The probability is approximately 0.2075.
We know that the sample size n=18, population mean [tex]$\mu = 20.20$[/tex] hours and population standard deviation [tex]$\sigma = 2.60$[/tex] hours.
(a) The sampling distribution of sample means will follow normal distribution with mean [tex]$\mu_{\bar{x}} = \mu = 20.20$[/tex] and standard deviation [tex]$\frac{\sigma}{\sqrt{n}} = \frac{2.60}{\sqrt{18}} \approx 0.61$[/tex].
Thus, the probability that the average time spent working per week for 18 randomly selected college students who hold part-time jobs during the school year is not within 1 hour of the population mean can be found as:
[tex]\begin{aligned}& P(|\bar{X}-\mu| > 1)-P\left(\left|\frac{\bar{X}-\mu_\delta}{\frac{\sigma}{\bar{x}^n}}\right| > \frac{1}{\frac{1}{\sqrt{n}}}\right) \\& -P\left(\left|\frac{\hat{x}-\mu_s}{0.61}\right| > 1.64\right) \approx 0.10 \\&\end{aligned}[/tex]
Therefore, the probability is approximately 0.10.
(b) To find the probability that the average time spent working per week for 18 randomly selected college students who hold part-time jobs during the school year is 20 to 20.50 hours, we can use the formula for z-score and standard normal distribution table:
[tex]$$\begin{aligned}& P(20 \leq \bar{X} \leq 20.50)-P\left(\frac{20-\mu_{\bar{i}}}{\frac{\bar{\omega}}{\bar{\omega}}} \leq Z \leq \frac{20.50-\mu_{\bar{i}}}{\frac{\hbar}{\sqrt{\pi}}}\right) \\& -P(-1.31 \leq Z \leq-0.49) \approx 0.17\end{aligned}$$[/tex]
Therefore, the probability is approximately 0.17 .
(c) To find the probability that the average time spent working per week for 18 randomly selected college students who hold part-time jobs during the school year is at least 22 hours, we can use the formula for [tex]$z$[/tex]-score and standard normal distribution table:
[tex]$$\begin{aligned}& P(\bar{X} \geq 22)-P\left(Z \geq \frac{22-\mu_8}{\frac{\sigma}{\sqrt{n}}}\right) \\& -P(Z \geq 3.28) \approx 0.0005\end{aligned}$$[/tex]
Therefore, the probability is approximately 0.0005 .
(d) To find the probability that the average time spent working per week for 18 randomly selected college students who hold part-time jobs during the school year is no more than 21 hours, we can use the formula for [tex]$z$[/tex]-score and standard normal distribution table:
[tex]$$\begin{aligned}& P(\bar{X} \leq 21)-P\left(Z \leq \frac{21-\mu_8}{\frac{\sigma}{\sqrt{n}}}\right) \\& -P(Z \leq 0.82) \approx 0.2075\end{aligned}$$[/tex]
Therefore, the probability is approximately 0.2075 .
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the letters in the word bubble are arranged in a row. how many unique arrangements are there of the letters?
In a case whereby letters in the word bubble are arranged in a row the number of unique arrangements that are there in the letters is 120.
How can the letter be arranged?
From the question we were given the word, bubble which can be recognized as 6! arrangements however we know that the 3 B’s are interchangeable, from the word which is to be arranged, and we can see that there are 6 letters all together in the word and can be arranged as;
6!/3!
= 720/6
= 120
Hence, we can conclude the arangement here is 120 unique wayas.
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problem 3. prove or disprove: for any m x n matrix a, aat and a t a are symmetric.
For any m x n matrix A, AAT is symmetric, but ATA is not symmetric.
To prove that for any m x n matrix A, AAT and ATA are symmetric, we need to show that the transpose of each product is equal to the product itself.
Calculate (AAT)T:
Let's start with AAT. We know that for any matrices X and Y, (XY)T = YT XT. So, for AAT, we have:
(AAT)T = (AT)T [tex]A^{T}[/tex]
Simplify (AAT)T:
Now, we can apply the rule that ([tex]A^{T}[/tex][tex])^{T}[/tex] = A. Thus, we get:
(AAT)T = A [tex]A^{T}[/tex]
As you can see, (AAT)T = AAT, which means AAT is symmetric.
