The kurtosis of an F-distribution with parameters r1 and r2 is given by: Kurtosis = [ 8(r2 + 2r1 - 1) ] / [ r2 (r1 - 2) (r1 - 4) ]
Assuming that r2 > 8, we can use the approximation given in the previous exercise to simplify this expression: Kurtosis ≈ 3 + [ 12 (r2 - 8) ] / [ (r2 - 6) (r2 - 4) ]
Therefore, the kurtosis of an F-distribution with parameters r1 and r2, assuming that r2 > 8, is approximately equal to 3 plus the expression above.
Kurtosis is a measure of the "peakedness" or "flatness" of a distribution compared to the normal distribution. It measures the degree to which a distribution has more or less weight in the tails compared to the normal distribution.
A distribution with kurtosis greater than 3 is said to be "leptokurtic," meaning it has heavier tails than the normal distribution. A distribution with kurtosis less than 3 is said to be "platykurtic," meaning it has lighter tails than the normal distribution. A distribution with kurtosis equal to 3 is said to be "mesokurtic," meaning it has tails that are similar in weight to the normal distribution.
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Which situation describes exponential decay? a. The value of a piece of machinery decreases by 19% every year. B. The value of a piece of machinery decreases by $5000 every year
The situation that describes exponential decay is (a) The value of a piece of machinery decreases by 19% every year.
What is the exponential function?
An exponential function is a mathematical function of the form:
f(x) = aˣ
where "a" is a constant called the base, and "x" is a variable. Exponential functions can be defined for any base "a", but the most common base is the mathematical constant "e" (approximately 2.71828), known as the natural exponential function.
Exponential decay is a mathematical term used to describe the process of decay that happens at an exponential rate.
In this case, the value of the machinery decreases by a percentage every year.
Since the percentage decrease is constant over time, this is an example of exponential decay.
On the other hand, option (b) describes a linear decrease in value since the value decreases by a constant amount of $5000 every year.
Hence, the situation that describes exponential decay is (a) The value of a piece of machinery decreases by 19% every year.
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answer all pls thx i need it answered
A dot plot can be created by drawing a number line and then adding one dot per value, the first statement is true but the second one is not.
How to draw and interpret a dot plot?The basic steps to draw this type of graph is:
Draw a number line that covers the range of the data, in this case from 18 to 32.Draw a dot for each value, for example, 25 is repeated two times, so two dots should be drawn.Now, let's interpret the information:
The class is an adult class - Yes- All of them are at least 18.Half of the students are between 2 and 3 years old - False- No one is this age.Learn more about dot plots in https://brainly.com/question/22746300
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A quadratic function f(x) is hidden from view. You must find all
intervals where f(x) is negative. Choose the form of the quadratic
function f(x) that you would like to see in order to answer the
question most efficiently.
Form: Standard Form, Factored form, or vertex form
To answer the question most efficiently , the standard form of the quadratic equation is the most suitable.
What is a quadrilateral function?A quadratic function is one of the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero.
The highest power of a quadratic function is 2.
To solve a quadratic function efficiently, we need to put the function in standard form. i.e We arrange the power in descending order.
For example, a quadratic functionf(x)= 5x-2 +x² is not in standard form. To solve this we need to put it in a form of f(x) = x²+5x -2.
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eliminate the parameter t for both parametrizations and write down the equation of the ellipse in terms of x and y only.
The differential equation of the ellipse in terms of x and y only: (x/2) ^2 + (y/3)^2 = 1. To eliminate the parameter t for a parametric equation of an ellipse, we can solve for t in terms of x or y and substitute it into the other equation.
Consider the parametrization of an ellipse given by x = 2cos(t) and y = 3sin(t).
We can eliminate t by solving for cos(t) and sin(t) in terms of x and y, respectively. From the first equation, we have cos(t) = x/2, so t = Arcos(x/2). Substituting this expression for t into the second equation, we get y = 3sin(arccos (x/2)) = 3sqrt(1-(x/2)^2). This gives us the equation of the ellipse in terms of x and y only:
(x/2)^2 + (y/3)^2 = 1.
This is the standard form equation of an ellipse with center at the origin and semi-axes of length 2 and 3 in the x and y directions, respectively.
In general, to eliminate the parameter t for a parametrization of an ellipse, we can use trigonometric identities to express cos(t) and sin(t) in terms of x and y, respectively. Substituting these expressions for cos(t) and sin(t) into the parametric equations and simplifying should give us the equation of the ellipse in terms of x and y only.
