Answer with Step-by-step explanation:
Part 1)
we know that
[tex]e^{i\theta }=cos(\theta )+isin(\theta )[/tex]
thus [tex]-i=e^{\frac{-i\times (4n-1)\pi }{2}}[/tex]
thus [tex](-i)^i=(e^{\frac{-i\times (4n-1)\pi }{2}})^i\\\\(-i)^i=e^{\frac{-i^2\times (4n-1)\pi }{2}}=e^{\frac{(4n-1)\pi }{2}}\\\\\therefore (-i)^i=e^{\frac{(4n-1)\pi }{2}}[/tex] where 'n' is any integer
Part 2)
We have [tex]-1=e^{(2n+1)\pi }\\\\\therefore (-1)^{i}=(e^{i(2n+1)\pi })^{i}\\\\(-1)^i=(e^{i^2(2n+1)\pi })\\\\(-1)^i=e^{-(2n+1)\pi }[/tex] where 'n' is any integer
One of the events at a swim meet is the 500 meter freestyle. Use conversion factors and dimensional analysis to determine the length of this race in: 2) a) feet. b) miles.
Answer:
1) Distance in feet equals 1640.4 feet
2) Distance in miles equals 0.3107 miles.
Step-by-step explanation:
From the basic conversion factors we know that
1 meter = 3.2808 feet
Thus by proportion we conclude that
[tex]500meters=500\times 3.2808feet\\\\\therefore 500meters=1640.4feet[/tex]
Similarly using the basic conversion factors we know that
1 mile = 1609.34 meters
thus we conclude that 1 meter =[tex]\frac{1}{1609.34}[/tex] miles
Hence by proportion 500 meters = [tex]500\times \frac{1}{1609.34}=0.3107[/tex] miles
If 30 participants each completed a memory task three times – once each after having no, moderate, or high levels of caffeine, calculate the total degrees of freedom.
Answer: 29
Step-by-step explanation:
In statistics, the degrees of freedom gives the number of values that have the freedom to vary.
If the sample size is 'n' then the degree of freedom is given by:-
df= n-1
Given: The number of participants for the sample =30
Then the degree of freedom for this sample=30-1=29
Hence, The total degree of freedom= 29
In the context of the experimental design, the total degrees of freedom would be calculated based on the between-group and within-group variations. With three groups and three repeats, the total degrees of freedom would be 62.
Explanation:When calculating the total degrees of freedom in an experiment where 30 participants each completed a memory task three times under different levels of caffeine, we can determine the degrees of freedom by considering the number of groups and the number of repeats within each group. Since there are three groups (no, moderate, and high levels of caffeine), and each participant completes the task three times, this setup hints at an Analysis of Variance (ANOVA) scenario with repeated measures. However, without the detailed design for the within-subject factors, we can only calculate the between-group degrees of freedom. For three groups, the between-group degrees of freedom would be the number of groups minus one, which would give us 2 degrees of freedom (dfbetween = number of groups - 1).
For within-group or repeated measures, you typically have dfwithin = (number of repeats - 1) × number of subjects. Since each participant repeats the task three times, and there are 30 participants, the calculation would be dfwithin = (3 - 1) × 30 = 2 × 30 = 60 degrees of freedom. Hence, the total degrees of freedom would be the sum of between-group and within-group degrees of freedom, which is 2 + 60 = 62.
An office has 30 computers. Seventeen of the 30 are Macintosh, and the remaining thirteen are windows. Two computers are randomly selected without replacement. What is the probability that the sample contains exactly one windows machine and exactly one Macintosh? If needed, round to FOUR decimal places. Pr(One Widows and One Macintosh) = ___________
To find the probability, you need to calculate the number of ways to choose one Macintosh and one Windows computer from the given options and divide it by the total number of ways to choose two computers. The probability is 0.507.
Explanation:To find the probability that the sample contains exactly one Windows machine and exactly one Macintosh machine, we can use the concept of combinations. There are a total of 30 computers, out of which 17 are Macintosh and 13 are Windows. The number of ways to choose one Macintosh and one Windows computer can be calculated by multiplying the number of ways to choose one Macintosh from 17 and the number of ways to choose one Windows from 13.
The number of ways to choose one Macintosh from 17 is C(17, 1) = 17 and the number of ways to choose one Windows from 13 is C(13, 1) = 13. Therefore, the total number of ways to choose one Macintosh and one Windows computer is 17 * 13 = 221.
The sample space is the total number of ways to choose two computers from 30, which is C(30, 2) = 435. So the probability of selecting exactly one Windows machine and exactly one Macintosh machine is 221 / 435 = 0.507.
Simplify this expression. -12 - 3 • (-8 +(-4)^2 - 6) + 2
Answer
-16
Step By Step explanation
Answer:
[tex] - 12 - 3 \times ( - 8 + ( { - 4)}^{2} - 6) + 2[/tex]
[tex] - 12 - 3 \times ( - 8 + 16 - 6) + 2[/tex]
[tex] - 12 - 3 \times (8 - 6) + 2[/tex]
[tex] - 12 - 3 \times 2 + 2[/tex]
[tex] - 12 - 6 + 2[/tex]
[tex] - 12 - 4[/tex]
[tex] - 16[/tex]
What steps do I take to solve this problem (cm) 8 + 27 =_____+18=___cm=____m
Answer:
[tex]8+27=17+18=35 cm=0.35 meter[/tex]
Step-by-step explanation:
[tex]8+27=35[/tex] cm
= [tex]17+18=35[/tex] cm as [tex]35-18=17[/tex] cm
Now as the final answer is in meters, so, we will convert 35 cm in meters.
