Step-by-step explanation:
Using z-score table the value is .9793 (97.93 %)
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. N = 1
∑ 2n/n!
The Ratio Test shows that the series [tex]\sum_{n=0}^{\infty} \frac{2^n}{n!}[/tex] converges absolutely.
We must determine the maximum ratio of successive terms before we can apply the ratio test to the series 2n/n!
[tex]\lim_{n \to \infty} \left| \frac{2(n+1)/(n+1)!}{2n/n!} \right|[/tex]
Simplifying the expression, we get:
[tex]\lim_{n \to \infty} \left| \frac{2(n+1)/(n+1)(n!)}{2n/n!} \right|[/tex]
[tex]\lim_{n \to \infty} \left| \frac{2(n+1)}{(n+1)} \right| \\[/tex]
[tex]\lim_{n \to \infty} 2 \\[/tex]
The Ratio Test informs us that the series absolutely converges because the limit is a positive finite constant (2). The Ratio Test, which compares the growth rates of successive terms, is an effective method for examining the convergence of series with positive terms. The series absolutely converges if the limit of the ratio of consecutive terms is smaller than 1, and it diverges if it is bigger than 1.
The Ratio Test is inconclusive if the limit is exactly 1 or if it does not exist, in which case we may need to use additional convergence tests. In this instance, the Ratio Test informs us that the series definitely converges, hence we do not need to take into account any other tests.
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Using the distributive property, which of the following is the expanded form of −14(−8x+12y)
?
Using the distributive property, the expanded form of (−1/4)(−8x+12y) is c) 2x - 3y.
The distributive property states that when you multiply a number or a variable expression by a sum or a difference, you can distribute the multiplication over each term within the parentheses.
So, to expand the expression (−1/4)(−8x+12y), we can apply the distributive property by multiplying -1/4 to each term inside the parentheses:
(−1/4)(−8x+12y) = (−1/4) × (−8x) + (−1/4) × (12y)
= (1/4) × 8x − (1/3) × 12y
= 2x − 3y
Therefore, the expanded form of (−1/4)(−8x+12y) is 2x − 3y. Hence, the correct answer is (c) 2x-3y.
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A bag contains 3 black, 2 white marbles, and 4 gray marbles. A marble Is replaced before picking a second marble. Whats the probability of selecting gray marble?
The value of the probability of selecting gray marble is,
⇒ 4 / 9
We have to given that;
A bag contains 3 black, 2 white marbles, and 4 gray marbles.
And, A marble Is replaced before picking a second marble.
Hence, We get;
Total marbles = 3 + 2 + 4
= 9
And, Number of gray marbles = 4
Thus, The value of the probability of selecting gray marble is,
⇒ 4 / 9
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Assume the sample is a random sample from a distribution that is reasonably normally distributed and we are doing inference for a sample mean. Find endpoints of a t-distribution with 1% beyond them in each tail if the sample has size n=18.Round your answer to three decimal places.
The endpoints of a t-distribution with 1% beyond them in each tail if the sample has size n=18 are 2.898 and -2.898.
To find the endpoints of the t-distribution with 1% beyond them in each tail for a sample of size n = 18, we need to find the t-values such that the area under the t-distribution curve beyond those values in each tail is 0.01.
Since the sample size is small (n < 30), we use the t-distribution instead of the normal distribution. The degrees of freedom for the t-distribution is n-1 = 18-1 = 17.
Using a t-table or a calculator, we find the t-value that has an area of 0.005 to the right of it in the t-distribution with 17 degrees of freedom:
[tex]t_0_._0_0_5_,_1_7[/tex]= 2.898
[tex]t_0_._0_0_5_,_1_7[/tex] = 2.898
To find the left endpoint, we take the negative of the right endpoint:
−2.898
−2.898
Therefore, the endpoints of the t-distribution with 1% beyond them in each tail for a sample of size n=18 are -2.898 and 2.898 (rounded to three decimal places).