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Which source of bias is most relevant to the following situation: Both members of a couple are asked to indicate if they have remained monogamous in their currently relationship self-interest study voluntary response blas nonrespolse bias or missing data perceived lack of anonymity loaded or leading question
The source of bias most relevant to this situation is "perceived lack of anonymity."
When both members of a couple are asked to indicate if they have remained monogamous in their current relationship, they may be hesitant to answer truthfully due to concerns about their privacy or potential repercussions. They might feel that their responses could be traced back to them, which could lead to negative consequences in their relationship. This fear of not being anonymous can result in participants providing inaccurate or dishonest responses.
In a study asking couples about their monogamy, the most relevant source of bias is perceived lack of anonymity, as it may affect the accuracy and honesty of participants' responses.
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why is it possible for condition means to differ at the end of an experiment even if the independent variable had no effect? type ii error has ocurrederror variancerandomizationa null finding has been obtained
Answer:
Answer:- 1) option b) error variance.Have a Nice Best Day : ) Please Give Me Brainliest
for the following factored polynomial, find all of the zeros and their multiplicities. f(x)=(x−5)5(x 1)7
the question is that the zeros of the polynomial f(x)=(x−5)5(x+1)7 are x=5 and x=-1, and their multiplicities are 5 and 7, respectively.
the zeros and their multiplicities is as follows:
To find the zeros of the polynomial, we set each factor equal to zero and solve for x.
For the factor (x−5)5, we get x=5 as the only zero.
For the factor (x+1)7, we get x=-1 as the only zero.
To determine the multiplicities of the zeros, we count the number of times each zero appears as a factor.
Since (x−5)5 is a factor of the polynomial, the zero x=5 has a multiplicity of 5.
Similarly, since (x+1)7 is a factor of the polynomial, the zero x=-1 has a multiplicity of 7.
the zeros of the polynomial f(x)=(x−5)5(x+1)7 are x=5 and x=-1, and their multiplicities are 5 and 7, respectively.
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Cynthia concludes that 8th graders are more likely than 9th graders to spend at most four hours per week studying because more 8th graders than 9th graders responded that they study this long.
Explain whether Cynthia is correct. Include all necessary work to support your answer.
find an equation of the plane. the plane through the point (9, −9, −6) and parallel to the plane 7x − y − z = 1
Using the normal vector of the parallel plane <7, -1, -1>. By Substituting the given points (9, -9, -6) in the plane equation and got the new plane equation: 7x - y - z = 69.
To find an equation of the plane passing through a given point and parallel to another plane, we need to use the normal vector of the parallel plane and substitute the given point in the equation of the plane.
A plane is defined by a point and a normal vector perpendicular to it. Therefore, to find an equation of the plane passing through the point (9, −9, −6), we need to determine its normal vector.
The given plane 7x − y − z = 1 can be written in the form Ax + By + Cz = D, where A = 7, B = -1, C = -1 and D = 1. The coefficients of x, y, and z in this equation give the components of the normal vector to the plane. Thus, the normal vector is N = (7, -1, -1).
Since the plane we want to find is parallel to the given plane, it has the same normal vector. Therefore, we can write its equation as 7x - y - z = D, where D is a constant. To determine D, we substitute the coordinates of the given point into the equation:
7(9) - (-9) - (-6) = 69
Thus, the equation of the plane passing through the point (9, −9, −6) and parallel to the plane 7x − y − z = 1 is 7x - y - z = 69.
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in general, there is more information provided by . a. a confidence interval than a p-value. b. a p-value than a confidence interval. c. a sample statistic than a confidence interval for the corresponding parameter. d. all of the above.
In general, a confidence interval provides more information than a p-value. A confidence interval provides more information than a p-value because it gives us an estimate of the parameter, a measure of uncertainty, and can be derived from a sample statistic.
A confidence interval is a range of values around an estimate of a population parameter that we are fairly certain contains the true value of the parameter. It provides both an estimate of the parameter and a measure of the uncertainty of the estimate. On the other hand, a p-value is a measure of the strength of evidence against a null hypothesis. It tells us the probability of observing a test statistic as extreme as the one we observed, or more extreme, if the null hypothesis were true. However, it does not tell us anything about the magnitude or direction of the effect, or the precision of the estimate.
Furthermore, a confidence interval can be derived from a sample statistic, whereas a p-value cannot. A confidence interval gives us an estimate of the population parameter based on the sample, while a p-value tells us how likely it is to observe a sample statistic as extreme as the one we observed, assuming the null hypothesis is true.
