83 of them Nevaeh would be expected to score less than 193.
We can solve this problem using the Z-score formula. The Z-score is a measure of how many standard deviations away a particular value is from the mean.
We can calculate the Z-score for 193, which is (193 - 165) / 13 = 1.38.
We can look up the Z-score of 1.38 in a normal distribution table to find the probability of a score less than 193. From the table, the probability is 0.9172.
We want to find out how many games out of 90 Nevaeh scored less than 193. We can calculate this using the probability, 0.9172.
90 × 0.9172 = 82.548
≈ 83
Therefore, 83 of them Nevaeh would be expected to score less than 193.
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We collect data about the characteristics of households and draw conclusions about the individuals in those households. This approach may suffer from:
While collecting data about the characteristics of households, the approach may suffer from:
ecological fallacy, generalization issues, lack of individual-level detail, within-household variability, reverse causality, self-reporting bias
What are the limitations the approach may suffer from?This approach may suffer from several limitations:
Ecological Fallacy:Drawing conclusions about individuals based on aggregated household-level data can lead to an ecological fallacy.
It assumes that individual characteristics align with the characteristics of the entire household, which may not be accurate for all individuals within the household.
Generalization Issues:Findings derived from household-level data may have limited generalizability to the larger population.
Household characteristics may not be representative of individuals outside the sampled households, resulting in potential biases and limited applicability of the conclusions.
Lack of Individual-level Detail:Analyzing household-level data may overlook important individual-level details.
Factors influencing individual behaviors, preferences, and decision-making processes may not be fully captured or accurately attributed when only considering household-level characteristics.
Within-household Variability:Households often consist of individuals with diverse characteristics and behaviors.
Treating all individuals within a household as homogeneous can mask variations and nuances that exist within the household, leading to potential inaccuracies in the conclusions drawn.
Reverse Causality:Inferring causal relationships between household characteristics and individual outcomes is challenging.
Without proper experimental design and control over confounding variables, making it difficult to establish a causal link.
Self-reporting Bias:The reliance on self-reported data in household surveys may introduce biases and inaccuracies.
Individuals may provide socially desirable responses or misrepresent their characteristics, which can affect the validity of the conclusions drawn.
It is important to consider these limitations when interpreting findings based on household-level data and to supplement the analysis with individual-level data whenever possible.
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Cory has a square piece of wood that is 4.5 feet by4.5 feet. He wants to make a circular tabletop out of it. What is the area of the largest circle that he can cut out of the wood
The area of the largest circle that can be cut out of the square piece of wood is approximately 31.8 square feet.
The diameter of the largest circle that can be cut out of the square piece of
wood will be equal to the diagonal of the square, which is found using the
Pythagorean theorem.
The diagonal of the square is
√(4.5² + 4.5²) = √(40.5) ≈ 6.36 feet.
The radius of the circle will be half of the diameter,
so r ≈ 6.36/2 = 3.18 feet.
The area of the circle is
A = πr² ≈ π(3.18)² ≈ 31.8 square feet.
Therefore, the area of the largest circle that can be cut out of the square
piece of wood is approximately 31.8 square feet.
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A circle is centered at (−8, −13) and has a radius of 13. What is the equation of the circle? Enter the equation using lowercase variables x and y in the box.
The equation of the circle is (x + 8)²+ (y + 13)² = 169.
A circle is a two-dimensional geometric shape that is defined as the set of all points in a plane that are at a fixed distance (called the radius) from a given point called the centre.
In other words, a circle is a closed curve that consists of all the points that are equidistant from a given point. The distance around the circle is called the circumference, and the distance across the circle through its centre is called the diameter.
The equation of a circle with centre (a, b) and radius r is given by:
(x - a)² + (y - b)² = r²
Substituting the given values:
(x - (-8))² + (y - (-13))² = 13²
Simplifying:
(x + 8)² + (y + 13)² = 169
Therefore, the equation of the circle is (x + 8)²+ (y + 13)² = 169.
