dependent variable, Y-variable, target, outcome

Independent variable

The variable used to predict the response.

Synonyms

X-variable, feature, attribute

Record

The vector of predictor and outcome values for a specific individual or case.

Synonyms

row, case, instance, example

Intercept

The intercept of the regression line—that is, the predicted value when

�

=

0

.

Synonyms

�

0

,

�

0

Regression coefficient

The slope of the regression line.

Synonyms

slope,

�

1

,

�

1

, parameter estimates, weights

Fitted values

The estimates

�

^

�

obtained from the regression line.

Synonyms

predicted values

Residuals

The difference between the observed values and the fitted values.

Synonyms

errors

Least squares

The method of fitting a regression by minimizing the sum of squared residuals.

Synonyms

ordinary least squares

TRUE

Step-by-step explanation:

The observed value of the Pearson Goodness of Fit test statistic is 5.23.

The correct answer is not among the answer choices.

To calculate the observed value of the Pearson Goodness of Fit test

statistic, we need to first calculate the expected frequencies for each

category (MM, FF, and MF).

The expected frequency for MM is:

[tex]E(MM) = 50 \times P(MM) = 50 \times a^2 = 50 \times 0.04a = 2a[/tex]

Similarly, the expected frequency for FF is:

[tex]E(FF) = 50 \times P(FF) = 50 x a^2 = 2a[/tex]

And the expected frequency for MF is:

[tex]E(MF) = 50 \times P(MF) = 50 x (1 - a^2) = 50 - 50a^2[/tex]

Using the formula for the Pearson Goodness of Fit test statistic:

[tex]x^2[/tex] =[tex]\sum (O-E)^2 / E[/tex]

where O is the observed frequency and E is the expected frequency.

We can calculate the observed value of the test statistic as follows:

[tex]x^2 = [(16 - 2a)^2 / 2a] + [(14 - 2a)^2 / 2a] + [(20 - 50a^2)^2 / (50 - 50a^2)][/tex]

Substituting the value of â = 0.2, we get:

[tex]x^2 = [(16 - 0.4)^2 / 0.4] + [(14 - 0.4)^2 / 0.4] + [(20 - 8)^2 / 42][/tex]

[tex]= 0.6^2 / 0.4 + 0.6^2 / 0.4 + 12^2 / 42[/tex]

= 0.9 + 0.9 + 3.43

= 5.23

Therefore, the observed value of the Pearson Goodness of Fit test

statistic is 5.23.

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D. The statement indicates that the interval 11% +4% is likely to contain the true population percentage of people that prefer chocolate pie.

Population is the total number of people or inhabitants of a particular area or place. It can refer to any living organism, but usually refers to humans. Population density, which is the number of people per unit of area, is another important factor in population. Population growth is the rate at which a population increases over time. Population growth is impacted by factors such as birth and death rates, immigration, and net migration. Population size and density can also be impacted by external factors such as climate change, natural disasters, and wars. Population data is used for a variety of purposes, including analyzing economic and environmental trends, forecasting, and policymaking.

The margin of error of +4 percentage points indicates that the study is confident that the true population percentage of people that prefer chocolate pie is within 11% +4%, or 11% - 4%. This means that the study is confident that the true population percentage of people that prefer chocolate pie is likely to be within this range.

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Time taken by the gas pump to fill the empty gas tank will be 6 minutes. First, we need to calculate the entire amount of petrol required—18 gallons—to fill the empty gas tank.

So, we must determine how many quarter gallons there are in 18 gallons:18 gallons times four quarter gallons per gallon equals 72 quarter gallons.

The next thing to determine is how many quarters of a gallon can be pumped by the gas pump in one second:

1/5 gallons per second multiplied by 4 quarters per gallon equals 0.8 quarters per second.

To sum up, we can apply the formula:

Time is equal to the gas consumption rate.

to determine the time needed to fill the petrol tank. 72 quarter gallons of gas are being pumped at a rate of 0.8 quarters per second:

90 seconds are equal to time divided by 72.

Therefore, it will take the gas pump 90 seconds to fill the empty gas tank. To convert this to minutes, we can divide by 60:

90 seconds / 60 seconds/minute = 1.5 minutes

So, the answer is not one of the options given. However, if we round up, the answer is A. 6 minutes.

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The derivative of the function. h(x) = 9/x^9 - 7/x^7 + 3√xh'(x) is [tex]h'(x) = -81/x^{10} + 49/x^8 + (3/2)x^{-1/2}[/tex]

To find the derivative of the function h(x) = 9/x^9 - 7/x^7 + 3√x, we will use the power rule and the chain rule.

First, using the power rule, we have:

h'(x) = [tex]d/dx [9/x^9] - d/dx [7/x^7] + d/dx [3√x][/tex]

[tex]= (-99)/x^{10} + (77)/x^8 + (3/2)x^{-1/2}[/tex]

For the third term 3√x, we use the chain rule, which states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x), where g'(h(x)) is the derivative of the outer function and h'(x) is the derivative of the inner function.

