Answer:
Maximum at [tex]x =\frac{\sqrt{7}}{3}[/tex]
Step-by-step explanation:
Given function,
[tex]f(x) = 14x - 6x^3[/tex]
Differentiating with respect to x,
[tex]f'(x) = 14 - 18x^2----(1)[/tex]
For critical values :
[tex]f'(x) = 0[/tex]
[tex]14 - 18x^2 =0[/tex]
[tex]14 = 18x^2[/tex]
[tex]x^2 = \frac{14}{18}[/tex]
[tex]x^2=\frac{7}{9}[/tex]
[tex]x = \pm \frac{\sqrt{7}}{3}[/tex]
Now, differentiating equation (1) again with respect to x,
[tex]f''(x) = -36x[/tex]
Since,
[tex]f''(\frac{\sqrt{7}}{3}) = -36(\frac{\sqrt{7}}{3}) < 0[/tex]
This means that the function is maximum at [tex]x=\frac{\sqrt{7}}{3}[/tex]
While,
[tex]f''(-\frac{\sqrt{7}}{3}) = 36(\frac{\sqrt{7}}{3}) > 0[/tex]
This means that the function is minimum at [tex]x=-\frac{\sqrt{7}}{3}[/tex]
−9x+2>18 OR 13x+15≤−4
Answer:
[tex]-1.78>x\leq -1.46[/tex]
Step-by-step explanation:
1. Understanding the type of statement
We are given an OR statement. A certain x-value is a set of solution of the statement if it satisfies both of the inequalities.
Therefore, the solution of this statement is the Union of set of the solutions of both inequalities.
2. Finding the solutions to the two inequalities
[tex]-9x+2>18\\-9x>18-2\\-9x>16\\x<-\frac{16}{9}\\ \\x<-1.78[/tex]
Now Solving for other equation we get,
[tex]13x+15\leq-4\\13x\leq -4-15\\13x\leq -19\\x\leq -\frac{19}{13}\\ \\x\leq -1.46[/tex]
3. The solution is:
[tex]-1.78>x\leq -1.46[/tex]
Answer:
[tex]x \leq - 1.462[/tex]
Step-by-step explanation:
Let solve each inequation:
[tex]-9\cdot x + 2 > 18[/tex]
[tex]-16 > 9\cdot x[/tex]
[tex]9\cdot x < - 16[/tex]
[tex]x < - \frac{16}{9}[/tex]
[tex]x < -1.778[/tex]
[tex]13\cdot x + 15 \leq -4[/tex]
[tex]13\cdot x \leq -19[/tex]
[tex]x \leq -\frac{19}{13}[/tex]
[tex]x \leq -1.462[/tex]
The boolean operator OR means that proposition is true if at least one equation is true. Then, the domain that fulfill the proposition is:
[tex]x \leq - 1.462[/tex]
For a special game Don has two 8-sided fair dice numbered from 1 to 8 on the faces. Just as with ordinary dice, Don rolls the dice and sums the numbers that appear on the top face, getting a sum from 2 to 16. What is the sum of the possible numbers which have a probability of 1/32 of appearing
Answer
The sum of the possible numbers are {(1,2), (2,1)}
Step-by-step explanation:
Probability for rolling two dice with the eight sided dots are 1, 2, 3, 4, 5, 6, 7 and 8 dots in each die.
When two dice are rolled or thrown simultaneously, thus number of event can be [tex]8^{2}[/tex]= 64 because each die has 1 to 8 numbers on its faces. Then the possible outcomes of the sample space are shown in the pdf document below .
As Don rolls the dice and sums the number that appear on the top face, he gets a sum from 2 to 16.
Assuming; getting sum of 2
Let E[tex]_{1}[/tex] = event of getting sum of 2
E[tex]_{1}[/tex] = { (1 , 1 ) }
Therefore, Probability of getting sum of 2 will be;
P ( E[tex]_{1}[/tex] ) = [tex]\frac{Number of favorable outcome}{Total number of possible outcome}[/tex]
= [tex]\frac{1}{64}[/tex]
GETTING SUM OF TWO ( i.e { (1 , 1 ) } ) will give the probability of 1/64 of appearing. But we are looking for the probability of 1/32 of appearing. Let look at the possibility of getting sum of 3.