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What is the formula for finding the circumstance of a circle
with just the diameter?
A.πx D
B.b x h
C.L×W×H
D.4x1
Answer: A.) π x D
Step-by-step explanation:
The diameter of a circle is 2x the radius
Answer:
A.πx D
Step-by-step explanation: The formula for the circumference of a circle is C=π×d, or it can be written as C=2×π×r.
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Latrell has $8 to spend on postcards. He wants to buy one large postcard and some small ones. Write an inequality to determine how many small postcards Latrell can purchase. Postcards Large $2. 00 Medium $1. 50 Small $1. 25
The inequality to determine how many small postcards Latrell can purchase is: 1.25s + 2.00 ≤ 8.00
In the inequality 1.25s + 2.00 ≤ 8.00 s represents the number of small postcards and the left-hand side of the inequality represents the total cost of the postcards, including one large postcard costing $2.00.
To solve for s, we can begin by subtracting 2.00 from both sides of the inequality to isolate the term with s:
1.25s ≤ 6.00
Then, we can divide both sides of the inequality by 1.25 to solve for s:
s ≤ 4.8
Since s represents a whole number, the largest number of small postcards that Latrell can purchase is 4.
In summary, the inequality 1.25s + 2.00 ≤ 8.00 can be used to determine the maximum number of small postcards (s) that Latrell can purchase with $8 while also buying one large postcard. By solving the inequality, we find that s ≤ 4.8, so the largest whole number of small postcards that Latrell can purchase is 4.
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A. y = sin(x - Tt/2)
C. y=sin(x + 1)
Find the equation.
1
NEN
2
k
B. y = sin x
D. y = sin(x +
TT/2)
The requried equation of the graph shown is y = sin (x - π/2).
The curve shown in the graph is of the sine function with some trasformation given as,
The graph of sinx is shifted π/2 units left, so the equation of the graph is given as,
y = sin (x - π/2)
Thus, the requried equation of the graph shown is y = sin (x - π/2).
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In a simple regression model: y=b0+b1∗x1+ei with a dummy variable ( 0 or 1 ) predictor x, the coefficient b1 may be interpreted as: a. the differential effect on the outcome of having the dummy variable equal to 1 , rather than 0 .b. the improvement in R-squared that we get from including x in the model. c. the predicted value of the outcome variable when the dummy variable is equal to 1 . d. the predicted value of the outcome variable when the dummy variable is equal to 0 .
In a simple regression model with a dummy variable predictor x, the coefficient b1 represents a) the differential effect on the outcome of having the dummy variable equal to 1, rather than 0.
Option b, the improvement in R-squared that we get from including x in the model, is not an accurate interpretation of b1. R-squared represents the proportion of variance in the outcome variable that is explained by the model, but it does not directly relate to the coefficient of a specific predictor.
Options c and d refer to the predicted values of the outcome variable when the dummy variable is either 1 or 0, but these values are not equivalent to b1. The predicted value of the outcome variable depends on the intercept term b0 and any other predictor variables in the model, as well as the coefficient b1.
In a simple regression model with a dummy variable predictor x, the coefficient b1 represents Option a, the differential effect on the outcome of having the dummy variable equal to 1, rather than 0. This means that b1 quantifies the change in the outcome variable when the dummy variable changes from 0 to 1, holding all other variables constant. This interpretation assumes that the dummy variable is coded in a meaningful way, such that 0 and 1 represent two distinct and relevant categories.
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(-2 3/4)^2
what does this equal?
Answer:
The answer is 121/16 or 7.5625
Step-by-step explanation:
(-2¾)²
(-11/4)²
=121/16 or 7.5625
thomas believes that a group is cohesive when it is marked by strong positive bonds between members of a group. thomas considers cohesion to be
Thomas considers cohesion to be the presence of strong positive bonds between members of a group. In his perspective, cohesion represents a sense of unity, attachment, and solidarity among individuals within the group.
Cohesion, according to Thomas, goes beyond mere cooperation or coordination of activities. It encompasses the emotional and social aspects of group dynamics, emphasizing the quality and strength of the relationships between members. The existence of strong positive bonds implies that individuals feel connected, trust each other, and have a shared sense of identity and purpose.
Thomas likely believes that cohesive groups are characterized by mutual support, collaboration, and a sense of belonging. Members are more likely to work together effectively, communicate openly, and have a higher level of commitment to group goals. Cohesion may also contribute to increased satisfaction, motivation, and overall well-being within the group.
