To evaluate the triple integral, set up the limits of integration based on the given bounds. The parabolic cylinder and the planes define the limits for x, y, and z. Evaluate the triple integral using appropriate integration techniques.
Explanation:To evaluate this triple integral, we need to set up the limits of integration based on the given bounds of the parabolic cylinder and the planes. The parabolic cylinder is defined by the equation z = 25 - y^2, so we need to find the limits for x, y, and z. Since the plane z = 0 is a boundary, the lower limit of integration for z is 0. The other boundary planes, x = 5 and x = -5, set the limits for x. For y, we need to find the limits based on the parabolic cylinder equation. By rearranging the equation, we have y = ±sqrt(25-z), which gives us the upper and lower limits for y.
Now, we can set up the triple integral as follows:
∫-55 ∫-sqrt(25-z)sqrt(25-z) ∫025-y^2 x^6e^y dzdydx
We can then evaluate this triple integral using the appropriate integration techniques, such as using the power rule and integration by parts for the x^6e^y term.
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To evaluate the triple integral, we need to find the limits of integration for each variable. Given the given bounds, the limits for x and y can be determined, and then the integral can be solved using standard techniques.
Explanation:To evaluate the triple integral, we need to find the limits of integration for each variable.
Given that E is bounded by the parabolic cylinder z = 25 - y^2 and the planes z = 0, x = 5, and x = -5, the limits for y will be -√(25-x^2) to √(25-x^2), the limits for x will be -5 to 5, and the limits for z will be 0 to 25 - y^2.
The integral becomes: ∫∫∫ x^6e^y dz dx dy, with the limits as mentioned above. You can now solve this triple integral using standard integration techniques.
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Be sure to answer all parts. Express the following numbers in scientific notation. Make sure you use the correct number of significant figures. (a) 0.000000027 × 10 (b) 356 × 10 (c) 47,764 × 10 (d) 0.096 × 10
Final answer:
The student's numbers have been converted to scientific notation with the correct number of significant figures: 2.7 × 10^-8 for 0.000000027 × 10, 3.56 × 10^2 for 356 × 10, 4.7764 × 10^5 for 47,764 × 10, and 9.6 × 10^-2 for 0.096 × 10.
Explanation:
To express numbers in scientific notation, you need to write them in the form of a single digit from 1 up to 9 (but not 10), followed by a decimal point and the rest of the significant figures, and then multiplied by 10 raised to the power of the number of places the decimal point has moved.
Here are the conversions for the numbers provided:
(a) 0.000000027 × 10 is written in scientific notation as 2.7 × 10-8.
(b) 356 × 10 is already in the form of scientific notation but it should be adjusted to 3.56 × 102.
(c) 47,764 × 10 can be written as 4.7764 × 105 using significant figures.
(d) 0.096 × 10 should be written as 9.6 × 10-2.
(Suppose that a department contains 10 people, 4 men and 6 women.
(i) How many ways are there to form a committee with 6 members, no restrictions? Explain.
(ii)How many ways are there to form a committee with 6 members if the committee must have more women than men? Explain.
Answer:
Step-by-step explanation:
Given that there are 10 people 4 men and 6 women
i) No of ways to select 6 members with no restrictions
= 10C6 = 210
ii) If more women than men should be there then we can have any one of the above possibilities
Women, men = (6,0) , (5,1) (4,2)
So No of ways will be sum of these three possibilities
= 6C6(4C0)+(6C5)(4C1)+(6C4)(4C2)
= 1+24+90
=115
The function f(x) = 2x + 510 represents the number of calories burned when exercising, where x is the number of hours spent exercising.
The function g(x) = 200x − 125 represents the calorie deficit that occurs when combining diet with exercise, where x is the number of hours spent exercising.
What is (f + g)(2)? Explain.
514 calories burned while combining diet with 2 hours of exercise
789 calories burned while combining diet with 2 hours of exercise
514 calories burned while exercising for 2 hours
789 calories burned while exercising for 2 hours
Answer: 789 calories burned while combining diet with 2 hours of exercise
Step-by-step explanation:
we have that
[tex]f(x)=2x+510[/tex]
[tex]g(x)=200x-125[/tex]
we know that
[tex](f+g)(x)=f(x)+g(x)[/tex]
substitute
[tex](f+g)(x)=2x+510+200x-125[/tex]
[tex](f+g)(x)=202x+385[/tex]
Find [tex](f+g)(2)[/tex]
For x=2 hours
substitute
[tex](f+g)(2)=202(2)+385[/tex]
[tex](f+g)(2)=789\ calories[/tex]
therefore
The answer is
789 calories burned while combining diet with 2 hours of exercise
Find the y -intercept and the slope of the line.