Assuming; getting sum of 3
Let E[tex]_{2}[/tex] = event of getting sum of 3
E[tex]_{2}[/tex] = { (1 , 2 ) (2 , 1) }
Therefore, Probability of getting sum of 3 will be;
P ( E[tex]_{2}[/tex] ) = [tex]\frac{Number of favorable outcome}{Total number of possible outcome}[/tex]
= [tex]\frac{2}{64}[/tex]
= [tex]\frac{1}{32}[/tex]
For all probability of getting the sum greater than 3 to 16 will be void because there wont be a chance for 1/32 to appear. Therefore, the sum of the possible numbers are: {(1,2), (2,1)} which have a probability of 1/32 of appearing.
I hope this comes in handy at the rightful time!
Maylin and Nina are making fruit baskets. They have 36 apples, 27 bananas, and 18 oranges. They want each basket tocontain the same amount of each fruit. Maylin believes the greatest number of baskets they can make is 6, and Ninabelieves the greatest number of baskets they can make is 9.
Answer:
The greatest number of baskets can be 81 and the least number of baskets can be 9.
Step-by-step explanation:
We have a constraint on our actions, that every basket should have the same number of fruits in each basket.
To find the highest number of fruits in each basket we have to find the Highest Common Factor( HCF) of the number of apples , bananas and oranges.
HCF of 36, 27 and 18 is 9.
Therefore the number of fruits in each basket is 9.
Apples will require 4 baskets, bananas will require 3 baskets and oranges will require 2 baskets. Thus Nina is right and total 9 baskets will be required.
9 is the least of number of baskets required
If we have to maximize the number of baskets, then we have to place the least number of same fruits in a basket ie. 1.
Therefore, he maximum number of baskets required is 36+27+18=81 baskets.
What probability should be assigned to the outcome of heads when a biased coin is tossed, if heads is three times as likely to come up as tails? What probability should be assigned to the outcome of tails?
Answer:
propability is the low of chance
Answer:
The probability of tail occurs = [tex]\frac{1}{4}[/tex] = 0.25
and
The probability of heads occurs = [tex]\frac{3}{4}[/tex] = 0.75
Step-by-step explanation:
Given:
Let P be the tail as outcome in a toss.
Heads is three times as likely to come up as tails.
So, Probability of getting heads = 3P
The total probability is 1
So, P + 3P = 1
4P = 1
P = [tex]\frac{1}{4}[/tex] = 0.25
We denoted P as the probability of getting tail in a toss.
So probability of getting heads = 1 - [tex]\frac{1}{4}[/tex] = [tex]\frac{3}{4}[/tex]
Therefore, the probability of tail occurs = [tex]\frac{1}{4}[/tex] = 0.25
and
The probability of heads occurs = [tex]\frac{3}{4}[/tex] = 0.75
5.
The present value of a sum of money is the
amount that must be invested now, at a given
rate of interest, to produce the desired sum at a
later date. Find the present value of 10,000 if
interest is paid at a rate of 6.2% compounded
weekly for 8 years.
Answer:
The present value of 10,000 if interest is paid at a rate of 6.2% compounded weekly for 8 years is 6097.56
Explanation:
We know that compound interest is given by
[tex]A=P\left(1+\frac{r}{n}\right)^{n t}[/tex]
Where ,
Where A = final amount (which is given to be = 10000)
P = Principal amount (which is the present amount which we have to find)
r = interest rate = 6.2 = 0.062
n = no. of times interest applied per time period = it is given that the interest is applied weekly, so in one year there are 52 weeks so n = 52
t = time period = 8 years
Substituting the given values, we get
[tex]10000=\mathrm{P}\left(1+\frac{6.2}{52}\right)^{52\times 8}[/tex]
P = 6097.5
We get, P = 6097.56 which is the present value of a sum of money
Mata exercises 30 minutes each day Greg exercises for 10 minutes each day how many more minutes that's Greg exercise in a month of 31 days than Martha
Answer:
620 minutes
Step-by-step explanation
In one day,Mata exercise 20minutes more than Greg
31 days so we have 20 *31=620
Amber had 3/8 of a cake left after her party she wrapped a piece that was 1/4 of the original cake for her best friend what freshener part did she have left over for herself
Answer:
Fraction of cake that was left for Amber = [tex]\frac{1}{8}[/tex]
Step-by-step explanation:
Given:
Amber had [tex]\frac{3}{8}[/tex] of a cake left after her party.
Amber wrapped a piece of cake for her friend that was [tex]\frac{1}{4}[/tex] of the original cake.