Thomas's perspective aligns with the socio-emotional aspect of group cohesion, emphasizing the interpersonal connections and social dynamics that contribute to the cohesiveness of a group.
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if r is the ring of integers, let u be the ideal consisting of all multiples of 17. prove that if v is an ideal of r and r ⊃ v ⊃ u then either v = r or v = u. generalize!
(a) is a prime ideal of R, and since v is contained in (a), we have either v = (a) or v = R. Therefore, The only ideals of R containing u are u and R itself.
To prove that if v is an ideal of r and r ⊃ v ⊃ u, then either v = r or v = u, we need to use the fact that the only ideals of r are the trivial ones (0 and r) and the maximal ones (prime ideals). Since u is not a prime ideal (since it contains multiples of 17), the only possibilities for v are u and r. To see this, suppose that v is a proper ideal of r containing u. Then there exists some non-zero element a in v that is not in u. Since a is not a multiple of 17, it must have a prime factor p that is not 17 (otherwise, it would be a multiple of 17). But then (a) is a prime ideal of r (since r is a UFD), and since v is contained in (a), we have either v = (a) or v = r. Therefore, we have shown that if v is an ideal of r and r ⊃ v ⊃ u, then either v = r or v = u. We can generalize this result to any ring R and any proper ideal u of R by using the same argument. If v is a proper ideal of R containing u, then there exists some non-zero element a in v that is not in u. But then (a) is a prime ideal of R, and since v is contained in (a), we have either v = (a) or v = R. Therefore, the only ideals of R containing u are u and R itself.
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Aaron’s major surgery resulted in $25,000 in bills. His insurance has an annual deductible of $5,500, an 80% coinsurance, and an out-of-pocket maximum of $12,000. How much will he be responsible for paying?
Aaron is responsible for paying the sum of $10,700 for his surgery.
How much will Aaron be responsible for paying?Aaron's deductible = $5,500. Thus means he is responsible for paying first $5,500 of his medical bills.
Coinsurance = 80%. Insurance cover 80% of remaining cost and he is responsible for 20%.
Aaron's out-of-pocket maximum is computed as follows
= Surgery cost - Deductible
= $12,000 - $5,500
= $6,500
Amount payable * Coinsurance
= $6,500 x 0.8
= $5,200
So, Aaron's insurance will cover the remaining $19,800 of his $25,000 surgery.
So, he is responsible for paying:
= $5,500 (deductible) + $5,200 (out-of-pocket maximum)
= $10,700
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suppose ()=100, ()=200, ()=300 (∩)=10, (∩)=15, (∩)=20 (∩∩)=5 (∪∪)= A. 400 B. 600 C. 300 D. NONE
The cardinality of the union is 240, none of the options listed.
How to find union cardinality?
Based on the symbols and values given, it seems like you are referring to a set notation problem. Here is what I understand from the symbols and values given:
()=100 means the cardinality (size) of set A is 100.
()=200 means the cardinality of set B is 200.
()=300 means the cardinality of set C is 300.
(∩)=10 means the intersection of sets A and B has a cardinality of 10.
(∩)=15 means the intersection of sets B and C has a cardinality of 15.
(∩)=20 means the intersection of sets A and C has a cardinality of 20.
(∩∩)=5 means the intersection of sets A, B, and C has a cardinality of 5.
To find the cardinality of the union of all three sets (i.e., (∪∪)), we can use the formula:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|
Substituting the given values, we get:
|A ∪ B ∪ C| = 100 + 200 + 300 - 10 - 15 - 20 + 5 = 240
Therefore, the answer is not one of the options given.
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Apples are $1.79 per lb at Food Lion. If I purchase 3.8 lbs, how much sure I expect to pay?
Answer:
$6.80
Step-by-step explanation:
1.79 x 3.8 = 6.802
So $6.80
Determine the missing angle measures, Can someone please answer this I need to turn this in Im in a hurry i beg
The measure of angles are;
angle Y = 29 degrees
angle W = 112
angle X = 39
we can write as
29 + angle Z + 112 = 180° (Being linear pair angles)
angle Z = 180 - 112 - 29
angle Z = 39
Therefore,
angle Z = angle X = 39 degrees (by vertically opposite angles)
angle Y = 29 degrees(by vertically opposite angles)
angle W = 112 degrees(by vertically opposite angles)
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for the linear program max x1 − 2x3 x1 − x2 ≤ 1 2x2 − x3 ≤ 1 x1, x2, x3 ≥ 0 prove that the solution (x1, x2, x3) = (3/2, 1/2, 0) is optimal.