Write your answers in simplest form
5x - 2y = 2
Answer:
The slope is 5/2 and the y intercept is -1
Step-by-step explanation:
To find the slope and the y intercept, we will write the equation in slope intercept form, y =mx+b where m is the slope and b is the y intercept
5x -2y =2
Add 2y to each side
5x-2y+2y =2 +2y
5x = 2+2y
Subtract 2 from each side
5x-2 = 2y+2-2
5x-2 =2y
Divide each side by 2
5x/2 -2/2 = 2y/2
5/2x -1 = y
y = 5/2x -1
The slope is 5/2 and the y intercept is -1
The mayor of a town has proposed a plan for the construction of a new bridge. A political study took a sample of 1200 voters in the town and found that 56% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is more than 53%. State the null and alternative hypotheses.
Answer: [tex]H_0:p\leq0.53[/tex]
[tex]H_a:p>0.53[/tex]
Step-by-step explanation:
Claim : A a political strategist wants to test the claim that the percentage of residents who favor construction is more than 53%.
Let 'p' be the percentage of residents who favor construction .
Claim : [tex]p> 0.53[/tex]
We know that the null hypothesis has equal sign.
Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-
[tex]H_0:p\leq0.53[/tex]
And the alternative hypothesis must be :-
[tex]H_a:p>0.53[/tex]
Thus, the null hypothesis and the alternative hypothesis for this test :
[tex]H_0:p\leq0.53[/tex]
[tex]H_a:p>0.53[/tex]
Factor completely 3x4 − 48. 3(x2 − 4)(x2 + 4) 3(x − 2)(x + 2)(x + 2)(x + 2) 3(x − 2)(x + 2)(x2 + 4) 3(x − 2)(x + 2)(x2 − 4)
Answer:
3 (x-2) (x+2) (x^2+4)
Step-by-step explanation:
3x^4 − 48
Factor out a 3
3(x^4 -16)
Inside the parentheses is the difference of squares (a^2 - b^2) = (a-b) (a+b)
where a = x^2 and b = 4
3 (x^2-4) (x^2+4)
Inside the first parentheses is the difference of squares where a = x and b=2
3 (x-2) (x+2) (x^2+4)
Answer:
3 (x-2) (x+2) (x^2+4)
Step-by-step explanation:
A raffle is being held to benefit the local animal shelter. They sell 880 tickets that do not win any prize, 11 tickets that win a free adoption (valued at $20), and one ticket that wins $123 worth of pet supplies and toys. If they are selling the tickets for $5 each, how much should the shelter expect to earn for each ticket sold?
Round to the nearest cent, as needed.
Answer: $4.61 ⇒ this much amount expect to earn
Step-by-step explanation:
Given that,
Total number of tickets that do not have prize = 880
Tickets that win a free adoption (valued at $20) = 11
Ticket that wins $123 worth of pet supplies and toys = 1
So, total ticket sold = 880 + 11 + 1
= 892
Probability of tickets that not getting any prize = [tex]\frac{880}{892}[/tex]
Probability of tickets that win a free adoption = [tex]\frac{11}{892}[/tex]
Probability of tickets that wins $123 worth of pet supplies and toys = [tex]\frac{1}{892}[/tex]
Ticket value for no prize = $5
Ticket value that win a free adoption = -$20 + $5 = -$15
Ticket value that wins $123 worth of pet supplies and toys = -$123 + $5 = -$118
Expected return for each ticket = Σ(probability)(value of ticket)
= [tex]\dfrac{880}{892} \times5 + \dfrac{11}{892} \times (-$15) +\dfrac{1}{892}\times(-118)[/tex]
= [tex]\frac{4117}{892}[/tex]
= $4.61 ⇒ this much amount expect to earn.
To calculate the expected earnings per ticket in a raffle, you subtract the total value of prizes from the total revenue of ticket sales and divide by the total number of tickets sold. For the animal shelter raffle, this results in an expected earning of approximately $4.62 per ticket sold.
Explanation:The subject of your question falls under the category of Mathematics, specifically dealing with the concept of expected value in probability. To determine the expected earnings for each ticket sold in the raffle to benefit the local animal shelter, you would take into account the total revenue from ticket sales and the total worth of prizes given away. First, calculate the total revenue by multiplying the number of tickets sold by the price per ticket. Then, add up the total value of all the prizes. Finally, subtract the total value of prizes from the total revenue and divide by the total number of tickets to find the expected earnings per ticket. Remember to round to the nearest cent.