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Match each for plot with its median
Answer:
youre in high school and ure asking this question pls go back to elementaary and think of what u js wrote
Step-by-step explanation:
Figure ABCDE is the result of a 180° rotation of figure LMNOP about point F.
Which angle corresponds to angle D?
The angle that corresponds to angle D is O.
We have,
If you rotate an object or a coordinate system by 180 degrees, you are flipping it upside down or reversing it.
If you have a coordinate system with the x-axis running from left to right and the y-axis running from bottom to top, rotating it by 180 degrees would cause the x-axis to now run from right to left and the y-axis to run from top to bottom.
Now,
Figure ABCDE is the result of a 180° rotation of figure LMNOP about point F.
So,
The angle that corresponds to angle A is L.
The angle that corresponds to angle B is M.
The angle that corresponds to angle C is N.
The angle that corresponds to angle D is O.
The angle that corresponds to angle E is P.
Thus,
The angle that corresponds to angle D is O.
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I need help with this question
The equivalent expressions are given as follows:
[tex]b^{\frac{2}{3}} = \sqrt[3]{b^2}[/tex][tex]\sqrt[2]{b^3} = b^{\frac{3}{2}}[/tex][tex]\sqrt[3]{b^5} = b^{\frac{5}{3}}[/tex][tex]b^{\frac{5}{2}} = \sqrt{b^5}[/tex]How to obtain the radical form of each expression?The general format of the exponential expression is given as follows:
[tex]a^{\frac{n}{m}}[/tex]
To obtain the radical form, we have that:
a is the radicand.n is the exponent.m is the root.Hence the radical form of the exponential expression is given as follows:
[tex]a^{\frac{n}{m}} = \sqrt[m]{a^n}[/tex]
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find a parametric representation using spherical-like coordinates for the upper half of the ellipsoid 4(x 1)2 9y2 36z2
A parametric representation using spherical-like coordinates for the upper half of the ellipsoid 4(x₁)² + 9y² + 36z² is given by:
x = 2r cosθ sinφ
y = 3r sinθ sinφ
z = 6r cosφ, where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π/2.
We want to find a parametric representation for the upper half of the ellipsoid 4(x₁)² + 9y² + 36z² = 36. To do this, we can use spherical-like coordinates, which are similar to spherical coordinates but with an additional parameter to account for the asymmetry of the ellipsoid.
We start by assuming that the ellipsoid is centered at the origin, so we can write it as:
(x₁/3)² + y²/4 + z²/1 = 1
We can then express x, y, and z in terms of the parameters r, θ, and φ:
x = r cosθ sinφ
y = r sinθ sinφ
z = r cosφ
We can use these equations to find r, θ, and φ in terms of x, y, and z, and substitute into the equation of the ellipsoid to obtain:
[(x₁/3)² + (y/2)² + z²]/1 = 1
Simplifying, we get:
r² = 36/(4 cos²θ sin²φ + 9 sin²θ sin²φ + 36 cos²φ)
We can then use the equations for x, y, and z in terms of r, θ, and φ to obtain the desired parametric representation:
x = 2r cosθ sinφ
y = 3r sinθ sinφ
z = 6r cosφ
We restrict φ to the range 0 ≤ φ ≤ π/2 to obtain only the upper half of the ellipsoid. The range of θ is 0 ≤ θ ≤ 2π, which allows us to cover the entire surface of the ellipsoid.
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The value of the integral ∫^2_0 t^2d(t - 4) Select one: O a. equals to 0.
O b. equals to 16. O c. equals to 1
O d. equals to 8/3
O e. does not exist.
The value of the integral ∫^2_0 t^2d(t - 4) equals to 8/3 (option d). To get the value of the integral, we will perform the following steps:
1. Rewrite the integral in terms of dt using the substitution method.
2. Evaluate the integral.
3. Substitute the limits of integration and calculate the value.
Step 1: Substitution
Let u = t - 4, so du = dt.