To find the fractional part of cake that was left for Amber.
Solution:
Fraction of cake left Amber had after party = [tex]\frac{3}{8}[/tex]
Fraction of cake she wrapped for her friend = [tex]\frac{1}{4}[/tex]
To find the fractional part of cake that was left for Amber we will subtract [tex]\frac{1}{4}[/tex] of the cake from [tex]\frac{3}{8}[/tex] of the cake.
∴ Fraction of cake that was left for Amber = [tex]\frac{3}{8}-\frac{1}{4}[/tex]
To subtract fractions we need to take LCD
⇒ [tex]\frac{3}{8}-\frac{1}{4}[/tex]
LCD will be =8 as its the least common multiple of 4 and 8.
To write [tex]\frac{1}{4}[/tex] as a fraction with common denominator 8 we multiply numerator and denominator with 2.
So, we have
⇒ [tex]\frac{3}{8}-\frac{1\times 2}{4\times 2}[/tex]
⇒ [tex]\frac{3}{8}-\frac{ 2}{8}[/tex]
Then we simply subtract the numerators.
⇒ [tex]\frac{3-2}{8}[/tex]
⇒ [tex]\frac{1}{8}[/tex]
∴ Fraction of cake that was left for Amber = [tex]\frac{1}{8}[/tex]
24. Suppose you throw two fair number cubes. What is the probability that the sum of the results of the throw is 4,5, or 6? Show your work and explain.
Answer:
0.33
Step-by-step explanation:
Two fair number cubes can be thought as dice with sides numbered from 1 to 6. The throw of two dice may result in one of the following combinations in which (d1,d2) are the results of die 1 and 2 respectively:
Ω={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
There are 36 as many possible combinations
The sum of both quantities will produce 11 possible results
S={2,3,4,5,6,7,8,9,10,11,12}
The combinations which produce a sum of 4 are (1,3)(2,2),(3,1), 3 in total
The combinations which produce a sum of 5 are (1,4)(2,3),(3,2),(4,1) 4 in total
The combinations which produce a sum of 6 are (1,5)(2,4),(3,3),(4,2),(5,1) 5 in total
If we want to know the probability that the sum of the results of the throw is 4,5, or 6, we compute the total ways to produce them
T=3+4+5=12 combinations
The probability is finally computed as
[tex]P=\frac{T}{36}=\frac{12}{36}=\frac{1}{3}=0.33[/tex]
Final answer:
To find the probability of getting a sum of 4, 5, or 6 when two dice are rolled, all possible combinations are listed, resulting in 12 favorable outcomes. Those outcomes are then divided by the total number of possible outcomes, 36, resulting in a probability of 1/3.
Explanation:
The question asks about the probability of getting a sum of 4, 5, or 6 when throwing two fair number cubes (dice). Each die has six faces, with numbers ranging from 1 to 6. When rolling two dice, there are 36 possible outcomes (6 outcomes from the first die multiplied by 6 outcomes from the second die).
To find the probability of obtaining a sum of 4, 5, or 6, we first want to identify all the possible combinations that lead to these sums:
For a sum of 4: (1,3), (2,2), (3,1)For a sum of 5: (1,4), (2,3), (3,2), (4,1)For a sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)Altogether, there are 3 + 4 + 5 = 12 outcomes that result in either a 4, 5, or 6 as the sum. Therefore, the probability is the number of favorable outcomes (12) divided by the total number of outcomes (36), which simplifies to 1/3. Hence, the probability is 1/3.
An alcohol awareness task force at a Big-Ten university sampled 200 students after the midterm to ask them whether they went bar hopping the weekend before the midterm or spent the weekend studying, and whether they did well or poorly on the midterm. The following result was obtained. (Please show the calculation process)
Answer:
a) 0.30
b) 0.90
c) 0.15
Step-by-step explanation:
When two events are independent then conditional probability take place.
P(x)P(x)P, (, x, )is a polynomial. P(x)P(x)P, (, x, )divided by (x+7)(x+7)(, x, plus, 7, )has a remainder of 555. P(x)P(x)P, (, x, )divided by (x+3)(x+3)(, x, plus, 3, )has a remainder of -4−4minus, 4. P(x)P(x)P, (, x, )divided by (x-3)(x−3)(, x, minus, 3, )has a remainder of 666. P(x)P(x)P, (, x, )divided by (x-7)(x−7)(, x, minus, 7, )has a remainder of 999. Find the following values of P(x)P(x)P, (, x, ). P(-3)=P(−3)=P, (, minus, 3, ), equals P(7)=P(7)=P, (, 7, ), equals
Answer:
P(-3)=-4
P(7) = 9
Step-by-step explanation:
Consider P(x) is a polynomial.