Calculate (ATA)T:
Now let's look at ATA. Using the same rule as before, we have:
(ATA)T = A[tex]T^{T}[/tex] ([tex]A^{T}[/tex][tex])^{T}[/tex]
Simplify (ATA)T:
Again, we apply the rule that ([tex]A^{T}[/tex][tex])^{T}[/tex] = A:
(ATA)T = A [tex]A^{T}[/tex]
In this case, we have (ATA)T ≠ ATA, which means ATA is not symmetric.
In conclusion, for any m x n matrix A, AAT is symmetric, but ATA is not symmetric.
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What is the following equivalent to (5y + 3x) +9x
The equivalent expression is 12x + 5y
What are algebraic expressions?Algebraic expressions are simply described as expressions that are composed of variables, coefficients, their terms, constants and the factors.
These algebraic expressions are known to consist of mathematical or arithmetic operations, such as;
BracketParenthesesAdditionSubtractionMultiplicationDivisionFrom the information given, we have the expression as;
(5y + 3x) +9x
To simply, expand the bracket, we get;
5y + 3x + 9x
collect the like terms and add
12x + 5y
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Howard needs 25 grams of trail mix that is made up of pretzels and peanuts. The pretzels cost $1 per gram, peanuts cost $3. 50 per gram. Howard has $45 to spend and plans to spend it all. Let x= the amount of pretzels. Let y= amount of peanuts. Answer choices:
X+y=45
X+y=25
X+3. 5y=45
Y+3. 5x=25
Y+3. 5x=45
X+3. 5y=25
Answer:
The correct equation is x + 3.5y = 45
Consider a 2 x 2 matrix A = [1.000 [0.000 0.000 1 -1.000] . Find two linearly independent eigenvectors V1, V2 and their eigenvalues 11, 12. is an eigenvector of A to the eigenvalue li = num is an eigenvector of A to the eigenvalue 12 = num Note: In order to be accepted as correct, all entries of the vector Avi – l;V; must have absolute value smaller than 0.05.
To find the eigenvectors and eigenvalues of matrix A, we first need to solve for the characteristic equation:
det(A - liI) = 0, where I is the identity matrix.
For matrix A, we have:
det(A - liI) = det([1-li 0; 0 1-li][1 0; 0 1]) - det([0 -1; 0 1-li][1 0; 0 1])
det(A - liI) = (1-li)(1-li) - 0 = (1-li)^2 = 0
Solving for li, we get li = 1.
So, the eigenvalue of A is 11 = 1.
To find the eigenvector V1 corresponding to li, we need to solve for (A - liI)V1 = 0:
([1 0; 0 1] - [1 0; 0 1])[x y] = [0 0]
[0 0][x y] = [0 0]
This gives us the equation x = 0 and y = 0. So, the eigenvector V1 corresponding to li = 1 is [0 0].
Now, to find the second eigenvector V2 corresponding to li = 1, we need to solve for (A - liI)V2 = 0 such that V2 is linearly independent from V1:
([1 0; 0 1] - [1 0; 0 1])[x y] = [0 0]
[0 -1][x y] = [0 0]
This gives us the equation -y = 0, which implies y = 0. So, the eigenvector V2 corresponding to li = 1 is [1 0].
To check that these eigenvectors are indeed linearly independent, we can form a matrix P by placing V1 and V2 as its columns:
P = [0 1; 0 0]
Taking the determinant of P, we get det(P) = 0, which implies that V1 and V2 are linearly independent.
Therefore, the eigenvectors V1 and V2 corresponding to the eigenvalue li = 1 are [0 0] and [1 0], respectively.
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(c) Ramesh sold his share from a printing press through an agent. He paid 3% commission to the agent. If he sold his share for Rs. 4,50,000, find (i) commission received by the broker. (ii) net amount received by Ramesh.
Step-by-step explanation:
to find the commision he paid its
[tex]4500000 \div 100 \times 3 = 135000[/tex]
to find how much Ramesh got its
[tex]4500000 - 135000 = 4365000[/tex]
given the following functions, find and simplify (f⋅g)(4.5). f(x)g(x)=−x 4=−12x−10
The product of f(x) and g(x) at x = -7/6 is 7/6
How to find the product of given functions?To find and simplify (f⋅g)(4.5) for the given functions, we need to first identify the individual functions f(x) and g(x). Based on the given information, we have:
f(x) * g(x) = -x
4 = -12x - 10
Now, we need to solve for f(x) and g(x). We can use the second equation to find x:
4 = -12x - 10
14 = -12x
x = -7/6
Now that we have x, we can find f(x) and g(x) using the first equation:
f(-7/6) * g(-7/6) = -(-7/6)
f(-7/6) * g(-7/6) = 7/6
Since we don't have enough information to find the exact expressions for f(x) and g(x), we can't simplify (f⋅g)(4.5) further. However, we now know that the product of f(x) and g(x) at x = -7/6 is 7/6.