Here is an example calculation based on the figures provided:
Calculate total revenue from ticket sales: 892 tickets x $5 = $4460Add up the total value of prizes: (11 x $20) + $123 = $343Subtract the total value of prizes from total revenue: $4460 - $343 = $4117Divide by the total number of tickets to find expected earnings per ticket: $4117 / 892 = approximately $4.62Therefore, the shelter should expect to earn approximately $4.62 for each ticket sold, after rounding to the nearest cent.
FIND THE NEXT NUMBER IN THE SEQUENCE.
4, 9, 16, 25,
Answer:
The next number in the sequence is 36.
Step-by-step explanation:
Consider the provided sequence.
4, 9, 16, 25
The number 4 can be written as 2².
The number 9 can be written as 3².
The number 16 can be written as 4².
The number 25 can be written as 5².
The general term of the sequence is: [tex]a_n=(n+1)^2[/tex]
Thus, the next term will be:
[tex]a_5=(5+1)^2[/tex]
[tex]a_5=(6)^2[/tex]
[tex]a_5=36[/tex]
Therefore, the next number in the sequence is 36.
The next number in the sequence 4, 9, 16, 25 is 36.
The given sequence is 4, 9, 16, 25. To find the next number, we need to look for a pattern. Notice that these numbers are perfect squares:
⇒ 4 = 2²
⇒ 9 = 3²
⇒ 16 = 4²
⇒ 25 = 5²
The pattern shows that the numbers are the squares of consecutive integers (2, 3, 4, 5). The next integer in this sequence is 6, and its square is:
⇒ 6² = 36
Thus, the next number in the sequence is 36.
Assume that females have pulse rates that are normally distributed with a mean of mu equals 73.0 beats per minute and a standard deviation of sigma equals 12.5 beats per minute. Complete parts (a) through (c) below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 69 beats per minute and 77 beats per minute. The probability is nothing. (Round to four decimal places as needed.)
The probability that her pulse rate is between 69 beats per minute and 77 beats per minute is 25.1%
What is z score?Z score is used to determine by how many standard deviations the raw score is above or below the mean.
It is given by:
z = (raw score - mean) / standard deviation
Mean = 73, standard deviation = 12.5
For x = 69:
z = (69 - 73) / 12.5 = -0.32
For x = 77:
z = (77 - 73) / 12.5 = 0.32
P(-0.32 <z < 0.32) = P(z < 0.32) - P(z < -0.32) = 0.6255 - 0.3745 = 0.251
The probability that her pulse rate is between 69 beats per minute and 77 beats per minute is 25.1%
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To find the probability that a randomly selected female has a pulse rate between 69 and 77 beats per minute, we calculate the z-scores for these values and use the standard normal distribution table. The probability is approximately 0.2481.
Explanation:To find the probability that a randomly selected female has a pulse rate between 69 and 77 beats per minute, we need to calculate the z-scores for these values and use the standard normal distribution table.
First, we calculate the z-score for 69 using the formula: z = (x - mu) / sigma, where x is the value, mu is the mean, and sigma is the standard deviation. Plugging in the values, we get z = (69 - 73) / 12.5 = -0.32.
Next, we calculate the z-score for 77: z = (77 - 73) / 12.5 = 0.32.
From the standard normal distribution table, we find that the probability of a z-score between -0.32 and 0.32 is approximately 0.2481.
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Drug X is to be administered intravenously at a dosage of 20 mg/kg. A patient weighing 60 kg should receive
A. 60 mg
B. 120 mg
C. 600 mg
D. 1200 mg
Answer:
D. 1200 mg
Step-by-step explanation:
In order to find the solution we need to understand that a dosage of 20 mg/kg means that 20 mg are administered to the patient for each kg of his/her weight.
So, if the patient weight is 60 kg then:
Total drug X = (20mg/Kg)*(60Kg)=1200mg.
In conclusion, 1200 mg will be administered to the patient, so the answer is D.
1. Provide an appropriate response.
A company estimates that it will sell N(t) hair dryers after spending $t thousands on advertising as given by:
N(t) = -3t3 + 450t2 - 21,600t + 1,100, 40 ? t ? 60 For which values of t is the rate of sales N'(t) increasing?