When t = 0, u = -4. When t = 2, u = -2.
Now we have: ∫^(-2)_{-4} (u + 4)^2 du.
Step 2: Evaluate the integral
We need to integrate (u + 4)^2 with respect to u.
∫ (u + 4)^2 du = (1/3)(u + 4)^3 + C, where C is the constant of integration.
Step 3: Substitute the limits of integration and calculate the value
[(1/3)(-2 + 4)^3 - (1/3)(-4 + 4)^3] = (1/3)(2^3) = 8/3.
The value of the integral ∫^2_0 t^2d(t - 4) equals to 8/3 (option d).
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What is the probability that either event will occur?
Now, find the probability of event B.
A
6
20
6
B
20
P(B) = [?]
Enter as a decimal rounded to the nearest hundredth.
Enter
The probability of either event A or event B occurring is 1 or 100%, since they are the only two possible outcomes. The probability of event B occurring alone is also 1 or 100%.
To find the probability that either event A or event B will occur, we can add their individual probabilities and then subtract the probability that both events occur, since we don't want to count this intersection twice. So, we have
P(A or B) = P(A) + P(B) - P(A and B)
Plugging in the given values
P(A or B) = 6/20 + 20/20 - 6/20 = 20/20 = 1
So the probability that either event A or event B will occur is 1 or 100%.
To find the probability of event B, we simply use the given probability
P(B) = 20/20 = 1
So the probability of event B is also 1 or 100%.
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Answer:
its 0.50
Step-by-step explanation:
Use the quadratic formula to find the solutions to the equation.
3x²10x+5=0
O A.
10 ± √40
6
2+√24
O B.
O c. 1± √/35
O D. 5 ± √15
3
Using the quadratic formula to find the solutions, we get x = (-10 ± √40)/6
Using the quadratic formula to find the solutions to the equation.From the question, we have the following parameters that can be used in our computation:
3x² + 10x + 5=0
Using the quadratic formula, we have
x = (-10 ± √(10² - 4 * 3 * 5))/(2 * 3)
Evaluate the products and the exponents
So, we have
x = (-10 ± √40)/6
Hence, the solution is x = (-10 ± √40)/6
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Which transformations take the graph of f(x) = 5^x to the graph of g(x) = 5^x+7 — 2? *Choose two correct answers.*
a) The graph is translated left 7 units. b) The graph is translated down 2 units c) The graph is translated to the right 7 units.
d) The graph is translated left 2 units.
The transformations that take the graph of f(x) = 5^x to the graph of g(x) = 5^x+7 — 2 are;
a) The graph is translated left 7 units.
b) The graph is translated down 2 units
What is the transformation of the function?The function originally is given as:
f(x) = 5^(x)
Now, after transformation it becomes:
g(x) = (5^(x + 7)) - 2
We know that:
If the function f(x) is translated by a units to the right, the we have: 5^(x - a)
If the function f(x) is translated by a units to the left, the we have: 5^(x +a)
If the function f(x) is translated by b units up , the we have: 5^(x) + b
If the function f(x) is translated by b units down , the we have: 5^(x) - b
Thus, the transformation occurring is:
The graph is translated left 7 units
The graph is translated down 2 units
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When is the sum of two rational numbers with different signs positive?
The sum of two rational numbers with different signs is positive whilst the absolute value of the variety with the larger absolute value is greater than the absolute value of the number with the smaller absolute value.
In different meaning, the bigger positive rational number is greater than absolutely the value of the smaller negative rational number.
For Example, if we add the rational figures together also the -2/ 3 and4/5, we have got
-2/3 + 4/5 = (-10/15) + (12/15) = 2/15
In this example, the sum is positive due to the fact absolutely the value of 4/5 (0.8) is greater than the absolute value of -2/3 (0.666...).
Therefore, the larger positive rational number (4/five) is greater than the absolute value of the smaller negative rational number (-2/3).