Simplifying this expression, we get:

[tex]h'(x) = -81/x^{10} + 49/x^8 + (3/2)x^{-1/2}[/tex]

Therefore, the derivative of h(x) is h'(x) = -81/x^10 + 49/x^8 + (3/2)x^(-1/2).

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Sampling is the use of a subset of the population to represent the whole population or to inform about processes that are meaningful beyond the particular cases, individuals or sites studied.

In non-probability sampling, the sample is selected based on non-random criteria, and not every member of the population has a chance of being included. Common non-probability sampling methods include convenience sampling, voluntary response sampling, purposive sampling, snowball sampling, and quota sampling.

The sum of the first 104 terms of the series is -798. 75

Note that the sum of all the terms in any sequence is the sum of the values from the first term to the last term.

Also note that an arithmetic sequence is defined as a sequence in which the consecutive terms differs with a common term called the common difference.

From the information given, we have that;

The sum of the terms takes the function;

an = -17.25(n-1)+978

Then, the sum of the first 104 terms would be;

a(104) = -17. 25 ( 104 -1 ) +978

expand the bracket

a(104) = -17. 25(103) + 978

a(104) = -1776. 75 + 978

add the values

a(104) = -798. 75

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The probability of having a girl or a boy is 1/2 or 0.5. Therefore, the probability of having three girls in a row is (0.5)^3 = 0.125.

However, we are given that the third child is a girl, so we can disregard the other two outcomes (GG and GB). Thus, the probability that all three children are girls given that the third one is a girl is simply 0.5 or 50%.

To find the probability of all three children being girls, we only need to consider the probabilities of the first two children being girls, as the third one is already given as a girl.

Step 1: Find the probability of the first child being a girl:
P(First child = Girl) = 1/2

Step 2: Find the probability of the second child being a girl:
P(Second child = Girl) = 1/2

Step 3: Find the probability of both the first and second child being girls:
P(All children = Girls | Third child = Girl) = P(First child = Girl) * P(Second child = Girl) = (1/2) * (1/2) = 1/4

So, the probability that all three children are girls, given that the third one is a girl, is 1/4 or 0.25 or 0.125

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The expression that shows how many toy cars were in each package is 10p

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

This means that

Expression = Number of toy cars * Number of packages

Substitute the known values in the above equation, so, we have the following representation

Expression = 10 * p

Evaluate

Expression = 10p

Hence. the expression is 10p

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A correlation coefficient of -0.96 tells you that there is a strong negative linear relationship between the two variables being analyzed.

In other words, as one variable increases, the other variable tends to decrease, and vice versa. The correlation coefficient ranges from -1 to 1, and a value close to -1 or 1 indicates a strong relationship, while a value close to 0 indicates a weak relationship. In this case, -0.96 is close to -1, signifying a strong negative relationship.

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No, this would not result in an unbiased sample because it only includes the students you happen to meet, which could introduce sampling bias.

It is possible that you would be more likely to encounter certain types of students.

Such as those who are more outgoing or those who are on campus more frequently, which could skew the results.

To obtain an unbiased sample, you would need to use a more systematic and representative sampling method.

Such as selecting a random sample of students from the school's records and asking them about their age.

Using an online survey to collect age data from all students enrolled in both traditional and online courses.

It's probable that you'd run into specific student types more frequently.

For instance, those who are more talkative or those who attend campus more regularly, which can distort the results.

You would need to employ a more methodical and representative sampling technique in order to achieve an impartial sample.

Using a random sample of kids and asking them about their ages from the student database at the school.

collecting age information from all students enrolled in both traditional and online courses using an online survey.

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The probability that the mean number of daily emergency admissions is between 75 and 95 is approximately 0.8990.

To find the probability that the mean number of daily emergency admissions is between 75 and 95, we can use the Z-score formula for sample means: Z = (X - μ) / (σ / √n), where X is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size.

First, calculate the Z-scores for both 75 and 95:
Z_75 = (75 - 85) / (37 / √30) ≈ -1.62
Z_95 = (95 - 85) / (37 / √30) ≈ 1.62

Now, use a Z-table to find the probabilities corresponding to these Z-scores. P(Z ≤ 1.62) ≈ 0.9474 and P(Z ≤ -1.62) ≈ 0.0526.

Finally, subtract the probabilities to find the probability between the two Z-scores:
P(-1.62 ≤ Z ≤ 1.62) = P(Z ≤ 1.62) - P(Z ≤ -1.62) ≈ 0.9474 - 0.0526 ≈ 0.8948

Among the given answer choices, the closest value is 0.8990.