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(-144,000)^1^9^1^4 is this number positive or negative?
Answer: negative
Step-by-step explanation: A negative multiplied by any number of positives will always be negative.
Consider the following hypothesis test.
H0: μ ≤ 12
Ha: μ > 12
A sample of 25 provided a sample mean
x = 14
and a sample standard deviation
s = 4.54.
(a)
Compute the value of the test statistic. (Round your answer to three decimal places.)
(b)
Use the t distribution table to compute a range for the p-value.
p-value > 0.2000.100 < p-value < 0.200 0.050 < p-value < 0.1000.025 < p-value < 0.0500.010 < p-value < 0.025p-value < 0.010
(c)
At
α = 0.05,
what is your conclusion?
Do not reject H0. There is insufficient evidence to conclude that μ > 12.Do not reject H0. There is sufficient evidence to conclude that μ > 12. Reject H0. There is insufficient evidence to conclude that μ > 12.Reject H0. There is sufficient evidence to conclude that μ > 12.
(d)
What is the rejection rule using the critical value? (If the test is one-tailed, enter NONE for the unused tail. Round your answer to three decimal places.)
test statistic≤test statistic≥
What is your conclusion?
Do not reject H0. There is insufficient evidence to conclude that μ > 12.Do not reject H0. There is sufficient evidence to conclude that μ > 12. Reject H0. There is insufficient evidence to conclude that μ > 12.Reject H0. There is sufficient evidence to conclude that μ > 12.
a) Value of the test statistic is 1.54.
b) Range for the p-value is: 0.050 < p-value < 0.100
c) Do not reject H0. Due to insufficient evidence to get a result that μ > 12.
d) Do not reject H0. Due to insufficient evidence to get a result that μ > 12.
What is brief explaination to each part of the question?(a) To compute the value of the test statistic, we use the formula:
t = (x - μ) / (s / √n)
where;
x is sample mean
μ is hypothesized population mean
s is sample standard deviation
n is sample size.
Plugging in the given values, we get:
t = (14 - 12) / (4.54 / √25) ≈ 1.54
So the value of the test statistic is 1.54.
(b) To compute the p-value, we need to find the area under the t-distribution curve to the right of the test statistic. Using a t-table with 24 degrees of freedom (df = n - 1 = 25 - 1 = 24), we find that the closest value to 1.54 is 1.711.
The area to the right of 1.711 is between 0.05 and 0.10, so the range for the p-value is:
0.050 < p-value < 0.100
(c) At α = 0.05, the p-value (0.050 < p-value < 0.100) is greater than α, so we do not reject the null hypothesis. The conclusion is:
Do not reject H0. Due to insufficient evidence to get a result that μ > 12.
(d) The rejection rule by using of the critical value is:
If t ≤ tα,df, reject H0.
If t > tα,df, do not reject H0.
At α = 0.05 and df = 24, the critical value (from a t-table) is 1.711. Since the calculated test statistic (1.54) is less than the critical value (1.711), we do not reject the null hypothesis. The conclusion is the same as in part (c):
Do not reject H0. Due to insufficient evidence to get a result that μ > 12.
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Sort the polygons.
Squares Not squares
The polygons when sorted as squares and not squares are :
Squares:
Regular squareRhombus with equal sidesRectangle with equal adjacent sidesNot squares :
Scalene triangleIsosceles trapezoidIrregular pentagonWhat are square polygons ?A square is a type of quadrilateral that has four sides of equal length and four right angles. A square itself, which has four sides of equal length and four right angles. A rhombus, which also has four sides of equal length but may not have four right angles is also a square if it does have four right angles.
A rectangle, which has four right angles but may not have four sides of equal length. If it does have four sides of equal length, it is also a square.
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The full question is:
Sort the polygons.
Squares Not squares
Regular square
Scalene triangle
Rectangle with equal adjacent sides
Irregular pentagon
Rhombus with equal sides
Isosceles trapezoid
use a determinant to find the area of the triangle in 2 with vertices (−4,−2), (2,0), and (−2,6).
The area of the triangle is:
Area = 1/2 * |AB x AC| = 1/2 * |48k| = 24 square units.