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When a $10 check written on the First National Bank is deposited in an account at the Second National Bank, then A) the liabilities of the First National Bank decrease by $10. B) the reserves of the First National Bank increase by $10. C) the liabilities of the Second National Bank decrease by $10. D) the assets of Second National Bank decrease by $10.
When a $10 check written on the First National Bank is deposited in an account at the Second National Bank, the correct option is C) the liabilities of the Second National Bank decrease by $10. This is because the Second National Bank's liability to the account holder increases by $10, while its liability to the First National Bank decreases by $10. However, none of the other options are correct.
Option A) is incorrect because the liabilities of the First National Bank do not decrease as a result of the check being deposited in the Second National Bank. The liability of the First National Bank to the account holder is transferred to the Second National Bank.
Option B) is incorrect because the reserves of the First National Bank do not increase by $10 when the check is deposited in the Second National Bank. Reserves refer to the amount of money that banks hold in order to meet their depositors' demands for cash withdrawals. When the check is deposited in the Second National Bank, the reserves of the First National Bank remain the same.
Option D) is incorrect because the assets of the Second National Bank do not decrease by $10 when the check is deposited. The asset of the Second National Bank increases by $10 due to the deposit of the check.
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What information do you need before you can decide which type of business might be the most successful?
Before deciding which type of business might be the most successful, you would need to gather a variety of information, including Market demand, Competitors, Industry trend, Financial projections and Target market:
Market demand: You need to identify the needs and wants of the target customers in the market.
Competitors: Analyze the competition in the market and determine what they offer, what their strengths and weaknesses are, and how you can differentiate your business from them.
Industry trends: Keep up with industry trends and identify any new or emerging technologies or trends that could affect your business.
Financial projections: Estimate the initial and ongoing costs of running your business, including overhead, staffing, marketing, and inventory. Create a financial projection that includes cash flow, income statements, and balance sheets to help determine if your business can be profitable.
Target market: Identify your target market and understand their demographics, preferences, and buying habits.
Legal requirements: Determine the legal and regulatory requirements for starting and operating a business in your location, including permits, licenses, and taxes.
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↑
The length of a box is 5 inches more than the width. The height is 3 inches less than the width.
Enter an equation that can be used to find the width, w, in inches, of the box when the total volume of the box is
1836 cubic inches.
The equation to find the volume of the box is w³ + 2w² - 15w - 1836 = 0.
How to find the equation of a figure?The length of a box is 5 inches more than the width. The height is 3 inches less than the width.
The box is a rectangular prism. Therefore,
volume of the box = lwh
where
l = lengthw = widthh = heightTherefore, the equation that can be used to find the width in inches of the box when the total volume is 1836 cubic inches can be calculated as follows:
l = 5 + w
h = w - 3
Therefore,
1836 = (5 + w)(w - 3)(w)
1836 = (5w - 15 + w² - 3w)w
1836 =(w² + 2w - 15)w
1836 = w³ + 2w² - 15w
w³ + 2w² - 15w - 1836 = 0
Hence, the equation is w³ + 2w² - 15w - 1836 = 0
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Which equation represents the slope-intercept form of the line below
An equation represents the slope-intercept form of the given line is y= 1/2 x+8. Therefore, option D is the correct answer.
Given that, y-intercept = (0, 8) and slope = 1/2.
The general equation of a straight line is y=mx+c, where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis.
Substitute m=1/2 and c=8 in y=mx+c, we get
y= 1/2 x+8
Therefore, option D is the correct answer.
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3 1 point The computing system of WITS is currently undergoing shutdown repairs. Previous shutdowns have been due to either hardware failure software
electronic failure. The system is forced to shut down 43% of the time when it experiences hardware problems, 9% of the time when it experlener
problems, and 95% of the time when it experiences electronic problems. Maintenance engineers have determined that the probabilityur the com
having hardware failure is 0.45, software and electronic failure 0.25 and 0.30, respectively. Given that there is a shutdown the probability that it
hardware failure is
The probability that a shutdown is due to hardware failure given that there is a shutdown is 0.346.