Therefore, the probability that the mean number of daily emergency admissions is between 75 and 95 is approximately 0.8990.

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The statement "The 90% confidence interval for the mean is (29.83, 50.1)" can be interpreted to mean that the probability that the mean lies in the range (29.83, 50.1) is 90%. True.

A 90% confidence interval is a range within which we can be 90% confident that the population mean lies. In this case, the interval is (29.83, 50.1).

It does not mean that there is a 90% chance that the mean lies in this range; rather, it indicates that if we were to repeatedly draw random samples from the population and construct confidence intervals in the same manner, 90% of those intervals would contain the true population mean.

This interpretation emphasizes the reliability of the estimation method over the probability of the mean falling within a specific range.

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The limit you are looking for is lim(x→0) (e^x - 1)/x. Using L'Hôpital's rule, since this is an indeterminate form of 0/0, we can find the limit by taking the derivative of both the numerator and denominator with respect to x.
The derivative of e^x is e^x, and the derivative of 1 is 0, so the derivative of the numerator is e^x. The derivative of x is 1.
Now, we have the limit lim(x→0) (e^x)/1. When x approaches 0, e^x approaches e^0 which is equal to 1. Therefore, the limit is:
1 (Answer C)

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When a transistor is selected at random from the bin and it does not immediately fail, the probability that it is acceptable is 0.6.

We know that, the bin contains 15 defective transistors, 20 partially defective transistors, and 30 acceptable transistors.

If we randomly select a transistor from the bin, the probability that it is not defective is:

P(not defective) = P(partially defective) + P(acceptable)

P(not defective) = 20/65 + 30/65

P(not defective) = 50/65

So, the probability that the selected transistor is acceptable, given that it is not defective, can be calculated using Bayes' theorem:

P(acceptable | not defective) = P(not defective | acceptable) x P(acceptable) / P(not defective)

P(not defective | acceptable) is simply 1,

since an acceptable transistor will not immediately fail when put into use.

So, we have:

P(acceptable | not defective) = 1 x 30/65 / (50/65)

P(acceptable | not defective) = 30/50

P(acceptable | not defective) = 0.6

Therefore, When a transistor is selected at random from the bin and it does not immediately fail, the probability that it is acceptable is 0.6.

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Since the projectile is fired at a speed of 300 m/s and lands at a distance of 8000 m away. At approximately 30.55° angle in degrees is the projectile fired

To solve this problem, we can use the formula for the range of a projectile:

R = (v^2/g)*sin(2θ)

where R is the range (in this case, 8000 m), v is the initial speed (300 m/s), g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle at which the projectile is fired.

Here,
R = 8000 m (distance)
v = 300 m/s (speed)
g = 9.8 m/s² (acceleration due to gravity)

We can rearrange this equation to solve for θ:
θ = 1/2 * sin^-1 (R*g/v^2)

Plugging in the values we know, we get:
θ = (1/2) * arcsin(8000 * 9.8 / (300^2))
θ = (1/2) * arcsin(78400 / 90000)
θ = (1/2) * arcsin(0.8711)
θ = (1/2) * 61.11°
θ ≈ 30.55°

Therefore, the projectile is fired at an angle of approximately 27.6 degrees.

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a) The points of intersection are (1, -8) and (-1, -8).

b) The volume of the solid obtained by rotating the region bounded by y = -8x² and y = x² - 9 about the c-axis is 884π/15.

a) To find the points of intersection between y = -8x² and y = x² - 9, we can set the two equations equal to each other and solve for x:

-8x² = x² - 9

9x² = 9

x² = 1

x = ±1

Plugging these values of x back into either equation, we can find the corresponding y-values:

When x = 1, y = -8(1)² = -8

When x = -1, y = -8(-1)² = -8

b) To find the volume of the solid obtained by rotating the region bounded by y = -8x² and y = x² - 9 about the c-axis, we can use the formula for volume of revolution:

V = π[tex]\int\limits^a_b[/tex] y² dx

where a and b are the x-coordinates of the points of intersection.

In this case, a = -1 and b = 1, so we have:

V = π∫[-1,1] (x²-9)² - (-8x²)² dx

Simplifying this expression and integrating, we get:

V = π∫[-1,1] (65x⁴ - 162x² + 81) dx

= π(65/5 - 162/3 + 81)(1 - (-1))

= 884π/15

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The value of ([tex]8^{-1[/tex])"(0) is 8.

(A) To show that f is 1-1, we need to show that if f(a) = f(b) for some a and b, then a = b.