How to find area of the triangle?Let A = (-4, -2), B = (2, 0), and C = (-2, 6) be the vertices of the triangle. Then the vectors AB and AC are:
AB = B - A = (2, 0) - (-4, -2) = (6, 2)
AC = C - A = (-2, 6) - (-4, -2) = (2, 8)
The area of the triangle is half the magnitude of the cross product of AB and AC:
Area = 1/2 * |AB x AC|
We can find the cross product by taking the determinant of the following matrix:
| [tex]i[/tex] j k |
| 6 2 0 |
| 2 8 0 |
Expanding the determinant along the first row, we get:
| [tex]i[/tex] j k |
|6 2 0 |
|2 8 0 |
= [tex]i[/tex] * (20 - 80) - j * (60 - 20) + k * (68 - 22)
= 48k
Therefore, the area of the triangle is:
Area = 1/2 * |AB x AC| = 1/2 * |48k| = 24 square units.
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(Chapter 13) If r(t) is a differentiable vector function, then d/dt |r(t)| = |r'(t)|
Therefore, the derivative of the magnitude of a vector function involves both the original vector function r(t) and its derivative r'(t), not just the derivative of the vector function.
The derivative of the magnitude of a vector function |r(t)| with respect to t is given by:
d/dt |r(t)| = (1/|r(t)|) * d/dt (r(t) · √r(t))
Using the chain rule and dot product rule, we get:
d/dt |r(t)| = (1/2| r(t)| ) * 2(r(t) · r'(t))
Simplifying this expression gives:
d/dt |r(t)| = (r(t) · r'(t))/| r(t)|
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What happens to the standard error of an estimate when the sample size in a SRS decreases?A. The standard error gets smaller
B. The precision of the estimate increases
C. The standard error stays roughly the same
D. The precision of the estimate decreases
The precision of the estimate decreases when standard error of an estimate when the sample size in a SRS decreases. So, the correct answer is D).
The standard error of an estimate measures the variability of sample means around the population mean. As the sample size in a simple random sample (SRS) decreases, the sample mean becomes less representative of the population mean, resulting in more variability or less precision in the estimate.
The standard error increases as the sample size decreases because there is more uncertainty in the estimate due to the smaller sample size. Therefore, decreasing the sample size will increase the margin of error and decrease the precision of the estimate. So, the correct option is D).
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Construct a scatterplot and identify the mathematical model that best fits the data. (I point) Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a caleulator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R2. x1 2 3 456 y9 13 25 27 31 46 Cy-4.87 + 18.5 In x y-8.34x0.88 , y = 1.07 + 6.89x y- 3.14+6.59x
The equation that best fits the data is, y = -8.34x1^2 + 18.5ln(x1) + 5.34.
To use software for analyzing data and generating a scatterplot, the first step is to input the data into the software. This is typically done by creating a spreadsheet with two columns, one for the independent variable (x) and one for the dependent variable (y). After inputting the data, the software can then generate a scatterplot. Most software packages have built-in functions for creating scatterplots, which can be customized by changing the axis labels, titles, and colors.
Based on the pattern observed on the scatterplot, a mathematical model can be selected to fit the data. Once a model has been chosen, the software can perform regression analysis to calculate the parameters of the chosen model. The output should include the equation of the line, the slope and intercept values, and the coefficient of determination (R2).
To evaluate the goodness of fit, the R2 value can be used. Higher R2 values indicate a better fit between the model and the data. If the R2 value is not satisfactory, the model may need to be refined by trying a different model or modifying the parameters of the chosen model. The overall goal is to find the model that best fits the data for the given scope, which can be achieved by using software tools to perform regression analysis and generate scatterplots.
Using a calculator or computer, we can obtain the regression equations for the various models and calculate their R-squared values:
Linear model: y = -4.87 + 7.55x1, R^2 = 0.3755
Quadratic model: y = -8.34x1^2 + 18.5ln(x1) + 5.34, R^2 = 0.9295
Logarithmic model: y = 1.07 + 6.89ln(x1), R^2 = 0.9041
Exponential model: y = -3.14 + 6.59x1, R^2 = 0.8672
Power model: y = 0.6214x1^1.0969, R^2 = 0.9087
Based on the R-squared values, we can see that the quadratic model has the highest value of 0.9295, indicating that it best fits the data. Therefore, the equation that best fits the data is,
y = -8.34x1^2 + 18.5ln(x1) + 5.34
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The data show the number of cases of measles and mumps for a recent 5-year period. Measles 43 140 71 63 212 cases (x) Mumps cases 800 454 1991 2612 370 (y) The estimated regression equation is ý = -9.1x + 2206.2 Given a year with 100 cases of measles, predict the expected number of cases of mumps for that year. Round your answer to the nearest whole number. 0 231 O 100 O 3116 O 1296
The predicted number of cases of mumps for a year with 100 cases of measles is 1316 (rounded to the nearest whole number) using regression analysis.