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By what number -5 be divided to get -2/9
The number that can divide -5 to get -2/9 is 45/2
By what number is -5 divided to get -2/9From the question, we have the following parameters that can be used in our computation:
Dividing -5 by a number to get -2/9
Represent the number with s
So, we have
Dividing -5 by x to get -2/9
Express as an equation
So, we have
-5/x = -2/9
This gives
5/x = 2/9
Make x the subject of the formula
So, we have
2x = 45
This gives
x = 45/2
Hence, the number is x = 45/2
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A 4-liter mixture of milk and coffee is 20% milk. A liter of pure coffee is added to this mixture. What percent of the new mixture is milk
The new mixture is 16% milk.
Initially, the 4-liter mixture of milk and coffee contains 20% milk, which
means that there are 0.2 × 4 = 0.8 liters of milk in the mixture.
When a liter of pure coffee is added to the mixture, the total volume
becomes 4+1=5 liters.
The amount of milk in the new mixture is still 0.8 liters, but it now
represents a smaller proportion of the total volume.
The new percentage of milk in the mixture can be calculated as follows:
(0.8 / 5) × 100% = 16%
Therefore, the new mixture is 16% milk.
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The average price of a certain model of pickup truck in 1991 was $19,500. In 2012, the average price of the pickup truck was $35,100. What is the percentage increase in the average price of the pickup truck?
The average price of the pickup truck increased by 80%.
To find the percentage increase in the average price of the pickup truck, we need to calculate the difference between the 2012 and 1991 prices, divide that difference by the 1991 price, and then multiply by 100 to get the percentage increase.
First, we need to find the difference between the two prices:
$35,100 - $19,500 = $15,600
Next, we divide the difference by the 1991 price:
$15,600 / $19,500 = 0.8
Finally, we multiply by 100 to get the percentage increase:
0.8 x 100 = 80%
Therefore, the average price of the pickup truck increased by 80%.
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The average time to serve a customer at a fast-food restaurant is 5 minutes. The standard deviation of the service time is 4 minutes. What is the coefficient of variation of the service time
=13.62
step-by-step explanation- We are given the interarrival time (a = 15 min), service time (p = 20 min), number of servers (m = 3 people), standard deviation of interarrival time (15 min) and standard deviation of service time (60 min). - Therefore, the coefficient of variation of arrival times is 15 / 15 = 1 and the coefficient of variation of service times is 60 / 20 = 3. Moreover, the utilization is 20 / (15 x 3) = 0.4444. Therefore, the average time in the queue is 6.6667 x 0.4086 x 5.0 = 13.6211 minutes, or 13.62 minutes rounded to two decimals
- We are given the interarrival time (a = 15 min), service time (p = 20 min), number of servers (m = 3 people), standard deviation of interarrival time (15 min) and standard deviation of service time (60 min). - Therefore, the coefficient of variation of arrival times is 15 / 15 = 1 and the coefficient of variation of service times is 60 / 20 = 3. Moreover, the utilization is 20 / (15 x 3) = 0.4444. Therefore, the average time in the queue is 6.6667 x 0.4086 x 5.0 = 13.6211 minutes, or 13.62 minutes rounded to two decim
Joseph built a model airplane. His model is 16 inches long, and the actual airplane is 128 feet long. What is the scale of his model airplane
Answer:
8
Step-by-step explanation:
16 inchers = 128 feet
128/16=8
so 1 inch = 8 feet
Answer:
16 inches:128 feet = 1 inch:8 feet
A survey of 76 commercial airline flights of under 2 hours resulted in a sample average late time for a flight of 2.33 minutes. The population standard deviation was 12 minutes. Construct a 95% confidence interval for the average time that a commercial flight of under 2 hours is late. What is the point estimate
The point estimate for the average time a commercial flight of under 2 hours is late is 2.33 minutes. The 95% confidence interval is 2.33 minutes ± 2.70 minutes, or approximately (-0.37, 5.03) minutes.
The point estimate for the average time a commercial flight of under 2 hours is late is 2.33 minutes, which is the sample average obtained from the survey of 76 flights. To construct a 95% confidence interval, we'll use the formula:
CI = sample mean ± (Z * σ / √n)
where CI is the confidence interval, the sample mean (2.33 minutes), Z is the Z-score for a 95% confidence level (1.96), σ is the population standard deviation (12 minutes), and n is the sample size (76 flights).