According to the remainder theorem, if a polynomial, P(x), is divided by a linear polynomial (x - c), then the remainder of that division will be equivalent to f(c).
Using the given information and remainder theorem we conclude,
If P(x) is divided by (x+7), then remainder is 5.
⇒ P(-7)=5
If P(x) is divided by (x+3), then remainder is -4.
⇒ P(-3)=-4
If P(x) is divided by (x-3), then remainder is 6.
⇒ P(3)=6
If P(x) is divided by (x-7), then remainder is 9.
⇒ P(7)=9
Therefore, the required values are P(-3)=-4 and P(7) = 9.
An able order to join a health club a star a fee of $30 is required along with a monthly fee of seven dollars right in equation that Mama knows this situation
Answer:
f(t) = 30 +7t
Step-by-step explanation:
The fee f(t) in terms of months of membership t can be modeled as ...
fee = startup fee + (monthly fee)×(number of months)
f(t) = 30 + 7t
Mrs Dang drove her daughter to school at the average speed of 45 miles per hour. She returned home by the same route at the average speed of 30 miles per hour. If the trip took one half hour, how long did it take to get to school? How far is the school from their home?
Answer: Time it took her to get to the school is 0.6 hours
The distance of the school from their home is 27 miles
Step-by-step explanation:
Mrs Dang drove her daughter to school at the average speed of 45 miles per hour.
Let x miles = distance from the school to their home.
Distance = speed × time
Time = distance / speed
Time used in going to school will be
x/45
She returned home by the same route. This means that distance back home is also x miles.
She returned at an average speed of 30 miles per hour.
Time used in returning home from school will be x/30
x/45
If the trip took one half hour, then the time spent in going to school and the time spent in returning is 1 1/2 hours = 1.5 hours. Therefore
x/30 + x/45 = 1.5
(15x + 10x) /450 = 1.5
15x + 10x = 450 × 1.5 = 675
25x = 675
x = 675/25 = 27
Time it took her to get to the school will be x/45
= 27/45 = 0.6 hours
which of the following statements are always true of parallelagrams? (there are check boxes by the answers)
Answer:
1, 3, and 5
Step-by-step explanation:
A student says that (0, -2) is a solution of 9x-6=3y Are they correct or incorrect? Why?
Answer:
at y-intercept: (0,2) is the correct
Step-by-step explanation:
We have equation 1
9x-6=3y
to find the x-intercept, substitute in 0 for y and solve for x
3(0)=9x+6
3(0)=9x+6
9x+6=0
Subtract 6 from both sides
9x=−6
So x = -6/9 = -2/3
Now to find for y-intercept, substitute in 0 for x and solve for y
3y=9(0)+6
3y=0+6
3y=6
These are the x and y intercepts of the equation 3y=9x+6
x-intercept: (−2/3,0)
y-intercept: (0,2)
Answer:
Correct.
Step-by-step explanation:
Check if x = 0 and y = -2 fits the equation:
9(0) - 6 = -6
3(-2) = -6.
They do so the student is correct.
In Jamie's class, 1/5 of the students are boys. What percent of the students in Jamie’s class are boys?
A) 1.5%
B) 5%
C) 15%
D) 20%
Answer:
The percentage of boys students in Jamie's class is 20 %
Step-by-step explanation:
Given as :
The total number of boys in the Jamie's class = [tex]\frac{1}{5}[/tex] of the total student
Let the total number of student's in the class = x
So, The number of boys = [tex]\frac{1}{5}[/tex] × x
I.e The number of boys = [tex]\dfrac{x}{5}[/tex]
So , in percentage the number of boys students in class = [tex]\dfrac{\textrm Total number of boy student}{\textrm Total number of students}[/tex] × 100
OR, % boys students = [tex]\dfrac{\dfrac{x}{5}}{x}[/tex] × 100
or, % boys students = [tex]\frac{100}{5}[/tex]
∴ % boys students = 20
Hence The percentage of boys students in Jamie's class is 20 % . Answer
Answer:
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A shortstop fields a grounder at a point one-third of the way from second base to third base. How far will he have to throw the ball to make an out at first base? Give the exact answer and an approximation to two decimal places.