To find the probability that the shutdown is due to hardware failure given that there is a shutdown, we need to use Bayes' theorem.
Let H be the event that the shutdown is due to hardware failure, and S be the event that there is a shutdown. Then we need to find P(H|S).
Using the formula for Bayes' theorem:
P(H|S) = P(S|H) * P(H) / P(S)
We already have P(H) = 0.45, and P(S|H) = 0.43, the probability of a shutdown given hardware failure.
To find P(S), we need to use the law of total probability:
P(S) = P(S|H) * P(H) + P(S|E) * P(E) + P(S|S) * P(S)
where E is the event of electronic failure, and S is the event of software failure.
We are given P(S|E) = 0.95, P(E) = 0.30, P(S|S) = 0.09, and P(S|H), P(H) as calculated above.
Substituting the values:
P(S) = 0.43 * 0.45 + 0.95 * 0.30 + 0.09 * 0.25 = 0.558
Finally, substituting all values into Bayes' theorem:
P(H|S) = 0.43 * 0.45 / 0.558 = 0.346
Therefore, the probability that a shutdown is due to hardware failure given that there is a shutdown is 0.346.
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Find the value of X. Please help im studying i dont know what i did wrong (number 7 on homework 8 unit 7)
The value of the variable x is 8
How to determine the valueTo determine the value, we have to take note that the sum of the angles in polygon is determined with the formula
(n -2 )180
Such that the variable, 'n' is the number of sides of the polygon
From the information given, we have that;
n = 4
Substitute the values
(4-2) 180 = 360
Now, equate the measures to the angle, we have;
9x + 5 + 14x + 3 + 15x -11 + 7x + 3 = 360
collect the like terms
9x + 14x + 15x + 7x = 360
Add the like terms
45x = 360
Make 'x' the subject of formula
x = 8
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Jason worked 3 hours more than Keith. Jason worked 12 hours. Which equation represents this situation, if k is the numbers of hours Keith worked?
The equation that represents this situation, if k is the number of hours Keith worked is C. k + 3 = 12
A statement that affirms the equivalence of two expressions that are joined by the equals sign "=" is known mathematically as an equation. If Jason worked 12 hours, 3 more than Keith did, and k is the amount of hours Keith worked, then k + 3 = 12 is the proper equation to reflect the circumstance.
According to this calculation of the equation, the total number of hours Keith worked (k) plus the additional three hours equals 12, which agrees with the fact that Jason put in three more hours of labour than Keith did and worked for a total of 12 hours.
Complete Question:
Jason worked 3 more hours than Keith. Jason worked 12 hours. Which equation represents this situation, if k is the number of hours Keith worked?
A. 12 + k = 3
B. 12k = 3
C. k + 3 = 12
D. 3k = 12
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Hey id love if yall could fill out this worksheet for me its pretty simple i just have some stuff to do and dont have time currently to work on it
To be categorized as unemployed, a person has to be without a process, presently available to work, and actively seeking out work within the previous 4 weeks.
Unemployment is a time period regarding people who are employable and actively looking for a process but are not able to discover a process. blanketed in this institution are the ones people in the group of workers who are working but no longer have the precise job.
The situation of now not having a task but being a member of the labor pressure. To be considered unemployed, someone must be jobless but actively trying to find a task by having searched within the last 4 weeks.
Hence, To be categorized as unemployed, a person has to be without a process, presently available to work, and actively seeking out work within the previous 4 weeks.
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complete question:
how do economists define unemployed workers? group of answer choices although currently working, they are not happy with their current job working on contract for a limited time not currently working not currently working but actively searching for work
a researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. subjects were randomly assigned to either a treatment group or a control group. the mean blood pressure was determined for each group, and a 95% confidence interval for the difference in the means for the treatment group versus the control group, , was found to be . give an interpretation of this confidence interval.