CI = 2.33 ± (1.96 * 12 / √76)
CI = 2.33 ± (1.96 * 12 / 8.72)
CI = 2.33 ± (1.96 * 1.38)
CI = 2.33 ± 2.70
The 95% confidence interval for the average time that a commercial flight of under 2 hours is late is 2.33 minutes ± 2.70 minutes, or approximately (-0.37, 5.03) minutes. This means that we are 95% confident that the true average late time for flights of under 2 hours is between -0.37 minutes (early) and 5.03 minutes (late).
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Complete Question:
A survey of 76 commercial airline flights of under 2 hours resulted in a sample average late time for a flight of 2.55 minutes. The population standard deviation was 12 minutes. Construct a 95% confidence interval for the average time that a commercial flight of under 2 hours is late. What is the point estimate? What does the interval tell about whether the average flight is late?
When testing the goodness of fit for the logistic regression model, if the obtained chi-square is less than the critical value, one would:
Thus, while a chi-square test for goodness of fit is a useful tool for assessing the overall fit of a logistic regression model, it is important to also consider other measures of model fit and to interpret the results in the context of the research question being addressed.
When testing the goodness of fit for the logistic regression model, if the obtained chi-square is less than the critical value, one would accept the null hypothesis, which states that the model fits the data well.
This means that there is no significant difference between the observed values and the values predicted by the model. However, it is important to note that this does not necessarily mean that the model is a perfect fit, as there may still be some minor discrepancies between the observed and predicted values. In addition, it is also important to consider other measures of model fit, such as the Hosmer-Lemeshow test, which assesses the agreement between the observed and predicted values for groups of individuals with similar predicted probabilities. The AIC and BIC are also useful measures of model fit that take into account both goodness of fit and model complexity.Overall, while a chi-square test for goodness of fit is a useful tool for assessing the overall fit of a logistic regression model, it is important to also consider other measures of model fit and to interpret the results in the context of the research question being addressed.Know more about the chi-square test
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A fence 8 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length (in feet) of the shortest ladder that will reach from the ground over the fence to the wall of the building
The length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 8.94 feet.
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). In this case, the fence, building, and ladder form a right triangle, where the fence and building are the legs, and the ladder is the hypotenuse.
We know that the fence is 8 feet tall and the building is 4 feet away from the fence, so the height of the right triangle (the distance from the ground to the top of the building) is also 8 feet.
To find the length of the ladder, we need to use the Pythagorean theorem:
ladder^2 = fence^2 + height^2
ladder^2 = 8^2 + 4^2
ladder^2 = 64 + 16
ladder^2 = 80
ladder ≈ 8.94 feet
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 8.94 feet.
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A researcher wants to be able to predict first semester college grade-point averages with as much accuracy as possible, so she would like to use both high school grade-point averages and SAT scores as predictor variables. Which technique would be most appropriate to make this prediction
The most appropriate technique for the researcher to make this prediction would be multiple regression analysis, which would allow her to examine the relationship between the two predictor variables (high school grade-point averages and SAT scores) and the outcome variable (first semester college grade-point averages) while controlling for their effects on each other. This technique would help the researcher to determine the most accurate way to combine these two predictors in order to make the most accurate prediction possible.
A researcher aiming to predict first semester college grade-point averages (GPAs) using high school GPAs and SAT scores as predictor variables should use the Multiple Linear Regression technique. This technique allows the researcher to analyze the relationship between multiple predictor variables (high school GPAs and SAT scores) and a single continuous outcome variable (college GPAs), which can lead to more accurate predictions.
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3. Roll the die on the game 8 times and record which car would move. What is the empirical probability of how many times the red car moves in 8 rolls
The empirical probability of the red car moving a specific number of times in 8 rolls of the die can be estimated by rolling the die many times and counting the number of times the red car moves a specific number of times, then dividing this count by the total number of rolls.
Assuming that the probability of the red car moving is independent and equal for each roll, the number of times the red car moves in 8 rolls can be modeled using a binomial distribution.