100 cm = 1 meter
So, 35 cm = [tex]\frac{35}{100}=0.35[/tex] meters
Therefore, we can write the final expression as:
[tex]8+27=17+18=35 cm=0.35 meter[/tex]
Use De Morgan's Law to negate the following statements. a. Vx(x >5) b. 3.(x2+2x + 1 = 0)
Answer:
a) [tex]\neg \forall x :x>5 \equiv \forall x:x\leq 5[/tex]
b) [tex]\neg [x^2+2x+1=0]\equiv x^2+2x+1\neq0[/tex] or the set [tex]\{x:x\neq-1\}[/tex]
Step-by-step explanation:
First, notice that in both cases we have to sets:
a) is the set of all real numbers which are higher than 5 and in
b) the set is the solution of the equation [tex]x^2+2x+1=0[/tex] which is the set [tex]x=-1[/tex]
De Morgan's Law for set states:
[tex]\overline{\rm{A\cup B}} = \overline{\rm{A}} \cap \overline{\rm{B}}\\[/tex], being [tex]\overline{\rm{A}}[/tex] and [tex]\overline{\rm{B}}[/tex] are the complements of the sets [tex]A[/tex] and [tex]B[/tex]. [tex]\cup[/tex] is the union operation and [tex]\cap[/tex] the intersection.
Thus for:
a) [tex]\neg \forall x :x>5 \equiv \forall x:x\leq 5[/tex]. Notice that [tex]\forall x:x\leq 5[/tex] is the complement of the given set.
b) [tex]\neg [x^2+2x+1=0]\equiv x^2+2x+1\neq0[/tex] which is the set [tex]B = \{x:x\neq-1\}[/tex]
You know of a relative who is addicted to pain killers. Due to his history, 2.3g of Tylenol per day is dangerous. He takes 200 mg scored Tylenol tablets every six hours. For each six hour period (he takes the same number of tablets every six hours), how many scored tablets can the relative take without the amount being dangerous
Answer:
relative can take 2 tablets every six hours or 8 tablets a day without the amount being dangerous
Step-by-step explanation:
Given:
Amount of Tylenol per day that is dangerous = 2.3 g
1 g = 1000 mg
thus,
= 2.3 × 1000 mg
= 2300 mg
Amount of scored Tylenol tablets every six hours = 200 mg
Now,
for 6 hours intervals, total intervals in a day = [tex]\frac{\textup{24}}{\textup{6}}[/tex] = 4
thus,
He takes Tylenol tablets 4 times a day
Now,
let x be the number of tablets taken in every interval
thus,
4x × 200 mg ≤ 2300
or
800x ≤ 2300
or
x ≤ 2.875
hence, relative can take 2 tablets every six hours or 8 tablets a day without the amount being dangerous
A person should take 2 tablets of 200mg Tylenol every six hours to keep the daily dose under the dangerous level of 2.3g.
Explanation:The question is asking how many 200 mg tablets of Tylenol can a person consume every six hours (which is four times a day), while keeping the daily dose under 2.3g, to avoid a risk level that can be dangerous. The first step is to convert the maximum safe dose from grams to milligrams, because the dose per tablet is given in milligrams. 2.3 g equals 2300 milligrams.
Then, to find out the number of safe tablets per dose, you divide the total safe amount by the number of doses per day. So, 2300 milligrams divided by 4 equals 575 milligrams per dose. Last but not least, to find out the number of tablets, you should divide the amount per dose by the amount in each tablet: 575 divided by 200 equals about 2.875.
Since you cannot take a fraction of a tablet, the safest number of tablets to take every six hours would be 2.
Learn more about Dosage Calculation here:https://brainly.com/question/31339825
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A = {a, b, c, d}
B = {c, d, e, f, g, h}
LaTeX: A\cup BA∪B is the set of all elements that belong to A or B.
When given this information Johnny stated that union of A and B is {a, b, e, f, g, h}.
POST (by Thursday): Is Johnny correct? If not, correct his answer. Either way give a every day example to justify your answer. After posting your response, please REPLY to the responses of your group members explaining whyyou agree or disagree with their response, or asking clarifying questions.
Answer:
Johnny is wrong.
[tex]A \cup B = \left\{a,b,c,d,e,f,g,h\right\}[/tex]
Step-by-step explanation:
Johnny is wrong.
A better definition would be: [tex]A \cup B[/tex] is the set of all elements that belong to at least one A or B.
So, the elements that belong to both A and B, like c and d in this exercise, also belong to [tex]A \cup B[/tex].
So:
[tex]A \cup B=\left\{a,b,c,d,e,f,g,h\right\}[/tex]
Calculate the Laplace transforms of the following from the definition. 1. y = t^2. y = t^3
Answer:
1) [tex]L(y)=\frac{2}{s^{3}}[/tex]
2) [tex]L(y)=\frac{6}{s^{4}}[/tex]
Step-by-step explanation:
To find : Calculate the Laplace transforms of the following from the definition ?