Based on the demonstration of feasibility and optimality, we conclude that the solution (x1, x2, x3) = (3/2, 1/2, 0) is indeed optimal for the given linear program.
What is Optimality?
Optimality refers to the quality or state of being the best or most favorable among a set of alternatives or solutions. In various contexts, optimality is often associated with achieving the maximum or minimum value of an objective function, reaching an optimal solution, or attaining the best possible outcome.
To prove that the solution (x1, x2, x3) = (3/2, 1/2, 0) is optimal for the given linear program, we need to demonstrate two conditions: feasibility and optimality.
Feasibility: We need to show that the solution satisfies all the constraints of the linear program.
From the given constraints:
x1 - x2 ≤ 1: (3/2) - (1/2) = 1, which satisfies the inequality.
2x2 - x3 ≤ 1: 2(1/2) - 0 ≤ 1, which satisfies the inequality.
x1, x2, x3 ≥ 0: All the variables are non-negative, so this constraint is satisfied.
Therefore, the solution (x1, x2, x3) = (3/2, 1/2, 0) is feasible.
Optimality: We need to demonstrate that the solution provides the maximum value for the objective function.
The objective function is max x1 - 2x3.
Plugging in the values from the solution: (3/2) - 2(0) = 3/2.
We need to show that no other feasible solution can yield a higher value for the objective function.
By examining the constraints, we find that the upper bounds for x1 and x2 are 1, and x3 is non-negative.
Considering all possible values within these constraints, we observe that (3/2, 1/2, 0) provides the maximum value of 3/2 for the objective function.
Thus, the solution (x1, x2, x3) = (3/2, 1/2, 0) is optimal.
Therefore, based on the demonstration of feasibility and optimality, we conclude that the solution (x1, x2, x3) = (3/2, 1/2, 0) is indeed optimal for the given linear program.
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Complete Question -
Question: Linear Programming: For The Linear Program: Max X1-2x3 X1-X2 <= 1 2x2-X3 ≪= 1 X1, X2, X3 >= 0 Prove That The Solution (X1, X2, X3) = (3/2, 1/2, 0) Is Optimal
Linear programming:
For the linear program:
max x1-2x3
x1-x2 <= 1
2x2-x3 <= 1
x1, x2, x3 >= 0
prove that the solution (x1, x2, x3) = (3/2, 1/2, 0) is optimal
HELP ASAP 100 POINTS AND BRAINLIEST IF CORRECT IF WRONG I WILL REPORT!!
The data that has a variation, or mean absolute deviation, similar to the data set in the given dot plot is option D.
Why is this so?Deviation describes the extent of the Stata change, do plot in D has the same distribution with the example given to they must have the most similar Mean Absolute deviation.
A data set's mean absolute deviation (MAD) is the average distance between each data value and the mean. A data set's fluctuation can be described using mean absolute deviation. The mean absolute deviation tells us how "spread out" the values in a data collection are.
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in 2000, the population of california was 29,816,591 and increased yearly by 1.28%. what will the population be in the year 2020?
The population of California in the year 2020 using the initial population for the year 2000 will become approximately 38453143.
Population of California in the year 2000 = 29,816,591
Yearly increased rate = 1.28%
use the formula for compound interest,
A = [tex]P\times (1 + r/n)^{(n\times t)}[/tex]
where A is the final amount,
P is the initial amount,
r is the annual interest rate (as a decimal),
n is the number of times the interest is compounded per year,
and t is the number of years.
In this case, P = 29,816,591,
r = 0.0128 (since the annual increase is 1.28%),
n = 1 (since the increase is yearly),
and t = 20 (since we want to know the population in 2020, which is 20 years after 2000).
Substituting these values into the formula, we get,
A = 29,816,591(1 + 0.0128/1)²⁰
A = 29,816,591(1.0128)²⁰
A = 29,816,591 × (1.2897)
A = 38453142.868
Therefore, the population of California in the year 2020 will be approximately 38453143 (rounded to the nearest whole number).
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if a is an m x n matrix, if the equation = has at least two different solutions, and if the equation = is consistent, then the equation = has many solutions. true or false
if the equation Ax=b has at least two different solutions and is consistent, then it has infinitely many solutions.