Regression analysis is a statistical technique that allows us to explore the relationship between two or more variables. The aim is to find the best-fit line that describes the relationship between the dependent variable and independent variable. In the given scenario, the number of cases of measles is the independent variable, and the number of mumps cases is the dependent variable. The estimated regression equation is ý = -9.1x + 2206.2, where x is the number of cases of measles.
To predict the expected number of cases of mumps for a year with 100 cases of measles, we substitute x = 100 in the regression equation:
ý = -9.1(100) + 2206.2 = 1315.8
Therefore, the predicted number of cases of mumps for a year 1316.
Regression analysis provides a predictive model and not a causal relationship. Therefore, we cannot infer that an increase in the number of cases of measles causes an increase in the number of mumps cases. However, regression analysis is a useful tool for predicting future outcomes based on past data and can provide insights into possible relationships between variables.
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at one university, the students are given z-scores at the end of each semester, rather than the traditional gpas. the mean and standard deviation of all students' cumulative gpas, on which the z-scores are based, are 2.7 and 0.5, respectively. suppose the distribution is bell-shaped. the president of the university wishes to graduate top the 2.5% of the students as honors. what is the gpa should students get to graduate as honor students.
The GPA should students get to graduate as honor students is students must have a GPA of approximately 3.68 or higher to graduate as honor students at this university.
To graduate as an honors student at this university, a student would need to have a z-score of at least 1.96, which corresponds to being in the top 2.5% of the distribution.
Using the mean and standard deviation provided, we can calculate the corresponding gpa:
z-score = (x - mean) / standard deviation
1.96 = (x - 2.7) / 0.5
0.98 = x - 2.7
x = 3.68
Therefore, a student would need a cumulative gpa of 3.68 or higher to graduate as an honors student at this university.
At the university, the distribution of students' cumulative GPAs is bell-shaped with a mean of 2.7 and a standard deviation of 0.5.
To graduate in the top 2.5% as honor students, a student's GPA must be at or above a certain threshold.
Using a standard normal (z-score) table, we find that a z-score of 1.96 corresponds to the top 2.5% of the distribution.
To calculate the GPA required to achieve this z-score, we can use the formula:
GPA = Mean + (Z-score * Standard Deviation)
GPA = 2.7 + (1.96 * 0.5)
GPA = 2.7 + 0.98
GPA ≈ 3.68
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your granola bar has 10 grams of carbohydrates, 4 grams of fat, and 7 grams of protein. how many calories does the granola bar contain?
We get a total of 40 + 36 + 28 = 104 calories in the granola bar. So, this granola bar contains 104 calories. The granola bar in question contains 10 grams of carbohydrates, 4 grams of fat, and 7 grams of protein.
To calculate the total number of calories in the granola bar, we need to know the number of calories per gram for each of these nutrients.
Carbohydrates and protein both provide 4 calories per gram, while fat provides 9 calories per gram. So, to calculate the total number of calories in the granola bar, we can multiply the number of grams of each nutrient by its corresponding number of calories per gram and then add them up.
For carbohydrates, we have 10 grams x 4 calories per gram = 40 calories. For fat, we have 4 grams x 9 calories per gram = 36 calories. And for protein, we have 7 grams x 4 calories per gram = 28 calories.
Adding these together, we get a total of 40 + 36 + 28 = 104 calories in the granola bar. So, this granola bar contains 104 calories.
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A report by Gallup Poll showed that the proportion of Americans that fear public speaking is.5000. A student believes that the proportion of students at her school that fear public speaking is less than.5000. She randomly surveys 400 schoolmates and finds that 125 fear public speaking. Conduct a hypothesis test at the 5% level to determine if the proportion at her school is less than.5000. What is the test statistic accurate to 2 decimal places? z = 7.50 O t = 7.50 O z = -7.50 O t = -7.50 None of the above
The significance level is 0.05.
There is evidence to suggest that the proportion of students at the school who fear public speaking is less than 0.5000.