Solution :
We know that,
Laplace transforms of [tex]t^n[/tex] is given by,
[tex]L(t^n)=\frac{n!}{s^{n+1}}[/tex]
1) [tex]y=t^2[/tex]
Laplace of y,
[tex]L(y)=L(t^2)[/tex] here n=2
[tex]L(y)=\frac{2!}{s^{2+1}}[/tex]
[tex]L(y)=\frac{2}{s^{3}}[/tex]
2) [tex]y=t^3[/tex]
Laplace of y,
[tex]L(y)=L(t^3)[/tex] here n=3
[tex]L(y)=\frac{3!}{s^{3+1}}[/tex]
[tex]L(y)=\frac{3\times 2}{s^{4}}[/tex]
[tex]L(y)=\frac{6}{s^{4}}[/tex]
Suppose that A and B are square matrices and that ABC is invertible. Show that each of A, B, and C is invertible.
Answer:
Step-by-step explanation:
Let A, B and C be square matrices, let [tex]D = ABC[/tex]. Suppose also that D is an invertible square matrix. Since D is an invertible matrix, then [tex]det (D) \neq 0[/tex]. Now, [tex]det (D) = det (ABC) = det (A) det (B) det (C) \neq 0[/tex]. Therefore,
[tex]det (A) \neq 0[/tex]
[tex]det (B) \neq 0 [/tex]
[tex]det (C) \neq 0[/tex]
which proves that A, B and C are invertible square matrices.
Find the number of 3-digit numbers formed using the digits 1 to 9, without repetition, such the numbers either have all digits less than 5 or all digits greater than 4.
Answer: 120
Step-by-step explanation:
The total number of digits from 1 to 9 = 10
The number of digits from less than 5 (0,1,2,3,4)=5
Since repetition is not allowed so we use Permutations , then the number of 3-digit different codes will be formed :-
[tex]^5P_3=\dfrac{5!}{(5-3)!}=\dfrac{5\times4\times3\times2!}{2!}=5\times4\times3=60[/tex]
The number of digits from greater than 4 (5,6,7,8,9)=5
Similarly, Number of 3-digit different codes will be formed :-
[tex]^5P_3=60[/tex]
Hence, the required number of 3-digit different codes = 60+60=120
Joe's annual income has been increasing each year by the same dollar amount. The first year his income was $17 comma 90017,900, and the 44th year his income was $20 comma 30020,300. In which year was his income $ 30 comma 700 question mark
Answer:
In 17th year, his income was $30,700.
Step-by-step explanation:
It is given that the income has been increasing each year by the same dollar amount. It means it is linear function.
Income in first year = $17,900
Income in 4th year = $20,300
Let y be the income at x year.
It means the line passes through the point (1,17900) and (4,20300).
If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the equation of line is
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
The equation of line is
[tex]y-17900=\frac{20300-17900}{4-1}(x-1)[/tex]
[tex]y-17900=\frac{2400}{3}(x-1)[/tex]
[tex]y-17900=800(x-1)[/tex]
[tex]y-17900=800x-800[/tex]
Add 17900 on both sides.
[tex]y=800x-800+17900[/tex]
[tex]y=800x+17100[/tex]
The income equation is y=800x+17100.
Substitute y=30,700 in the above equation.
[tex]30700=800x+17100[/tex]
Subtract 17100 from both sides.
[tex]30700-17100=800x[/tex]
[tex]13600=800x[/tex]
Divide both sides by 800.
[tex]\frac{13600}{800}=x[/tex]
[tex]17=x[/tex]
Therefore, in 17th year his income was $30,700.
Final answer:
Joe's annual income increases by approximately $55.81 each year, starting at $17,900 in the first year. By dividing the desired income minus the starting income by the annual increase, we find that Joe will reach an income of $30,700 in his 230th year.
Explanation:
To determine in which year Joe's income reached $30,700, we need to establish a pattern of how his income has increased over the years. Given that Joe's income started at $17,900 in the first year and was $20,300 in the 44th year, we can calculate the annual increase in his income.
The total increase over 43 years (from year 1 to year 44) is $20,300 - $17,900 = $2,400. To find the annual increase, divide this by the number of years the increase occurred over, which is one less than the total number of years, because the increase starts after the first year. That is:
Annual Income Increase = Total Increase / Number of Years
Annual Income Increase = $2,400 / 43
Annual Income Increase = approximately $55.81 (rounded to two decimal places)
Now we need to calculate the number of years it would take for his income to reach $30,700, starting from $17,900 and increasing at a rate of approximately $55.81 per year.
Years Needed = (Desired Income - Starting Income) / Annual Increase
Years Needed = ($30,700 - $17,900) / $55.81
Years Needed = approximately 229.84, which we round up to 230, because you can't have a partial year in this context.
Therefore, Joe will reach an income of $30,700 in his 230th year of work (adding 230 to the first year).
Mrs. Brown needs $5,000 in three years. If the interest rate on her investment account is 8.4% compounded monthly, how much should she put into her account at the end of each month to have $5,000 in three years?