Let's say that the probability of the red car moving in a single roll is p, and we want to find the empirical probability of the red car moving k times in 8 rolls.
To find the empirical probability, we would need to roll the die 8 times and record how many times the red car moves. We can repeat this process many times to collect a large sample of outcomes and estimate the probability based on the proportion of times the red car moves k times in 8 rolls.
For example, if we roll the die 8 times and observe that the red car moves 4 times, we would record that as 4 occurrences of the red car moving in 8 rolls. We can repeat this process many times and record the number of occurrences for each possible value of k (from 0 to 8).
Then, we can calculate the empirical probability of the red car moving k times in 8 rolls as:
The empirical probability of k red car moves = (number of occurrences of k red car moves) ÷ (total number of trials)
For each value of k, we would calculate this empirical probability based on the collected sample.
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Binita is going to buy 450 envolopes
Buying from Value World would cost Binita £0.72 less (in pounds) than buying from Letters2go.
To compare the costs of envelopes from both shops, we need to calculate the price per envelope.
At Letters2go, the price per envelope is:
3.29 pounds / 25 envelopes = 0.1316 pounds per envelope
At Value World, the price per envelope is:
(1.95 pounds * 2 packs) / (10 envelopes * 2 packs + 1 free pack) = 0.1300 pounds per envelope
So the price per envelope at Value World is slightly less than the price per envelope at Letters2go.
To buy 450 envelopes, Binita would need to purchase:
18 packs of 25 envelopes at Letters2go (18 x 25 = 450 envelopes)
45 packs of 10 envelopes at Value World (45 x 10 = 450 envelopes)
At Letters2go, the cost of 18 packs of 25 envelopes would be:
18 packs x £3.29 per pack = £59.22
At Value World, the cost of 45 packs of 10 envelopes (with 15 packs being free) would be:
30 packs x £1.95 per pack = £58.50
Therefore, buying from Value World would cost Binita £59.22 - £58.50 = £0.72 less than buying from Letters2go.
Correct Question :
Binita is going to buy 450 envelopes.
Here is some information about the cost of envelopes in two shops,
Letters2go : Pack of 25 envelopes for £3.29
Value World : Pack of 10 envelopes for £1.95, Buy 2 packs get 1 pack free
How much less, in pounds (£), will it cost Binita to buy the envelopes from Value World.
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Please help me out with this!
Based on the information on the graph, the total number of cups the students used is 3 cups.
How to calculate the number of cups?The total number of cups can be expressed using the following mathematical expression.
[tex]Total = (\frac{1}{8} x 6) + (\frac{1}{4} x 2) + (\frac{3}{8} x 3) + (\frac{5}{8} x 1)[/tex]
This expression is the result of multiplying the amount of water used by the number of students who used that specific volume of water. Knowing this expression, let's solve it to find the total of cups the students used. The step-by-step is shown below:
Total = 0.75 + 0.5 + 1.125 + 0.625
Total = 3 cups
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When Dunkin Donuts offers free samples of its new products to potential customers, such sampling is a form of
Product sampling is a common and effective marketing strategy used by many companies to introduce new products to the market and increase sales. marketing or promotion called "product sampling".
Product sampling is a marketing strategy that involves providing free samples of a product to potential customers, with the hope that they will try the product, enjoy it, and then purchase it in the future.
Dunkin Donuts' free samples of new products to potential customers is a way to generate interest in their new offerings, create brand awareness, and encourage customers to come back to their store and purchase the products. It can also help to gather feedback from customers, which can be used to improve the product or the overall customer experience. Overall, product sampling is a common and effective marketing strategy used by many companies to introduce new products to the market and increase sales.
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Vic wants to fence in his rectangular garden. He has 128 feet of fence to use, and he wants the length of the fence to be 6 feet longer than the width. What are the dimensions of the fence
Answer:
P = 2l + 2w
l = 6 + w
2(6 + w) + 2w = 128
6 + w + w = 64
2w = 58, so w = 29 and l = 35
The length of the fence is 35 feet, and the width of the fence is 29 feet.