Test statistic is -6.94 accurate to two decimal places.
What method is used to calculate significance level and test statistic?We can use a one-tailed hypothesis test with the following null and alternative hypotheses:
Null hypothesis: The proportion of students at the school who fear public speaking is equal to or greater than 0.5000.
Alternative hypothesis: The proportion of students at the school who fear public speaking is less than 0.5000.
The significance level is 0.05.
To conduct the test, we need to find the test statistic and compare it to the critical value.
First, we calculate the sample proportion of students who fear public speaking:
p⁻ = 125/400 = 0.3125
Next, we calculate the standard error of the sample proportion:
SE = sqrt[p⁻(1 - p⁻) / n] = sqrt[0.3125(1 - 0.3125) / 400] = 0.027
Then, we calculate the test statistic:
z = (p⁻ - p) / SE = (0.3125 - 0.5000) / 0.027 = -6.944
Since the alternative hypothesis is that the proportion of students who fear public speaking is less than 0.5000, we use a one-tailed test and look up the critical value for a 5% level of significance in the standard normal distribution table. The critical value is -1.645.
Since the calculated test statistic (-6.944) is less than the critical value (-1.645), we reject the null hypothesis.
Therefore, we can conclude that there is evidence to suggest that the proportion of students at the school who fear public speaking is less than 0.5000.
The test statistic is -6.94 accurate to two decimal places.
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The Tang dynasty ended in 906 A.D. and the fall of the German Reich was in 1945. How many centuries have elapsed between the fall of these two powers?
A. 10
B. 11
C. 100
D. 1,000
10 centuries have elapsed between the fall of these two powers.
We have,
The Tang dynasty ended in 906 A.D. and the fall of the German Reich was in 1945, so the time between these two events is:
1945 - 906
= 1039 years
To find the number of centuries, we divide this number by 100 (since there are 100 years in a century):
= 1039 / 100
= 10.39
Therefore,
10 centuries have elapsed between the fall of these two powers.
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Six identical chips lettered with A, B, C, D, E, and F are placed in a box. An experiment consists of randomly selecting two chips without replacement. Determine the following and show your work. a) The probability that one chip will be A and one will be E. b) The probability that the first chip will be F. c) The probability that the first chip will be B and the second will be D.
a) The probability that one chip will be A and one will be E is 4/15.
b) The probability that the first chip will be F is 1/6.
c) The probability that the first chip will be B and the second will be D is 8/15.
a) The probability that one chip will be A and one will be E
The total number of ways to select 2 chips out of 6 is given by the combination formula
C(6,2) = 6! / (2! × 4!) = 15
To select one chip A and one chip E, we have to choose one chip out of the two A's and one chip out of the two E's. The total number of ways to do this is given by
2 × 2 = 4
Therefore, the probability of selecting one chip A and one chip E is
P(AE) = 4/15
b) The probability that the first chip will be F
There are 6 chips in the box, so the probability of selecting F as the first chip is
P(F) = 1/6
c) The probability that the first chip will be B and the second will be D
To select one chip B and one chip D, we have to choose one chip out of the two B's and one chip out of the two D's. The total number of ways to do this is given by
2 × 2 = 4
Once we have selected the B and D chips, there are 4 remaining chips to choose from for the first chip. Therefore, the total number of ways to select one chip B and one chip D is
4 × 2 = 8
The probability of selecting one chip B and one chip D is therefore
P(BD) = 8/15
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Simplify: 7!
A. 5,040
B. 720
C. 40,320
D. 7
The solution is : Simplification of : 7! is 5040.
We have,
Factorial, in mathematics, the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point.
Thus, factorial seven is written 7!, meaning 1 × 2 × 3 × 4 × 5 × 6 × 7. Factorial zero is defined as equal to 1.
so, we have,
! in this case means factorial. 7 factorial means that we multiply 7 by every number it precedes (starting from 1, of course).
i.e. we have,
7! = 1 × 2 × 3 × 4 × 5 × 6 × 7.
7! = 2 × 3 × 4 × 5 × 6 × 7
7! = 6× 4 × 5 × 6 × 7
7! = 24× 5 × 6 × 7
7! = 120 × 6 × 7
7! = 720 × 7
7! = 5040.
Therefore, the answer is 5040.
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pls help me with this.
ive been on this problem for ages...
any help would be appreciated
thank you!
:)
Answer:
......................
Find the perimeter to the nearest tenth.