Give answer in US dollars and cents rounded to the nearest cent. Do NOT enter "$" sign
Answer:
ohhh you have the right time and I am so glad to see it in a better time and if it was not to do you have no doubt
Step-by-step explanation:
your body has to have your body to have the best results you need for kids in a group that can help with a lot to learn about it as you do so much of the time it works out to you to get the job you need for the next few years or 8īijjhguu is not just the best person but the meaning and skills of a person is the meaning of life and the meaning is the only 3AM in a school that you need a lot of a job 3to to do with you need to get your mind off to get your job done with your mind and then you have to do it right now and.
A movie theater manager wants to determine whether popcorn sales have increased since the theater switched from using "butter-flavored topping" to real butter. Historically the average popcorn revenue per weekend day was approximately $3,500. After the theater started using real butter, the manager randomly sampled 12 weekend days and calculated the sample’s summary statistics. The average revenue per weekend day in the sample was approximately $4,200 with a standard deviation of $140. Select the function that would correctly calculate the 90% range of likely sample means.A. 3,500±CONFIDENCE.T(0.10,140,12)B. 4,200±CONFIDENCE.T(0.10,140,12)C. 3,500±CONFIDENCE.NORM(0.10,140,12)D. 4,200±CONFIDENCE.NORM(0.10,140,12)
Answer:
B. 4,200±CONFIDENCE.T(0.10,140,12)
Step-by-step explanation:
We are in posession's of the sample standard deviation, so the t-distribution is used.
The confidence interval is a function of the sample mean and the margin of error.
That is:
[tex]C.I = S_{M} \pm M_{E}[/tex]
In which [tex]S_{M}[/tex] is the sample mean, and the [tex]M_{E}[/tex] is the margin of error, related to the confidence level, the sample's standard deviation and the sample size.
So the correct answer is:
B. 4,200±CONFIDENCE.T(0.10,140,12)
A firm produces a product that has the production cost function
C(x)equals=195195xplus+88408840
and the revenue function
R(x)equals=260260x.
No more than
229229
units can be sold. Find and analyze the break-even quantity, then find the profit function.
Answer:
136 units
65x - 8840
Step-by-step explanation:
Given,
The production cost function is,
[tex]C(x) = 195x + 8840[/tex]
Revenue function,
[tex]R(x)=260x[/tex]
So, profit would be,
P(x) = Revenue - cost
= 260x - 195x - 8840
= 65x - 8840
In break even condition,
Profit, P(x) = 0
65x - 8840 = 0
65x = 8840
⇒ x = 136.
Hence, the break even quantity is 136 units.
The break-even quantity is 1360 units, where the revenue equals the cost. The profit function is given by P(x) = R(x) - C(x).
Explanation:To find the break-even quantity, we need to find the point where the revenue function equals the production cost function. So we set R(x) equal to C(x) and solve for x:
260x = 195x + 88408
Simplifying the equation, we get:
65x = 88408
x = 1360
Now we analyze the break-even quantity. The break-even quantity is the point at which the firm's revenue equals its cost. In this case, it occurs at 1360 units. At this quantity, the firm's total revenue will equal its total cost, resulting in zero profit.
To find the profit function, we subtract the cost function C(x) from the revenue function R(x). The profit function can be expressed as:
P(x) = R(x) - C(x)
Gandalf the Grey started in the Forest of Mirkwood at a point P with coordinates (3, 0) and arrived in the Iron Hills at the point Q with coordinates (5, 5). If he began walking in the direction of the vector v - 3i + 2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn?
Answer:
Turning point has coordinates [tex]\left(\dfrac{27}{13},\dfrac{8}{13}\right)[/tex]
Step-by-step explanation:
Gandalf the Grey started in the Forest of Mirkwood at a point P(3, 0) and began walking in the direction of the vector [tex]\vec{v}=-3i+2j.[/tex] The coordinates of the vector v are (-3,2). Then he changed the direction at a right angle, so he was walking in the direction of the vector [tex]\vec{u}=2i+3j[/tex] (vectors u and v are perpendicular).
Let B(x,y) be the turning point. Find vectors PB and BQ:
[tex]\overrightarrow{PB}=(x-3,y-0)\\ \\\overrightarrow {BQ}=(5-x,5-y)[/tex]
Note that vectors v and PB and vectors u and BQ are collinear, so
[tex]\dfrac{x-3}{-3}=\dfrac{y}{2}\\ \\\dfrac{5-x}{2}=\dfrac{5-y}{3}[/tex]
Hence
[tex]2(x-3)=-3y\Rightarrow 2x-6=-3y\\ \\3(5-x)=2(5-y)\Rightarrow 15-3x=10-2y[/tex]
Now solve the system of two equations:
[tex]\left\{\begin{array}{l}2x+3y=6\\ -3x+2y=-5\end{array}\right.[/tex]
Multiply the first equation by 3, the second equation by 2 and add them:
[tex]3(2x+3y)+2(-3x+2y)=3\cdot 6+2\cdot (-5)\\ \\6x+9y-6x+4y=18-10\\ \\13y=8\\ \\y=\dfrac{8}{13}[/tex]
Substitute it into the first equation:
[tex]2x+3\cdot \dfrac{8}{13}=6\\ \\2x=6-\dfrac{24}{13}=\dfrac{54}{13}\\ \\x=\dfrac{27}{13}[/tex]
Turning point has coordinates [tex]\left(\dfrac{27}{13},\dfrac{8}{13}\right)[/tex]
The coordinates of the point where Gandalf makes the turn are (5, 5).