Answer: d = 94.87 ft
Step-by-step explanation:
In a baseball game distances between bases is equal to 90 feet
Then if a shortstop get the ball one third of second base ( in the way from second base to third base) shortstop got the ball at 30 ft from second base
Now between the above mentioned point, first base and second base, we have a right triangle. In which distance between shortstop and first base is the hypotenuse. Then
d² = (90)² + (30)² d = √ 8100 + 900 d = 94.87 ft
Final answer:
Using the Pythagorean theorem, the exact distance a shortstop must throw to make an out at first base is the square root of 11700 feet, approximately 108.17 feet.
Explanation:
The question pertains to the distance a shortstop must throw a baseball to make an out at first base. To answer this question, we need to use the geometry of a baseball diamond, which is a square with 90 feet between each base. The Pythagorean theorem can be applied to find the distance from the shortstop to first base. Specifically, when the shortstop is one-third of the way from second to third base, we treat the path from the shortstop to first base as the hypotenuse of a right-angled triangle, with the two sides being from the shortstop to second base and from second base to first base. A full side is 90 feet, so one-third of the way is 30 feet (one-third of 90), making the side from the shortstop to second base 60 feet (90 - 30). The other side is a full 90 feet.
We then calculate the hypotenuse (the throw distance) using the Pythagorean theorem (a² + b² = c²), where a is 60 feet and b is 90 feet:
a² = 60² = 3600b² = 90² = 8100c² = a² + b² = 3600 + 8100 = 11700c = √11700 ≈ 108.1665 feetThe exact distance the shortstop must throw the ball is the square root of 11700, which is an irrational number, and an approximation to two decimal places is 108.17 feet.
A company manufactures and sells video games. A survey of video game stores indicated that at a price of $66 each, the demand would be 400 games, and at a price of $36 each, the demand would be 1,300 games. If a linear relationship between price and demand exists, which of the following equations models the price-demand relationship? (Let x represent the price per video game and y represent the demand.)
Answer:
[tex]y=-30x+2380[/tex]
Step-by-step explanation:
[tex]x\rightarrow[/tex] represent price per video game.
[tex]y\rightarrow[/tex] represent demand.
The linear equation in slope intercept form can be represented as:
[tex]y=mx+b[/tex]
where [tex]m[/tex] is slope of line or rate of change of demand of game per dollar change in price and [tex]b[/tex] is the y-intercept or initial price of game.
We can construct two points using the data given.
When price was $66 each demand was 400. [tex](66,400)[/tex]
When price was $36 each demand was 1300. [tex](36,1300)[/tex]
Using the points we can find slope [tex]m[/tex] of line.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{1300-400}{36-66)}[/tex]
[tex]m=\frac{900}{-30}[/tex]
[tex]m=-30[/tex]
Using point slope form of linear equation to write the equation using a given point.
[tex]y-y_1=m(x-x_1)[/tex]
Using point [tex](66,400)[/tex].
[tex]y-400=-30(x-66)[/tex]
⇒ [tex]y-400=-30x+1980[/tex] [Using distribution]
Adding 400 to both sides:
⇒ [tex]y-400+400=-30x+1980+400[/tex]
⇒[tex]y=-30x+2380[/tex]
The linear relationship between price and demand can be written as:
[tex]y=-30x+2380[/tex]
A greenhouse in a tri-county area has kept track of its customers for the last several years and has determined that it has about 10,000 regular customers. Of those customers, 28% of them plant a vegetable garden in the spring. The greenhouse obtains a random sample of 800 of its customers. Is it safe to assume that the sampling distribution of , the sample proportion of customers that plant a vegetable garden, is approximately normal? Answer Yes or No.
Answer:
No
Step-by-step explanation:
N = 10,000
n= 800
p = .28
Sampling distribution drawn from specific population. So it is not safe to assume that sampling proportion of customers is approximately normal.
Final answer:
The sampling distribution of the sample proportion of customers planting a vegetable garden may not be approximately normal due to insufficient sample size, according to the central limit theorem.
Explanation:
No, it is not safe to assume the sampling distribution of the sample proportion of customers that plant a vegetable garden is approximately normal.