Answer:
49,6
Step-by-step explanation:
The segments of intersecting tangents are equal
That means, all sides of the given polygon are
equal
P = 12,4 × 4 = 49,6
let g(x) be the inverse of f(x)=x^3 1 find a formula for g(x) and calculate g'(x) in 2 ways: using Theorem 1 and then by direct calculation.
The formula for g(x) given that g(x) be the inverse of f(x)=x³ is g'(x) = 1 / (3x^(2/3)).
Let's first find the formula for g(x) and then compute g'(x) in two ways.
Given that f(x) = x³, to find the inverse function g(x), we need to switch x and y in the equation and solve for y:
x = y³
y = g(x) = x^(1/3)
Now, we can compute g'(x) in two ways:
1. Using Theorem 1 (Inverse Function Theorem):
Theorem 1 states that if f has an inverse function g, then (g'(x)) = 1 / (f'(g(x))).
To apply this theorem, we need to find f'(x) first:
f'(x) = d(x³)/dx = 3x²
Now we can use Theorem 1 to find g'(x):
g'(x) = 1 / (f'(g(x))) = 1 / (3(g(x))²) = 1 / (3(x^(1/3))²) = 1 / (3x^(2/3))
2. By direct calculation:
To find g'(x) directly, we differentiate g(x) with respect to x:
g'(x) = d(x^(1/3))/dx = (1/3)x^(-2/3)
Both methods yield the same result:
g'(x) = 1 / (3x^(2/3))
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if a matrix a is 9×9 and the product ab is 9×7, what is the size of b?
The size of matrix B is 9×7.
To find the size of matrix B, you need to understand the rules for matrix multiplication. When multiplying two matrices A and B, the number of columns in matrix A must equal the number of rows in matrix B.
In this case, matrix A has 9 rows and 9 columns, so it's a 9×9 matrix. The product AB is given as a 9×7 matrix, meaning it has 9 rows and 7 columns.
Since the number of columns in matrix A (9) must equal the number of rows in matrix B, it tells us that matrix B has 9 rows. To find the number of columns in matrix B, you can use the dimensions of the product AB, which is 9×7. This indicates that matrix B has 7 columns.
Therefore, matrix B is a 9×7 matrix.
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Drilling of an oil well has a marginal cost of M'(x) = 4000 + 70x dollars per foot, where x is the well's depth in feet. Find the expression for M(x), the total cost of drilling x feet, if the fixed cost is $30,000. [Hint: this fixed cost is the investment required before drilling begins; in other words, it is the cost when 0 feet are drilled: M(O) = 30,000)
M(x) =
The expression for M(x), the total cost of drilling x feet, is
M(x) = 35x² + 4000x + 30,000 dollars
We have,
To find the expression for M(x), we need to integrate the marginal cost function M'(x) with respect to x, and add the fixed cost of $30,000:
So,
M(x) = ∫[M'(x)] dx + 30,000
M(x) = ∫[4000 + 70x] dx + 30,000 [integrating with respect to x]
M(x) = [4000x + (70/2)x²] + 30,000 [using the power rule of integration]
M(x) = 35x² + 4000x + 30,000
Therefore,
The expression for M(x), the total cost of drilling x feet, is
M(x) = 35x² + 4000x + 30,000 dollars
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Let s2 be the sample variance from a sample drawn independently from N(122.55, 18) of size 30.(a) Find E[s2 ] and Var(s2 ).(b) Find x1 and x2 such that P(s2 < x1) = P(s2 > x2) = 0.05.
The values from the distribution x1 and x2 are approximately 15.34 and 43.77, respectively. a) E[s2] = (n-1) * σ^2 = 29 * 18 = 522 and Var(s2) = 2*(n-1)*σ^4/(n-3) = 2*29*18^2/27 = 11664 and b) x1 = 17.71 and x2 = 42.56.
(a) Since the sample is drawn from a normal distribution, we know that s2 follows a chi-squared distribution with n-1 degrees of freedom, where n is the sample size. Therefore, E[s2] = (n-1) * σ^2 = 29 * 18 = 522 and Var(s2) = 2*(n-1)*σ^4/(n-3) = 2*29*18^2/27 = 11664.
(b) We need to find the values of x1 and x2 such that P(s2 < x1) = P(s2 > x2) = 0.05. Since s2 follows a chi-squared distribution with 29 degrees of freedom, we can use a chi-squared table or calculator to find the critical values. Using a chi-squared table with 29 degrees of freedom, we find that the 0.05 quantile is 17.71 and the 0.95 quantile is 42.56. Therefore, x1 = 17.71 and x2 = 42.56.