Explanation:To find the point where Gandalf makes a right angle turn, we need to find the intersection of the line formed by the vector v and the line connecting points P and Q. The equation of the line formed by the vector v is given by y = 2x - 3. The equation of the line connecting points P and Q is given by y = x. To find the intersection point, we can solve these two equations simultaneously. Substituting y = 2x - 3 into y = x, we get x = 5. Substituting x = 5 into y = x, we get y = 5. Therefore, the coordinates of the point where Gandalf makes the turn are (5, 5).
Find the solutions of the quadratic equation 3x^2-5x+1=0.
Answer:
The solutions of the quadratic equation are [tex]x_{1} = \frac{5 + \sqrt{13}}{6}, x_{2} = \frac{5 - \sqrt{13}}{6}[/tex]
Step-by-step explanation:
This is a second order polynomial, and we can find it's roots by the Bhaskara formula.
Explanation of the bhaskara formula:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = (x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}[/tex]
[tex]\bigtriangleup = b^{2} - 4ac[/tex]
For this problem, we have to find [tex]x_{1} \text{and} x_{2}[/tex].
The polynomial is [tex]3x^{2} - 5x +1[/tex], so a = 3, b = -5, c = 1.
Solution
[tex]\bigtriangleup = b^{2} - 4ac = (-5)^{2} - 4*3*1 = 25 - 12 = 13[/tex]
[tex]x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a} = \frac{-(-5) + \sqrt{13}}{2*3} = \frac{5 + \sqrt{13}}{6}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a} = \frac{-(-5) - \sqrt{13}}{2*3} = \frac{5 - \sqrt{13}}{6}[/tex]
The solutions of the quadratic equation are [tex]x_{1} = \frac{5 + \sqrt{13}}{6}, x_{2} = \frac{5 - \sqrt{13}}{6}[/tex]
Rewrite the set T = { 5a + 2b; a,b belong Z} as a list of its elements
Answer:
T={...,-3,-2,-1,0,1,2,3,4,....}
Step-by-step explanation:
We are given that T={5a+2b; a,b belongs Z}
We have to rewrite the given set T as a list of its elements
Substitute a=0 and b=0
Then we get 5(0)+2(0)=0
Substitute a=-1 and b=2 then we get
5(-1)+2(2)=-1
Substitute a=2 and b=0
Then , 5(2)+2(0)=10
If a=0 and b=1
Then, 5(0)+2(1)=2
Substitute a=0 and b=2
Then, 5(0)+2(2)=4
If substitute a=1 and b=1
5(1)+2(1)=7
If substitute a=-1 and b=3
Then, we get 5(-1)+2(3)=1
Then, T={...,-3,-2,-1,0,1,2,3,4,....}
In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set 2, 2, 3, 6, 10. (a) Compute the mode, median, and mean. (b) Add 5 to each of the data values. Compute the mode, median, and mean. (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set?
Answer:
a) Mode: 2 Median: 3 Mean: 4.6
b) Mode: 7 Median: 8 Mean: 9.6
c) Just added 5 to values. General below.
Step-by-step explanation: 2, 2, 3, 6, 10
a) Mode: 2 (Most apperances)
Median: 3 (odd data, middle number)
Mean: (2+2+3+6+10)/5 = 23/5 = 4.6
b) + 5
Data: 7,7,8,11,15
Mode: 7 (Most apperances)
Median: 8 (odd data, middle number)
Mean: (7+7+8+11+15)/5 = 48/5 = 9.6
c) The results from (b) is (a) + 5
In general: Let's add x to the same data provided:
2+x, 2+x, 3+x, 6+x, 10+x,
For the mode, it does not matter, the number with most apperances will continue to be the mode + x
For the median, same thing. It is just the median + x
For the mean, same thing. For the set of 5 numbers:
(2+x + 2+x + 3+x + 6+x + 10+x)/5 =
(23+5x)/5
23/5 + 5x/5 =
23/5 + x
For example, If it was 6 numbers, we would add 6 times that number and divide it by 6, adding x to the mean.
To compute the mode, median, and mean of a data set, count the frequency of each number, arrange the data in order, and find the middle value. Adding the same constant to each data value affects the mean but does not change the mode or median.
Explanation:To compute the mode, median, and mean of the data set {2, 2, 3, 6, 10}, we can follow these steps:
To find the mode, count the frequency of each number and identify the number(s) with the highest frequency. In this case, the mode is 2, as it appears twice.To find the median, arrange the data in ascending order and find the middle value. In this case, the median is 3.To find the mean, add up all the numbers and divide by the total count. In this case, the mean is (2+2+3+6+10)/5 = 23/5 = 4.6.After adding 5 to each data value, the new data set becomes {7, 7, 8, 11, 15}.
The mode remains the same, which is 7.The median remains the same, which is 8.The mean is calculated as (7+7+8+11+15)/5 = 48/5 = 9.6.In general, when the same constant is added to each data value in a set, the mode remains unchanged, the median remains unchanged, and the mean is affected by adding the constant to each value. The mean increases when the constant is positive and decreases when the constant is negative.