This situation involves calculating the normality of a sample proportion. The **central limit theorem** states that the sampling distribution of a sample proportion will be approximately normal if the sample size is large enough, specifically n * p >= 10 and n * (1-p) >= 10, where n is the sample size and p is the probability of success.
In this case, the sample size is 800 and the probability of success (customers planting a garden) is 28%, so 800 * 0.28 = 224, which is less than 10. Thus, the sampling distribution may not be normal.
It takes the high-speed train x hours to travel the z miles from Town A to Town B at a constant rate, while it takes the regular train y hours to travel the same distance at a constant rate. If the high-speed train leaves Town A for Town B at the same time that the regular train leaves Town B for Town A, how many more miles will the high-speed train have traveled than the regular train when the two trains pass each other?
(A) z(y – x)/x + y
(B) z(x – y)/x + y
(C) z(x + y)/y – x
(D) xy(x – y)/x + y
(E) xy(y – x)/x + y
Answer:
B
Step-by-step explanation:
To solve this, we use ratio.
Firstly, we need to know the number of hours traveled. The total number of hours traveled = x+y
Ratio of this used by high speed train = x/(x +y).
Total distance traveled before they meet = [x/(x + y)] × z
For low speed train = [y/(x + y)] × z.
The difference would be distance by high speed train - distance by low speed train.
= z [ (x - y)/x + y)]
Jayden gets a piece of candy for every 15 minutes he spends reading each day. The number of pieces of candy he receives each day is shown in the chart Monday = * * * Tuesday = * * Wednesday = * * * * * Thursday = Friday = * * * * If he spends 300 minutes reading during the week, how many pieces of candy did Jayden get on Thursday?
Answer:
20 peices
Step-by-step explanation:
There are x number of students at helms. If the number of students increases by 7.8% each year, how many students will be there next year. Write an equation to express this.
There will be 1.078x students next year and equation is number of students in next year = x + 7.8% of x
Solution:Given, There are "x" number of students at helms.
The number of students increases by 7.8% each year which means if there "x" number of students in present year, then the number of students in next year will be x + 7.8% of x
Number of students in next year = number of students in present year + increased number of students.
[tex]\begin{array}{l}{\text { Number of students in next year }=x+7.8 \% \text { of } x} \\\\ {\text { Number of students in next year }=x\left(1+\frac{7.8}{100}\right)} \\\\ {\text { Number of students in next year }=x(1+0.078)=1.078 x}\end{array}[/tex]
Thus there will be 1.078x students in next year
expand and simplify 5(3m-2)+3(m+4)
Answer:
The answer to your question is 18m + 2
Step-by-step explanation:
5(3m - 2) + 3(m + 4)
Multiply 5 by 3m and -2 and 3 by m and 4
15m - 10 + 3m + 12
Simplify like terms
15m +3m - 10 + 12
Result 18m + 2
Rewrite the formula for area of a circle to find the radius of a circle.
The area of a circle (A) is given by the formula A=πr^2 where r is the circle's radius. The formula to find r is (1)_____. If A=54 centimeters^2 and π=22/7, r is (2)_____ centimeters.
1. (A/pi)^1/2; A^2/pi; pi/A
2. 4.15; 2.33; 17.2
Answer:
The correct answers are: Part A. A/π^1/2 = r and Part B. 4.15 centimeters.
Step-by-step explanation:
1. Let's review all the information provided for solving this question:
Area of the circle = π*r² where r is the circle's radius
2. Let's find the solution for r for part A and for part B:
Part A:
A = π*r²
A/π = r²
√A/π = r
A/π^1/2 = r
Part B:
If A = 54 centimeters² and π = 22/7, what is the value of r in centimeters?
Using the result of part A and replacing with the real values, we have:
√A/π = r
√54/(22/7) = r
√54 * 7/22 = r
√378/22 = r
√17.1818 = r
4.1451 = r
4.15 = r (Rounding to two decimal places)
A researcher measures IQ and weight for a group of college students. What kind of correlation is likely to be obtained for these two variables?
a) a positive correlation
b) a negative correlation
c) a correlation near zero*
d) a correlation near one
Answer:
Option b
Step-by-step explanation:
Given that a researcher measures IQ and weight for a group of college students.
In general, we think that the weight has nothing to do with IQ of a person and hence not correlated.
But if we go deep, we find that after a certain weight, the person becomes lazy and inactive with a chance to have reduced IQ
Weight gain causes also health problems including less activity of both brain and body and hence there is a chance for less IQ
So we find that as weight increases iq decreases and when weight decreases, IQ increases.