A researcher conducted a study to investigate if a specific diet can help reduce blood pressure in people with high blood pressure. Participants were randomly assigned to a treatment group (following the diet) or a control group (not following the diet).
The mean blood pressure was calculated for both groups, and a 95% confidence interval for the difference in the means between the treatment and control groups was determined.
The interpretation of this 95% confidence interval is that, in 95 out of 100 similar experiments, the true difference in mean blood pressure between the treatment and control groups would fall within the calculated range. If the confidence interval does not include zero, it suggests that there is a significant difference between the treatment and control groups, meaning the diet may have a positive effect on reducing blood pressure. If the confidence interval includes zero, it indicates that the difference may not be statistically significant, and further research may be needed.
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Which ratio would be used to find x?
A. sin (20°) = x/54
B. cos (20°) = x/54
C. cos (20°) = 54/x
D. sin (20°) = 54/x
Answer:
The ratio that would be used to find x is C. cos (20°) = 54/x.
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
*REMEMBER*:
Sin: opposite/hypotenuse
Cos: adjacent/hypotenuse
Tan: opposite/adjacent
Using this, we have to find x, which is the hypotenuse of this right triangle. This immediately eliminates using a tangent equation.
We also don't know the opposite side of 20 degrees, so this also eliminates the sine.
Now, we know that the equation has to be either B or C, as those have cosines. The cosine of 20 degrees is equal to adjacent/hypotenuse, which would give us the equation:
cos(20)=54/x, meaning C is the correct option.
Hope this helps! :)
(Cost or debt) Carraway Seed Company is issuing a $1,000 par value bond that pays 12 percent annual interest and matures in o years. Investors are willing to pay
$935 for the bond. Flotation costs will be 13 percent of market value. The company is in a 20 percent tax bracket. What will be the firm's after-tax cost of debt on the
bond!
The firm's after-tax cost of debt on the bond will be%. (Round to two decimal places.)
The firm's after-tax cost of debt on the bond will be 11.90% .
To calculate the firm's after-tax cost of debt on the bond, we need to follow these steps:
1. Determine the bond's yield to maturity (YTM) given its price, par value, annual interest, and time to maturity.
2. Calculate the bond's pre-tax cost of debt.
3. Adjust for flotation costs.
4. Apply the tax bracket to find the after-tax cost of debt.
1. The bond's YTM can be calculated using a financial calculator or software. Given the bond's price of $935, par value of $1,000, annual interest of 12% ($120), and a maturity of 0 years, the YTM is approximately 12.83%.
2. The pre-tax cost of debt is the YTM, which is 12.83%.
3. Flotation costs are 13% of the bond's market value ($935). Therefore, flotation costs are $121.55 ($935 * 0.13). The adjusted bond price, taking into account flotation costs, is $813.45 ($935 - $121.55). To find the adjusted YTM, we can use the adjusted bond price, keeping other factors constant. The adjusted YTM is approximately 14.87%.
4. The company is in a 20% tax bracket. To find the after-tax cost of debt, we need to apply the tax bracket: After-tax cost of debt = Adjusted YTM * (1 - Tax Rate) = 14.87% * (1 - 0.2) = 14.87% * 0.8 = 11.90%.
The firm's after-tax cost of debt on the bond will be 11.90% (rounded to two decimal places).
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How many different subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ contain at least one element in common with each of the sets $\{2, 4, 6, 8, 10, 12\}$, $\{3, 6, 9, 12\}$ and $\{2, 3, 5, 7, 11\}\,?$
The number of different subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ is 13.
We are given that;
Subset = $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$
Now,
To apply the principle of inclusion-exclusion, we need to find the number of elements in each set and each intersection of sets. We have:
∣A∣=∣B∣=∣C∣=6
∣A∩B∣=∣A∩C∣=∣B∩C∣=2
∣A∩B∩C∣=1
Using the principle of inclusion-exclusion, we get:
∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣
Plugging in the values we have found above, we get:
∣A∪B∪C∣=6+6+6−2−2−2+1=13
Therefore, by the subset the answer will be 13.