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A survey of 510 adults aged 18-24 year olds was conducted in which they were asked what they did last Friday night. It found: 161 watched TV 196 hung out with friends 161 ate pizza 28 watched TV and ate pizza, but did not hang out with friends 29 watched TV and hung out with friends, but did not eat pizza 47 hung out with friends and ate pizza, but did not watch TV 43 watched TV, hung out with friends, and ate pizza How may 18-24 year olds did not do any of these three activities last Friday night?
Answer:
182 of these adults did not do any of these three activities last Friday night.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the adults that watched TV
-The set B represents the adults that hung out with friends.
-The set C represents the adults that ate pizza
-The set D represents the adults that did not do any of these three activities.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of adults that only watched TV, [tex]A \cap B[/tex] is the number of adults that both watched TV and hung out with friends, [tex]A \cap C[/tex] is the number of adults that both watched TV and ate pizza, is the number of adults that both hung out with friends and ate pizza, and [tex]A \cap B \cap C[/tex] is the number of adults that did all these three activies.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,D,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There were 510 adults suveyed. This means that:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 510[/tex]
We start finding the values from the intersection of three sets.
Solution:
43 watched TV, hung out with friends, and ate pizza:
[tex]A \cap B \cap C = 43[/tex]
47 hung out with friends and ate pizza, but did not watch TV:
[tex]B \cap C = 47[/tex]
29 watched TV and hung out with friends, but did not eat pizza:
[tex]A \cap B = 29[/tex]
28 watched TV and ate pizza, but did not hang out with friends:
[tex]A \cap C = 28[/tex]
161 ate pizza
[tex]C = 161[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]161 = c + 28 + 47 + 43[/tex]
[tex]c = 43[/tex]
196 hung out with friends
[tex]B = 196[/tex]
[tex]196 = b + 47 + 29 + 43[/tex]
[tex]b = 77[/tex]
161 watched TV
[tex]A = 161[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]161 = a + 29 + 28 + 43[/tex]
[tex]a = 61[/tex]
How may 18-24 year olds did not do any of these three activities last Friday night?
We can find the value of D from the following equation:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 510[/tex]
[tex]61 + 77 + 43 + D + 29 + 28 + 47 + 43 = 510[/tex]
[tex]D = 510 - 328[/tex]
[tex]D = 182[/tex]
182 of these adults did not do any of these three activities last Friday night.
To find the number of 18-24 year olds who did not do any of the three activities last Friday night, we can use the principle of inclusion-exclusion. By subtracting the number of people who did at least one of the activities from the total number of participants, we find that 455 individuals did not participate in watching TV, hanging out with friends, or eating pizza.
Explanation:To find the number of 18-24 year olds who did not do any of the three activities (watch TV, hang out with friends, eat pizza), we need to subtract the number of people who did at least one of these activities from the total number of participants. We can use the principle of inclusion-exclusion to solve this problem.
Let's define:
A = number of people who watched TVB = number of people who hung out with friendsC = number of people who ate pizzaFrom the given information, we know:
A = 161B = 196C = 161A ∩ C' (watched TV and ate pizza, but did not hang out with friends) = 28A ∩ B' (watched TV and hung out with friends, but did not eat pizza) = 29B ∩ C' (hung out with friends and ate pizza, but did not watch TV) = 47A ∩ B ∩ C (watched TV, hung out with friends, and ate pizza) = 43To find the number of people who did not do any of these activities, we can use the formula:
n(A' ∩ B' ∩ C') = n(U) - n(A) - n(B) - n(C) + n(A ∩ B) + n(A ∩ C) + n(B ∩ C) - n(A ∩ B ∩ C)
Substituting the known values, we have:
n(A' ∩ B' ∩ C') = 510 - 161 - 196 - 161 + 28 + 29 + 47 - 43
n(A' ∩ B' ∩ C') = 455
Therefore, there were 455 18-24 year olds who did not do any of the three activities last Friday night.
In how many ways can the digits 0,1,2,3,4,5,6,7,8,9 be arranged so that no prime number is in its original position?
I get the answer 1348225 by subtracting the number of derangements with fixed points 4,3,2 and 1 from 10! (the number of ways to arrange the numbers with none fixed).
Answer: 2399760
Step-by-step explanation:
The concept we use here is Partial derangement.
It says that for m things , the number of ways to arrange them such that k things are not in their fixed position is given by :-
[tex]m!-^kC_1(m-1)!+^kC_2(m-2)!-^kC_3(m-3)!+........[/tex]
Given digits : 0,1,2,3,4,5,6,7,8,9
Prime numbers = 2,3,5,7
Now by Partial derangement the number of ways to arrange 10 numbers such that none of 4 prime numbers is in its original position will be :_
[tex]10!-^4C_1(9)!+^4C_2(8)!-^4C_3(7)!+^4C_4(6)!\\\\=3628800-(4)(362880)+\dfrac{4!}{2!2!}(40320)-(4)(5040)+(1)(720)\\\\=3628800-1451520+241920-20160+720\\\\=2399760[/tex]
Hence, the number of ways can the digits 0,1,2,3,4,5,6,7,8,9 be arranged so that no prime number is in its original position = 2399760
Find the derivative of the following functions:
(i) f(x) = −x 2 + 10x + 4
(ii) f(x) = 20 − 1 x+2
(iii) f(x) = x 4 e −2x 2
(iv) f(x) = ln(x 2 + 2x + 2).
using the differentiation rules :Derivative definition and operation rules (Sum, subtraction, multiplication, quotient), Derivative of many basic functions (power, quadratic, exponential and natural logarithm functions), Function of Functions: (Chain Rule).