A company produces very unusual CD's for which the variable cost is $ 9 per CD and the fixed costs are $ 45000. They will sell the CD's for $ 69 each. Let x be the number of CD's produced. Write the total cost C as a function of the number of CD's produced.
The total cost C as a function of the number of CDs produced is given by C(x) = 45000 + 9x, and the profit P(x) as a function of the number of CDs sold is given by P(x) = 60x - 45000.
The total cost C for producing x number of CDs can be expressed as the sum of fixed costs and variable costs:
C(x) = Fixed costs + Variable costs
C(x) = 45000 + 9x
The fixed cost is [tex]$45,000[/tex] and the variable cost is [tex]$9[/tex] per CD, so the total variable cost for producing x CDs is 9x.
The total cost C(x) is the sum of the fixed cost and the variable cost.
The revenue generated by selling × CDs can be expressed as the product of the selling price per CD and the number of CDs sold:
R(x) = Selling price per CD × Number of CDs sold
R(x) = 69x
The profit P(x) can be calculated by subtracting the total cost from the revenue:
P(x) = R(x) - C(x)
P(x) = 69x - (45000 + 9x)
P(x) = 60x - 45000
To determine the number of CDs that need to be sold to break even (i.e., the profit is zero), we set P(x) equal to zero and solve for x:
0 = 60x - 45000
60x = 45000
x = 750
The company needs to sell 750 CDs to break even.
If they sell more than 750 CDs, they will make a profit, and if they sell fewer than 750 CDs, they will incur a loss.
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Suppose that you were offered a lottery with a 0.50 probability of winning $500 and a 0.50 probability of winning nothing or a guaranteed payoff of $200. If you choose the guaranteed payoff, you would be considered ___.
If you choose the guaranteed payoff of $200 instead of taking a chance on the lottery, you would be considered risk-averse. This means that you prefer a certain outcome (the $200) over an uncertain outcome with potentially higher gains (the lottery).
A risk-averse individual tends to prioritize avoiding losses or negative outcomes over seeking potential gains.
In this scenario, a risk-averse person may choose the guaranteed payoff because they would rather have a sure $200 than take a 50/50 chance of getting nothing at all. On the other hand, a risk-seeking person may choose the lottery because they are willing to take on the risk of winning nothing in order to potentially win $500.
Ultimately, the decision to choose the lottery or the guaranteed payoff depends on the individual's personal risk tolerance and preference for certainty.
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Can someone please help me ASAP? It’s due tomorrow!! I will give brainliest if it’s all correct. Select all that apply
The possible combinations of the number of samples and sample sizes she could use to gather data for 200 students will be:
8 samples of 25 students
10 samples of 20 students
5 samples of 40 students
How to calculate the valueThe possible combinations of the number of samples and sample sizes she could use to gather data for 200 students:
8 samples of 25 students
= 8 × 25.
= 200
10 samples of 20 students = 200
5 samples of 40 students = 200
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A determined gardener has 98 ft of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at least 510 square feet. What range of values is possible for the length of her garden
The gardener has 98 ft of deer-resistant fence to enclose a rectangular vegetable garden.
Let's say the length of the garden is l and the width is w. We know that the area of a rectangle is A = l x w. Therefore, we can solve for w in terms of l: w = A/l.
The gardener wants to enclose an area of at least 510 square feet. We can plug this in for A and get: w = 510/l.
We also know that the perimeter of the garden is 98 ft. The perimeter of a rectangle is P = 2l + 2w. We can substitute in w = 510/l and simplify to get: P = 2l + 1020/l.
We want to find the range of possible values for l, so we need to consider the constraints. Since l must be positive, we can take the derivative of P with respect to l and set it equal to zero to find the minimum value of P. We get: P' = 2 - 1020/l^2 = 0, which gives us l = sqrt(510).
So the range of possible values for the length of the garden is l > sqrt(510), or approximately l > 22.6 ft.
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draw the influence lines for the bar forces in members ab, bk, bc, and lk if the live load is applied to the truss through the lower chord.
To draw the influence lines for the bar forces in members ab, bk, bc, and lk, we need to first understand what influence lines are. Influence lines are graphical representations of the effect of a unit load applied at any point on the structure.