Thus we can say that there is a negative correlation but not necessarily near to one.
Hence option b is right
The most likely correlation between IQ and weight among college students is near zero, indicating no meaningful relationship between these variables.
Explanation:When it comes to the likelihood of obtaining a correlation between IQ and weight among college students, the correlation is expected to be c) a correlation near zero. This is because there is no theoretical basis or empirical evidence to suggest that these two variables are related in any systematic way. A correlation coefficient is a statistical measure that describes the strength and direction of a relationship between two variables. A coefficient close to 0 indicates a very weak or no correlation, whereas one closer to +1 or -1 indicates a strong positive or negative correlation, respectively. In our scenario, since weight and IQ are not assumed to be related, the correlation would likely be close to 0, suggesting no meaningful relationship.
What is the length of CD in the figure below? Show your work.
Answer:
5
Step-by-step explanation:
First, notice that these two triangles are similar using AA.
Because sides BC and EC are corresponding, you can divide 24 by 8, to determine that the ratio of similitude is 3.
That means that because sides AC and DC are corresponding, 25 - 2x divided by x is 3.
25 - 2x / x = 3
25 - 2x = 3x
25 = 5x
x = 5
x is the same as side CD, so CD = 5.
Using similar triangles and proportions, we can determine that the length of CD in the figure is 10/3 units.
In order to determine the length of CD in the figure, we can use the properties of similar triangles. Since triangles ABC and CDE are similar, we can set up a proportion using their corresponding side lengths:
δ CD / δ AB = CD / AB
Then, we can substitute the given values:
4 / 12 = CD / 20
Next, we can cross multiply and solve for CD:
12 * CD = 4 * 20
CD = (4 * 20) / 12
CD = 40 / 12
CD = 10 / 3
Therefore, the length of CD is 10/3 units.
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For x, y ∈ R we write x ∼ y if x − y is an integer. a) Show that ∼ is an equivalence relation on R. b) Show that the set [0, 1) = {x ∈ R : 0 ≤ x < 1} is a set of representatives for the set of equivalence classes. More precisely, show that the map Φ sending x ∈ [0, 1) to the equivalence class C(x) is a bijection.
Answer:
A. It is an equivalence relation on R
B. In fact, the set [0,1) is a set of representatives
Step-by-step explanation:
A. The definition of an equivalence relation demands 3 things:
The relation being reflexive (∀a∈R, a∼a)The relation being symmetric (∀a,b∈R, a∼b⇒b∼a)The relation being transitive (∀a,b,c∈R, a∼b^b∼c⇒a∼c)And the relation ∼ fills every condition.
∼ is Reflexive:
Let a ∈ R
it´s known that a-a=0 and because 0 is an integer
a∼a, ∀a ∈ R.
∼ is Reflexive by definition
∼ is Symmetric:
Let a,b ∈ R and suppose a∼b
a∼b ⇒ a-b=k, k ∈ Z
b-a=-k, -k ∈ Z
b∼a, ∀a,b ∈ R
∼ is Symmetric by definition
∼ is Transitive:
Let a,b,c ∈ R and suppose a∼b and b∼c
a-b=k and b-c=l, with k,l ∈ Z
(a-b)+(b-c)=k+l
a-c=k+l with k+l ∈ Z
a∼c, ∀a,b,c ∈ R
∼ is Transitive by definition
We´ve shown that ∼ is an equivalence relation on R.
B. Now we have to show that there´s a bijection from [0,1) to the set of all equivalence classes (C) in the relation ∼.
Let F: [0,1) ⇒ C a function that goes as follows: F(x)=[x] where [x] is the class of x.
Now we have to prove that this function F is injective (∀x,y∈[0,1), F(x)=F(y) ⇒ x=y) and surjective (∀b∈C, Exist x such that F(x)=b):
F is injective:
let x,y ∈ [0,1) and suppose F(x)=F(y)
[x]=[y]
x ∈ [y]
x-y=k, k ∈ Z
x=k+y
because x,y ∈ [0,1), then k must be 0. If it isn´t, then x ∉ [0,1) and then we would have a contradiction
x=y, ∀x,y ∈ [0,1)
F is injective by definition
F is surjective:
Let b ∈ R, let´s find x such as x ∈ [0,1) and F(x)=[b]
Let c=║b║, in other words the whole part of b (c ∈ Z)
Set r as b-c (let r be the decimal part of b)
r=b-c and r ∈ [0,1)
Let´s show that r∼b
r=b-c ⇒ c=b-r and because c ∈ Z
r∼b
[r]=[b]
F(r)=[b]
∼ is surjective
Then F maps [0,1) into C, i.e [0,1) is a set of representatives for the set of the equivalence classes.