Explain carefully what rule(s) you have used and where you have applied it(them)
I will only do the first-three. You can repost number 4.
Question 1
f(x) = -x^2 + 10x + 4
dy/dx = -2x + 10
Question 2
f(x) = 20 - x + 2
dy/dx = -1
Question 3
f(x) = ex^4 - 2x^2
dy/dx = e•4x^3 + x^4 - 4x
dy/dx = 4ex^3 + x^4 - 4x
Consider the function fx) = -3.15x + 723.45. Graph it on the interval (0,25), and then answer Questions 8 - 11 below. Question 8 (1 point) What is the domain of the function (the entire function, not just the part you graphed)? O [-3.15, 7.42) [-10, 10] [7.42, c) O 10,co)
Answer:
Domain : D{-∞,∞} the reals.
Step-by-step explanation:
The function is plotted in the image.
[tex] f(x) = -3.15 * x + 723.45 [tex]
the linear functions usually have a domain from - infinite to infinite, the domain when is a piece wise function or discontinuous, the domain is defined in the pieces where is defined.
In this case there is no restriction so the function is continuous.
Hillary, Meredith, and Aly are sitting in their favorite coffee shop when their waiter asks: "Does everyone want coffee?" Hillary replies "I don't know." Meredith then replies "I don't know" as well. Finally, Aly says "Not everyone wants coffee." The waiter comes back and gives a coffee to each person that wants one.
Answer the following question:
(a) Did Hillary get a coffee?
(b) Did meredith get a coffee?
Answer:
a) Yes.
b) Yes.
Step-by-step explanation:
Meredith and Hillary both want coffe, but they don't know if the other two people do, therefore they can't know if everyone want coffee. If they didn't want coffee, their answer would have been just "no".
Aly knows that she doesn't want coffee, therefore she knows that not everyone wants coffee.
Hillary and Meredith said 'I don't know' which implies they don't know if everyone wants coffee because they themselves do not want it. Aly confirmed that not everyone wants coffee. Therefore, neither Hillary nor Meredith got a coffee.
We have a logical puzzle where Hillary, Meredith, and Aly are deciding whether they want coffee. The key to solving this puzzle is understanding the implications of their statements to the waiter's question: "Does everyone want coffee?"
Hillary says, "I don't know." This means Hillary cannot be sure that everyone wants coffee, so there are two possibilities: either she does not want coffee or she doesn't know the preferences of the others. Meredith also responds with "I don't know," implying the same possibilities for her.
Finally, Aly states, "Not everyone wants coffee." This is the definitive answer that tells us at least one person does not want coffee. Since Aly knows for sure that not everyone wants coffee, it implies that either she does not want coffee herself or knows of someone else who doesn’t. Given that Hillary and Meredith both said they did not know, they could not have communicated their preference to Aly.
Therefore:
Hillary did not get a coffee, because if she did want coffee, she would have known that at least she herself wants coffee and would not have said, "I don't know."Meredith did not get a coffee either for the same reason as Hillary.Find the effective rate of the compound interest rate or investment. (Round your answer to two decimal places.) A $50,000 zero-coupon bond maturing in 8 years and selling now for $43,035. %
Answer:
Ans. Effective annual rate=1.8928%
Annual Compound semi-annually=1.8839%
Step-by-step explanation:
Hi, this is the formula to find the effective annual rate for this zero-coupon bond.
[tex]EffectiveAnnualRate=\sqrt[n]{\frac{FaceValue}{Price} } -1[/tex]
n= years to maturity
That is:
[tex]EffectiveAnnualRate=\sqrt[8]{\frac{50,000}{43,035} } -1=0.018928[/tex]
Means that the effective interest rate is 1.8928% effective annual
Now, let´s find the compound interest rate.
First, we have to turn this rate effective semi-annually
[tex]Semi-AnnualRate=(1+0.018928)^{\frac{1}{2} } -1=0.00942[/tex]
0.942% effective semi annual
To turn this into a semi-annual, compounded semi-annually, we just have to multiply by 2, so we get.
1.8839% compounded semi-annually
Best of luck
What horsepower is required to lift an 8,000 pound aircraft six feet in two minutes?
Answer:
The horsepower required is 235440 watt.
Step-by-step explanation:
To find : What horsepower is required to lift an 8,000 pound aircraft six feet in two minutes?
Solution :
The horsepower formula is given by,
[tex]W=\frac{mgh}{t}[/tex]
Where, W is the horsepower
m is the mass m=8000 pound
g is the gravitational constant g=9.81
t is the time t= 2 minutes
h is the height h=6 feet
Substitute all values in the formula,
[tex]W=\frac{8000\times 9.81\times 6}{2}[/tex]
[tex]W=\frac{470880}{2}[/tex]
[tex]W=235440[/tex]
Therefore, The horsepower required is 235440 watt.