If the equation Ax=b has at least two different solutions, then there exists more than one vector x that satisfies the equation. This means that the system of equations Ax=b has more than one solution. Since the system is consistent, there exists at least one solution.
If we have at least two solutions to the system Ax=b, say x1 and x2, then any linear combination of x1 and x2 is also a solution. That is, if we take any scalar values c1 and c2, then the vector y=c1x1+c2x2 is also a solution to the equation Ax=b.
Therefore, if the equation Ax=b has at least two different solutions and is consistent, then it has infinitely many solutions.
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Two taps running at the same rate can fill a tank in 45 mins. How long will it take one tap to fill the same tank?
Answer:
90 minutes
Step-by-step explanation:
it will take twice as long therefore 90 minutes
Find the image of the given point
under the given translation.
P(2, 5)
T(x, y) = (x-6, y + 2)
P' = ([?], [])
Answer:Usually you should just use these two rules:
T(x)+T(y) = T(x+y)
cT(x) = T(cx)
Where T is your transformation (in this case, the scaling matrix), x and y are two abstract column vectors, and c is a constant.
If these two rules work, then you have a linear transformation :)
Step-by-step explanation:
a motorist travelled from Ibadan to Lagos ,a distance of 142 km,at an average speed of 60km/h. He spent 5/2 hours in Lagos and then returned to Ibadan at an average speed of 80km/h .(a).At what time did the man arrive back in Ibadan? .(b). find his average speed for the total journey.
To solve the given problem, let's break it down into two parts:
(a) At what time did the man arrive back in Ibadan?
We can calculate the time taken for the initial journey from Ibadan to Lagos using the formula:
Time = Distance / Speed
Time = 142 km / 60 km/h
Time = 2.37 hours
The motorist then spent an additional 5/2 hours in Lagos.
To calculate the total time taken for the return journey from Lagos to Ibadan, we use the formula:
Time = Distance / Speed
Time = 142 km / 80 km/h
Time = 1.775 hours
Adding the time spent in Lagos to the return journey time:
Total time = 2.37 hours + 5/2 hours + 1.775 hours
Total time = 6.145 hours
Therefore, the man arrived back in Ibadan approximately 6.145 hours after he left.
(b) To find the average speed for the total journey, we use the formula:
Average Speed = Total Distance / Total Time
The total distance covered in the round trip is 2 times the distance between Ibadan and Lagos, which is:
Total Distance = 2 x 142 km
Total Distance = 284 km
Average Speed = 284 km / 6.145 hours
Average Speed = 46.22 km/h
Therefore, the average speed for the total journey is approximately 46.22 km/h.
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You are a gift wrapper at a department store. You are wrapping a box that has a top and a bottom
that are both 8 inches by 3 inches, a front and a back that are both 8 inches by 2 inches, and sides
that are both 3 inches by 2 inches. What is the surface area of the box?
A 26 square inches
B. 48 square inches
C. 68 square inches
D. 92 square inches
2 in
3 in
8 in
find the exact length of the curve. x = 6 6t2, y = 5 4t3, 0 ≤ t ≤ 1
To find the length of the curve given by x=6t^2 and y=5t^3, the exact length of the curve x=6t^2 and y=5t^3 for 0≤t≤1 is approximately 7.697 units long.
To find the length of the curve given by x=6t^2 and y=5t^3, we need to use the arc length formula, which is:
L = ∫[a,b] √[dx/dt]^2 + [dy/dt]^2 dt
Here, a=0 and b=1 since 0≤t≤1. Also, dx/dt = 12t and dy/dt = 15t^2. Substituting these values in the above formula, we get:
L = ∫[0,1] √(12t)^2 + (15t^2)^2 dt
Simplifying, we get:
L = ∫[0,1] √(144t^2 + 225t^4) dt
Taking out the common factor of t^2 inside the square root, we get:
L = ∫[0,1] t√(144 + 225t^2) dt
To evaluate this integral, we need to make a substitution u = 144 + 225t^2, which gives du/dt = 450t. Substituting these values, we get:
L = (1/450) ∫[144,369] √u du
Now, we can evaluate the integral using the power rule of integration:
L = (1/450) * [(2/3) u^(3/2)]_144^369
Simplifying, we get:
L = (1/450) * [(2/3) * (369^(3/2) - 144^(3/2))]
L = 7.697
Therefore, the exact length of the curve x=6t^2 and y=5t^3 for 0≤t≤1 is approximately 7.697 units long.
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