Assume that f(a) = f(b), then we have:

a ln(a) - a = b ln(b) - b

Using the fact that eln(x) = x, we can rewrite this as

(b/a) * (a/b) = (a+b)

Taking the natural logarithm of both sides, we get:

a ln(b/a) + b ln(a/b) = ln(e^(a+b))

Simplifying, we get:

a ln(b/a) + b ln(a/b) = a + b

Substituting u = b/a, we can rewrite this as:

a ln(u) + b ln(1/u) = a + b

Using the fact that ln(1/x) = -ln(x), we can simplify this to:

a ln(u) - b ln(u) = a - b

Simplifying, we get:

(a - b) ln(u) = a - b

Since a and b are both positive, we can divide by a - b to get:

ln(u) = 1

Using the fact that eln(x) = x, we can rewrite this as:

u = e

Therefore, if f(a) = f(b), then b = a e, which shows that f is 1-1.

(B) To find (F)'(0), we need to find the inverse function F1 and then evaluate its derivative at 0.

To find [tex]F^{-1[/tex], we need to solve for x in the equation F(x) = y, where F(x) = x ln(x) - x.

We have:

y = x ln(x) - x

Rearranging, we get:

y + x = x ln(x)

Using the Lambert W function, we can solve for x to get:

To find ([tex]F^{-1[/tex])'(0), we need to evaluate the derivative of F^-1 at y = 0:

Using the derivative of the Lambert W function, we have:

W'(z) = W(z) / (z (1 + W(z)))

(C) To find (8)"(0), we need to find the second derivative of the function f(x) = x ln(x) - x evaluated at x = 8-1.

We have:

f(x) = x ln(x) - x

f'(x) = ln(x)

f''(x) = 1/x

Therefore,

(8)"(0) = f''(8) = 1/(8) = 8.

Hence, ([tex]8^{-1[/tex])"(0) = 8.

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A 90% confidence interval for the proportion of defective devices will be constructed as 0.026 to 0.036.

To construct a 90% confidence interval for the proportion of defected electronic devices, we can use the formula:
CI = p ± z*^(p(1-p)/n)
Where:
- CI is the confidence interval
- p is the sample proportion of defected devices (310/10000 = 0.031)
- z is the z-score corresponding to the confidence level (90% = 1.645)
- n is the sample size (10000)
Substituting the values:
CI = 0.031 ± 1.645*^(0.031(1-0.031)/10000)
CI = 0.031 ± 0.005
CI = (0.026, 0.036)
Therefore, we can say with 90% confidence that the true proportion of defected electronic devices from the assembly line falls within the interval of 0.026 to 0.036.

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Answer:

The square root of a negative number is an imaginary number. The square root of -100 is 10i where i is the imaginary unit. Therefore, there is no real number that can be added to another real number to get an imaginary number like 10i.

Step-by-step explanation:

Answer: The square root of a negative number is an imaginary number. The square root of -100 is 10i where i is the imaginary unit. Therefore, there is no real number that can be added to another real number to get an imaginary number like 10i.

According to unitary method, Tommy found 68 acorns on the nature hike.

In this case, we know that Patrick found 83 acorns, which is 15 more than Tommy found. So, we can use the unitary method to find out how many acorns Tommy found.

First, we need to find the value of one unit, which is the number of acorns that Tommy found. Let's call this value "x." Since Patrick found 15 more acorns than Tommy, we can write an equation to represent this:

83 = x + 15

To solve for x, we need to isolate it on one side of the equation. We can do this by subtracting 15 from both sides:

83 - 15 = x

68 = x

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The values of simple interests in ascending order will give the result (Case 1 < Case 3 < Case 2).

To calculate simple interest we use the formula as:

[tex]simple \; interest= (P*R*T)/100[/tex]

where,

Principle is represented by 'P'

Rate is represented by 'R'

Time is represented by 'T'

Now for given problem 3 cases are given as

Using formula of simple interest,

Case:1

[tex]SI=(P*R*T)/100=(2000*10*3)/100\\SI=600[/tex]

Case:2

[tex]SI=(P*R*T)/100=(2000*10*9)/100\\SI=1800[/tex]

Case:3

[tex]SI=(P*R*T)/100=(2000*20*3)/100\\SI=1200[/tex]

Thus, putting values of simple interests in ascending order will give the result Case 1 < Case 3 < Case 2

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There must be at least one age with at least 8 people and there must be at least one age with at least 9 people is the right statement

If there are no 8 people with the same age, then the maximum number of people with the same age is 7 (since there are only 83 people in total).

Assume that each age appears only 7 times at the most.

A total of 7× 10=70 people with different ages.

However, that would mean that there are only 13 ages taken, and so at least one age is missing.

This is a contradiction, since we know that all the ages from 24 to 33 are present.

Therefore, there must be at least one age with at least 8 people.

(b) A better statement would be that at least 9 people must have the same age.

If there are at most 8 people with the same age, then the maximum number of people with different ages is 83-8=75.

This means that at most 7 ages are missing, but we know that all the ages from 24 to 33 are present.

Therefore, there must be at least one age with at least 8 people and there must be at least one age with at least 9 people.

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