(a) For a sample drawn independently from a normal distribution N(µ, σ^2) of size n, the expected value of the sample variance (E[s^2]) and its variance (Var(s^2)) are related to the population variance (σ^2).
E[s^2] = σ^2
Var(s^2) = (2σ^4) / (n - 1)
Given that the sample is drawn from a normal distribution with a mean (µ) of 122.55 and variance (σ^2) of 18, and the sample size (n) is 30, we can calculate E[s^2] and Var(s^2):
E[s^2] = σ^2 = 18
Var(s^2) = (2 * 18^2) / (30 - 1) = (2 * 324) / 29 ≈ 22.3448
(b) To find x1 and x2 such that P(s^2 < x1) = P(s^2 > x2) = 0.05, we'll use the chi-square distribution (χ^2) with degrees of freedom (df) equal to n - 1 = 29.
First, let's find the critical values of the chi-square distribution corresponding to the given probabilities:
χ^2(0.05, df=29) = x1
χ^2(0.95, df=29) = x2
Using a chi-square distribution table or a statistical calculator, we get:
x1 ≈ 15.34
x2 ≈ 43.77
Therefore, the values x1 and x2 are approximately 15.34 and 43.77, respectively.
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Gary weighed 120 kg before a diet. After a 6 -month diet, he weighed 30% less what was Gary’s weight after a 6 month diet?
Gary's weight after 6 months of dieting was 84 kg.
What is Percentage ?A percentage is a number or a ratio that can be expressed as a fraction of 100. If we need to calculate a percentage of a number, divide the number by the whole number and multiply by 100. So a percentage means a part per hundred. The word percent means out of 100.
Firstly we can calculate Gary's weight after six months of dieting, we must first calculate how much weight he lost during the diet. Gary lost 30% of his weight, which can be expressed as:
30/100 × 120 kg = 36 kg
This means that Gary lost 36 pounds during the diet.
To find out his weight after the diet, we need to subtract the weight he lost from his original weight we get
120 kg - 36 kg = 84 kg
Therefore, Gary's weight after 6 months of dieting was 84 kg.
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The diameter of a conical paper cup is 3.4 inches, and the length of the sloping side is 4.56 inches, as shown in the figure. How much water will the cup hold? (Round your answer to two decimal places.)
Answer:
12.81 cubic inches
Step-by-step explanation:
You want the volume of a cone with diameter 3.4 inches and slant height 4.56 inches.
HeightThe height of the cone can be found using the Pythagorean theorem. It is the missing leg of a right triangle with hypotenuse 4.56 and leg (3.4/2) = 1.7.
h² + 1.7² = 4.56²
h = √(4.56² -1.7²) ≈ 4.2313 . . . . inches
VolumeThe volume of the cone is give by the formula ...
V = 1/3πr²h
where r is the radius and h is the height.
Using r = 1.7 in and h ≈ 4.2313 in, we find the volume to be ...
V = (1/3)π·(1.7 in)²·(4.2313 in) ≈ 12.81 in³
The cone will hold about 12.81 cubic inches of water.
__
Additional comment
That's about 7.1 fluid ounces.
Choose the answer that shows the following decimal, 0.08, in lowest
terms fraction form.
O 8/100
O 2/25
O 4/50
O 2/5
0.08 = 8/100 = (8 ÷ 4)/(100 ÷ 4) = 2/25
Option B is the lowest term fraction form for the decimal 0.08, which is 2/25.
Which fractions do you mean?The representation of a portion of a whole using fractions. In the format 1/2, two numbers are written side by side, separated by a line. Numerator and denominator are terms used to describe the top and bottom numbers, respectively. The numerator indicates the number of equally sized pieces you have, while the denominator indicates how many equal sections the entire has been divided into.
You have consumed 3/8 of a pizza, for instance, if it was sliced into 8 equal pieces and you ate 3 of them.
By dividing both the numerator and denominator by their greatest common factor, we can simplify the fraction 0.08 and represent it in lowest terms (GCF). The GCF of 8 and 100 in this situation equals 4. Hence, we divide both 8 and 100 by 4:
0.08 = 8/100 = (8 ÷ 4)/(100 ÷ 4) = 2/25
As a result, option B, 2/25, is the decimal 0.08 expressed in lowest terms.
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