In this case, we want to draw the influence lines for the bar forces in members ab, bk, bc, and lk if the live load is applied to the truss through the lower chord. This means that we need to determine the effect of a unit load applied at different points along the lower chord on the bar forces in these members.
To draw the influence line for member ab, we need to consider a unit load applied at different points along the lower chord and determine the corresponding force in member ab. We can do this by analyzing the truss using the method of joints and solving for the force in member ab. We repeat this process for different points along the lower chord and plot the results on a graph to obtain the influence line for member ab.
Similarly, we can draw the influence lines for members bk, bc, and lk by considering unit loads applied at different points along the lower chord and determining the corresponding forces in these members.
In summary, to draw the influence lines for the bar forces in members ab, bk, bc, and lk if the live load is applied to the truss through the lower chord, we need to analyze the truss using the method of joints and consider unit loads applied at different points along the lower chord to determine the corresponding forces in these members. We can then plot the results on a graph to obtain the influence lines for each member.
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Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 4? Responses x2=−2(y−2) x squared equals negative 2 open parenthesis y minus 2 close parenthesis x2=−8y x squared equals negative 8 y x2=−2y x squared equals negative 2 y, , x2=−8(y−2) x squared equals negative 8 open parenthesis y minus 2 close parenthesis
Write the repeating decimal as a fraction.
.941141141...
A visualization technique to view a distribution of the popularity of words in a bag-of-words model is:
A visualization technique to view a distribution of the popularity of words in a bag-of-words model is a word cloud.
A common technique for visualizing the distribution of the popularity of words in a bag-of-words model is the word cloud.
A word cloud is a visual representation of text data, where the size and prominence of each word in the cloud correspond to its frequency or importance in the given dataset.Word clouds display the most frequently occurring words in a text or corpus, with the size of each word corresponding to its frequency. This allows users to quickly identify the most common words in a dataset and get a sense of the overall distribution of word frequencies. Another approach is to use a histogram, which displays the frequency of each word or group of words in a bar chart. This technique can help identify patterns in the distribution of word frequencies, such as whether the distribution is skewed or bimodal. Both word clouds and histograms can be useful tools for exploratory data analysis and can help researchers understand the characteristics of a text or corpus.Know more about the histogram
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The experimental probability of Tyrese playing volleyball and being able to hit the ball is 43%. Out of 15 rounds, about how many can he predict he will make?
Answer:
6.45, or rounded down to 6.
Step-by-step explanation:
We need to find 43% of 15.
This is found using the equation: 0.43 x 15 = 6.45
A rectangular piece of sheet metal with an area of 1800 in2 is to be bent into a cylindrical length of stovepipe having a volume of 900 in3. What are the dimensions of the sheet metal
The dimensions of the sheet metal are 10 inches by 2.86 inches.
To find the dimensions of the sheet metal, we need to use the formulas for the area and volume of a cylinder.
The formula for the volume of a cylinder is:
[tex]V = πr^2h[/tex]
where V is the volume, r is the radius, and h is the height.
Since we want the volume of the stovepipe to be 900 in^3, we can plug in the values and solve for the height:
[tex]900 = πr^2h[/tex]
[tex]h=\frac{900}{πr^{2} }[/tex]
Now, the formula for the lateral surface area of a cylinder is:
A = 2πrh
where A is the lateral surface area.
We know that the sheet metal has an area of 1800 in^2, so we can set up an equation:
1800 = 2πrh
Substituting h from the first equation, we get:
[tex]1800=2πr(\frac{900}{πr^{2} } )[/tex]
Simplifying, we get:
r = 10
So the radius of the cylinder is 10 inches.
Substituting this value of r into the equation for h, we get:
[tex]h=\frac{900}{π(10^{2}) }[/tex]
h =2.86
So the height of the cylinder is approximately 2.86 inches.
Therefore, the dimensions of the sheet metal are 10 inches by 2.86 inches.