A survey of 1,107 tourists visiting Orlando was taken. Of those surveyed:
268 tourists had visited the Magic Kingdom
258 tourists had visited Universal Studios
68 tourists had visited both the Magic Kingdom and LEGOLAND
79 tourists had visited both the Magic Kingdom and Universal Studios
72 tourists had visited both LEGOLAND and Universal Studios
36 tourists had visited all three theme parks
58 tourists did not visit any of these theme parks
How many tourists only visited the LEGOLAND (of these three)?
Answer:
602 tourists visited only the LEGOLAND.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the tourists that visited LEGOLAND
-The set B represents the tourists that visited Universal Studios
-The set C represents the tourists that visited Magic Kingdown.
-The value d is the number of tourists that did not visit any of these parks, so: [tex]d = 58[/tex]
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of tourists that only visited LEGOLAND, [tex]A \cap B[/tex] is the number of tourists that visited both LEGOLAND and Universal Studies, [tex]A \cap C[/tex] is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and [tex]A \cap B \cap C[/tex] is the number of students that visited all these parks.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There were 1,107 tourists suveyed. This means that:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]
We start finding the values from the intersection of three sets.
The problem states that:
36 tourists had visited all three theme parks. So:
[tex](A \cap B \cap C) = 36[/tex]
72 tourists had visited both LEGOLAND and Universal Studios. So:
[tex](A \cap B) + (A \cap B \cap C) = 72[/tex]
[tex](A \cap B) = 72 - 36[/tex]
[tex](A \cap B) = 36[/tex]
79 tourists had visited both the Magic Kingdom and Universal Studios
[tex](B \cap C) + (A \cap B \cap C) = 79[/tex]
[tex](B \cap C) = 79 - 36[/tex]
[tex](B \cap C) = 43[/tex]
68 tourists had visited both the Magic Kingdom and LEGOLAND
[tex](A \cap C) + (A \cap B \cap C) = 68[/tex]
[tex](A \cap C) = 68 - 36[/tex]
[tex](A \cap C) = 32[/tex]
258 tourists had visited Universal Studios:
[tex]B = 258[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]258 = b + 43 + 36 + 36[/tex]
[tex]b = 143[/tex]
268 tourists had visited the Magic Kingdom:
[tex]C = 268[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]268 = c + 32 + 43 + 36[/tex]
[tex]c = 157[/tex]
How many tourists only visited the LEGOLAND (of these three)?
We have to find the value of a, and we can do this by the following equation:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]
[tex]a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107[/tex]
[tex]a = 602[/tex]
602 tourists visited only the LEGOLAND.
The Schuller family has five members. Dad is 6ft 2in tall. Mom is 3 inches shorter than Dad, but 2 inches taller than Ivan. Marcia is 5 inches shorter than Ivan, but twice as tall as Sally-Jo. What is the mean height of the Schuller family?
Answer:
62 inches
Step-by-step explanation:
Let x be the height ( in inches ) of Ivan,
∵ Marcia is 5 inches shorter than Ivan,
height of Marcia = x - 5,
Marcia is twice as tall as Sally-Jo,
height of sally-jo = [tex]\frac{x-5}{2}[/tex]
Mom is 2 inches taller than Ivan.
⇒ height of mom = x + 2,
Mom is 3 inches shorter than Dad,
height of dad = x + 2 + 3 = x + 5,
So, mean height of family is,
[tex]\frac{x+x-5+\frac{x-5}{2}+x+2+x+5}{5}[/tex]
[tex]=\frac{2x+2x-10+x-5+2x+4+2x+10}{10}[/tex]
[tex]=\frac{9x-1}{10}[/tex]
According to the question,
x + 5 = 74 ( 1 ft = 12 in )
x = 69
Hence, mean height of the family = [tex]\frac{9\times 69-1}{10}[/tex]
[tex]=\frac{620}{10}[/tex]
= 62 inches
-. Mario walks 7 blocks from his home to
a restaurant. He then walks back toward
home for five blocks, where he stops at
a bookstore. How many blocks is Mario
from his home?
Answer:
the correct answer is 2 blocks
Mario is 2 blocks away from his home after walking 7 blocks to a restaurant and then 5 blocks back towards home with help of subtraction.
The number of blocks that Mario is from his home can be calculated by finding the difference between the number of blocks he walked away from home and the number of blocks he walked back home. If Mario starts by walking 7 blocks from his home to a restaurant, he is initially 7 blocks away from home. If he then walks 5 blocks back toward home, he is effectively reducing the distance from his home by 5 blocks. You can calculate the new distance to his home by subtracting 5 from 7, which equals 2 blocks. Therefore, Mario is 2 blocks away from his home when he stops at the bookstore.
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(b) "If x > 0 and y > 0 then xy > 0" where x,y are real numbers.
Answer:
Step-by-step explanation:
We know that multiplication product of two given positive real number will always be positive real number and if one of the real number is negative, the multiplication product will always be negative.
so for the given condition, if [tex]x > 0[/tex] and [tex]y > 0[/tex], [tex]x[/tex] and [tex]y[/tex] are both positive real numbers
hence their multiplication product [tex]xy[/tex] will also be a positive number.
∴ [tex]xy > 0[/tex]