Determine whether Rolle's Theorem can be applied to the function on the given interval; if so, find the value(s) of c guaranteed by the theorem. (Enter your answers as a comma-separated list. If Rolle's Theorem does not apply, enter DNE.) f(x) = x (5 − x) on [0, 5]
Step-by-step explanation:
1) Check if the function is differentiable on that interval. In this case, yes, because all polynomials are differentiable.
2) plug in the bounds of the interval to see if the y-values equal 0.
f(0)=0
f(5)=0
since the last 2 conditions are satisfied, DNE will not be an answer choice.
3)take derivative and make it equal to 0
f' (×) = 5- 2x
0 = 5- 2x
x = 5/2
4) at c = 5/2, f(x) satisfies rolle's theorem.
Rolle's Theorem can be applied to the function f(x) = x(5 - x) on the interval [0, 5], and the value of c guaranteed by the theorem is c = 2.5.
Explanation:The function f(x) = x(5 - x) on the interval [0, 5] is continuous on the closed interval and differentiable on the open interval (0, 5). To check if Rolle's Theorem can be applied, we first need to verify that the function is continuous on [0, 5] and differentiable on (0, 5). Both of these conditions are satisfied by the given function.
To find the value(s) of c guaranteed by Rolle's Theorem, we need to find the values of x where the derivative of the function is zero. Let's find the derivative of f(x):
f'(x) = 5 - 2x
Setting f'(x) = 0 and solving for x:
5 - 2x = 0
2x = 5
x = 2.5
Therefore, Rolle's Theorem can be applied to the function f(x) = x(5 - x) on the interval [0, 5], and the value of c guaranteed by the theorem is c = 2.5.
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Solve the system. Show your work using Graphing OR Substitution OR Elimination.
Check your answer by showing your solution works in both original equations.
y = 2x -6
y = -½ x +4
The solution is x = 4 and y = 2
Explanation:We have the following system of two linear equations in two variables:
[tex]\begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}y=2x-6\\y=-\frac{1}{2}x+4\end{array}\right.[/tex]
Subtract (2) from (1):
[tex]\begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}y=2x-6\\ -\left(y=-\frac{1}{2}x+4\right)\end{array}\right \\ \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ \\ y-y=2x-6-(-\frac{1}{2}x+4) \\ \\ 0=2x-6+\frac{1}{2}x-4 \\ \\ Combine \ like \ terms: \\ \\ 2x+\frac{1}{2}x-6-4=0 \\ \\ 2.5x-10=0 \\ \\ 2.5x=10 \\ \\ x=\frac{10}{2.5} \\ \\ x=4[/tex]
Substituting the x-value into (1):
[tex]y=2(4)-6 \\ \\ y=8-6 \\ \\ y=2[/tex]
So the solution to this system is:
[tex]\boxed{x=4 \ and \ y=2}[/tex]
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The following sum is a partial sum of an arithmetic sequence; use either formula for finding partial sums of arithmetic sequences to determine its value.
-9+1+...+561
Answer:
16008
Step-by-step explanation:
Sum of an arithmetic sequence is:
S = (n/2) (2a₁ + (n−1) d)
or
S = (n/2) (a₁ + a)
To use either equation, we need to find the number of terms n. We know the common difference d is 1 − (-9) = 10. Using the definition of the nth term of an arithmetic sequence:
a = a₁ + (n−1) d
561 = -9 + (n−1) (10)
570 = 10n − 10
580 = 10n
n = 58
Using the first equation to find the sum:
S = (n/2) (2a₁ + (n−1) d)
S = (58/2) (2(-9) + (58−1) 10)
S = 29 (-18 + 570)
S = 16008
Using the second equation to find the sum:
S = (n/2) (a₁ + a)
S = (58/2) (-9 + 561)
S = 16008
Answer:
16008
Step-by-step explanation:
Act scene where Macbeth and last Macbeth plan to kill king Duncan
Answer:
Step-by-step explanation:
inside the castle