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Geometry - Area of a regular polygons
Answer:
19888 cm²--------------------
The area of a regular polygon is half the product of its apothem and its perimeter:
A = aP/2Given is a pentagon, hence:
P = 5sWe are given the lengths:
a = 74 cms = 107.5 cmCalculate the area:
A = 5*107.5*74/2 = 19887.5 ≈ 19888 cm² (rounded)The standard deviation of a binomial distribution is: A. square of npq B. npq C. np D. square root of npq
The standard deviation for a binomial probability distribution is : √npq.
The correct option is (D) i.e., square root of npq
The standard deviation of a random variable, sample, statistical population, data set or probability distribution is the square root of its variance.
For a binomial distribution,
µ = np, which signifies the expected number of successes.
[tex]\sigma^2[/tex] = npq , [tex]\sigma^2[/tex] is the variance.
Since, the standard deviation is the square root of the variance,
Therefore, σ = Standard deviation = √npq
Thus, the standard deviation for a binomial probability distribution is given by √npq.
The correct option is (D)
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Suppose a coin is tossed 14 times and there are 3 heads and 11 tails. How many such sequences are there in which there are at least 6 tails in a row
There are at least i tails remaining in the sequence is
[tex]|A_7| = \sum(k=1 to 8) 2^{(k-1)} \times 2^{(7-k)} \times C(11-k,1) = 3200\\|A_8| = \sum (k=1 to 7) 2^{(k-1)} \times 2^{(6-k)} \times C(11-k,2) = 13552\\|A_9| = \sum (k=1 to 6) 2^{(k-1)} \times 2^{(5-k)} \times C(11-k,3) = 32704\\|A_10[/tex]
To solve this problem, we can use the concept of complementary counting. That is, we can count the total number of sequences of 14 coin tosses with 3 heads and 11 tails, and then subtract the number of sequences that do not have 6 tails in a row.
Let's first count the total number of sequences with 3 heads and 11 tails. Each toss can result in either a head or a tail, so there are 2 possibilities for each toss. Therefore, there are [tex]2^{14[/tex] possible sequences of 14 coin tosses. To count the number of sequences with 3 heads and 11 tails, we need to choose 3 out of the 14 tosses to be heads. This can be done in C(14,3) = 364 ways. Therefore, there are 364 sequences with 3 heads and 11 tails.
Now, let's count the number of sequences that do not have 6 tails in a row. To do this, we can use the technique of inclusion-exclusion. Let A be the set of all sequences with at least one run of 6 tails in a row, and let A_i be the set of sequences with a run of i tails in a row for i = 6, 7, 8, 9, 10, 11, or 12. Then, we can use the principle of inclusion-exclusion to count the number of sequences that are not in A:
|A| = |A_6 ∪ A_7 ∪ A_8 ∪ A_9 ∪ A_10 ∪ A_11 ∪ A_12|
= ∑|A_i| - ∑|A_i ∩ A_j| + ∑|A_i ∩ A_j ∩ A_k| - ...
= |A_6| - |A_6 ∩ A_7| + |A_6 ∩ A_7 ∩ A_8| - ...
To count the size of each set A_i, we can fix the position of the first run of i tails in the sequence, and then count the number of ways to fill in the rest of the sequence. For example, to count |A_6|, we can fix the position of the first run of 6 tails, say it starts at position k. Then, the first k-1 tosses can be either heads or tails [tex](2^{(k-1)[/tex] possibilities), and the next 6 tosses must be tails. The remaining 14-k-6 = 8-k tosses can be either heads or tails (2^(8-k) possibilities). Therefore,[tex]|A_6| = \sum (k=1 to 9) 2^{(k-1)} \times 2^{(8-k) }= 511.[/tex]
Using a similar approach, we can count the sizes of the other sets A_i. Note that for i ≥ 7, the first run of i tails can start at any of the 11 positions where there are at least i tails remaining in the sequence. Therefore, we have:
[tex]|A_7| = \sum(k=1 to 8) 2^{(k-1)} \times 2^{(7-k)} \times C(11-k,1) = 3200\\|A_8| = \sum (k=1 to 7) 2^{(k-1)} \times 2^{(6-k)} \times C(11-k,2) = 13552\\|A_9| = \sum (k=1 to 6) 2^{(k-1)} \times 2^{(5-k)} \times C(11-k,3) = 32704\\|A_10[/tex]
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