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NEED HELP ASAP +POINTS
a swimmer preparing for a competition should train for no more than 60% of her normal training time. enter an inequality that represents all possible values for the time, t, in minutes, that the swimmer should train given her normal training time is 90 minutes
Answer: t [tex]\leq[/tex] 54
Step-by-step explanation: Normally, she trains for 90 minutes, so we need to figure out what 60% of 90 is.
90 x .60 = 54.
SO then we can make the inequality, t [tex]\leq[/tex] 54
Find the slope for the line that passes through the points (6,-1) and (2,2)
write the recursive rule for the sequence shown in each table.
position, n 1 2 3 4 5
term, f(n) 5 18 31 44 57
1 2 3 4 5 6 7
65 54 43 32 21 10 -1
1 2 3 4 5 6 7
-9 6 21 36 51 66 81
1 2 3 4 5 6 7 8
17 13 9 5 1 -3 -7 -11
The recursive rule for the sequence given is aₙ = aₙ₋₁ + 13.
Given that, a sequence,
position, n = 1 2 3 4 5
term, f(n) = 5 18 31 44 57
We need to write the recursive rule for the sequence,
So,
a₁ = 5, a₂ = 18
18-5 = 13
a₃ = 31, a₄ = 44
44-31 = 13
Therefore,
We see that, the preceding term is 13 less than the succeeding term,
We can write,
aₙ = aₙ₋₁ + 13
Hence, the recursive rule for the sequence given is aₙ = aₙ₋₁ + 13.
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if two numbers have a difference of 56 and we let x represent the smaller of these two numbers, then the larger of these two numbers is given by the following.
If x represents the smaller of the two numbers, then the larger number can be represented as x + 56 (since the difference between the two numbers is 56).
Large numbers are numbers that are larger than those used in everyday life, such as simple calculations or currency operations, and often appear in fields such as mathematics, cosmology, cryptography, and similar mathematical machine. These are usually good numbers or more likely really good numbers, but there may be other numbers in other situations.
Given that the difference between the two numbers is 56, and x represents the smaller number, the larger number can be represented as x + 56.
This is because the larger number is 56 units greater than the smaller one (x). So, the larger number is given by the expression: x + 56.
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Prove Euler's Rotation Theorem: When a sphere is moved about its center it is always possible to find a diameter of the sphere whose direction in the displaced position is the same as in the initial position.
Prove using something related to orthogonal properties
To prove Euler's Rotation Theorem using orthogonal properties, we will follow these steps:
Step 1: Consider a sphere with center O and radius R. Let A and A' be the initial and final positions of a point on the sphere after rotation.
Step 2: Since the sphere is rotated about its center, the distance between O and A remains equal to the radius R in both the initial and displaced positions (OA = OA' = R).
Step 3: Let B and B' be the initial and final positions of another point on the sphere after the same rotation. Again, the distances OB and OB' both equal the radius R.
Step 4: Euler's Rotation Theorem states that there exists a diameter of the sphere whose direction is the same in both the initial and displaced positions. Let C and C' be the endpoints of this diameter in the initial and displaced positions, respectively.
Step 5: Now, consider the plane formed by the points A, B, and C in the initial position, and the plane formed by the points A', B', and C' in the displaced position. These two planes are called "orthogonal planes" as they are perpendicular to the diameter CC'.
Step 6: Since the rotations of points A and B are both orthogonal to CC', the rotation axis must be along the diameter CC'. Therefore, the direction of the diameter CC' remains unchanged during the rotation.
In conclusion, we have proven Euler's Rotation Theorem using orthogonal properties. When a sphere is moved about its center, it is always possible to find a diameter of the sphere whose direction in the displaced position is the same as in the initial position.
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