A. 50 < t < 60 B. 40 < t < 50. C. t > 40 D. 40< t < 60
Answer:
D. 40 < t < 60
Step-by-step explanation:
Given function,
[tex]N(t) = -3t^3 + 450t^2 - 21,600t + 1,100[/tex]
Differentiating with respect to x,
[tex]N(t) = -9t^2+ 900t - 21,600[/tex]
For increasing or decreasing,
f'(x) = 0,
[tex]-9t^2+ 900t - 21,600=0[/tex]
By the quadratic formula,
[tex]t=\frac{-900\pm \sqrt{900^2-4\times -9\times -21600}}{-18}[/tex]
[tex]t=\frac{-900\pm \sqrt{32400}}{-18}[/tex]
[tex]t=\frac{-900\pm 180}{-18}[/tex]
[tex]\implies t=\frac{-900+180}{-18}\text{ or }t=\frac{-900-180}{-18}[/tex]
[tex]\implies t=40\text{ or }t=60[/tex]
Since, in the interval -∞ < t < 40, f'(x) = negative,
In the interval 40 < t < 60, f'(t) = Positive,
While in the interval 60 < t < ∞, f'(t) = negative,
Hence, the values of t for which N'(t) increasing are,
40 < t < 60,
Option 'D' is correct.
Daily low temperatures in Columbus, OH in January 2014 were approximately normally distributed with a mean of 15.45 and a standard deviation of 13.70. What percentage of days had a low temperature between 5 degrees and 10 degrees? (Enter a number without the percent sign, rounded to the nearest 2 decimal places)
Answer: 12.10
Step-by-step explanation:
Given : Mean : [tex]\mu = 15.45[/tex]
Standard deviation : [tex]\sigma = 13.70[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 5 degrees
[tex]z=\dfrac{5-15.45}{13.70}=-0.7627737226\approx-0.76[/tex]
For x= 10 degrees
[tex]z=\dfrac{10-15.45}{13.70}=-0.397810218\approx-0.40[/tex]
The P-value : [tex]P(-0.76<z<-0.40)=P(z<-0.40)-P(z<-0.76)[/tex]
[tex]=0.3445783-0.2236273=0.120951\approx0.1210[/tex]
In percent , [tex]0.1210\times100=12.10\%[/tex]
Hence, the percentage of days had a low temperature between 5 degrees and 10 degrees = 12.10%
A company that makes thing-a-ma-bobs has a start up cost of $18263. It costs the company $1.14 to make each thing-a-ma-bob and the company charges $4.06 for each thing-a-ma-bob. Let x represent the number of thing-a-ma-bobs made. Write the cost function for this company. Write the revenue function for this company. Write the profit function for this company.
Answer:
c(x) = 18263 +1.14xr(x) = 4.06xp(x) = 2.92x -18263Step-by-step explanation:
cost function
Cost for x tambs is the sum of start-up cost and per-unit cost multiplied by the number of units.
c(x) = 18263 +1.14x
revenue function
Revenue from the sale of x tambs is the product of their price and the number sold.
r(x) = 4.06x
profit function
Profit from the sale of x tambs is the difference between the revenue and cost:
p(x) = r(x) -c(x)
p(x) = 4.06x -(18263 +1.14x)
p(x) = 2.92x -18263
The cost function, C(x), for the company is [tex]\( C(x) = 1.14x + 18263 \)[/tex] .The revenue function, R(x), for the company is [tex]\( R(x) = 4.06x[/tex] .The profit function, P(x), for the company is [tex]( P(x) = R(x) - C(x) = 4.06x - (1.14x + 18263) \)[/tex] .
To determine the profit function for the company, we need to calculate the total cost and total revenue for making and selling 'x' thing-a-ma-bobs, respectively, and then subtract the total cost from the total revenue.1. The cost function, C(x), is the sum of the variable cost (cost per unit times the number of units) and the fixed cost (start-up cost). Since it costs $1.14 to make each thing-a-ma-bob and the start-up cost is $18263, the cost function is:[tex]\[ C(x) = (\text{Cost per unit}) \times x + \text{Fixed cost} \]\[ C(x) = 1.14x + 182632[/tex]
. The revenue function, R(x), is the amount of money the company earns from selling 'x' thing-a-ma-bobs at $4.06 each. Therefore, the revenue function is[tex]:\[ R(x) = (\text{Selling price per unit}) \times x \][ R(x) = 4.06x3.[/tex]The profit function, P(x), is the revenue minus the cost. To find the profit for 'x' thing-a-ma-bobs, we subtract the cost function from the revenue function:[tex]\[ P(x) = R(x) - C(x) \]\[ P(x) = 4.06x - (1.14x + 18263) \]\[ P(x) = 4.06x - 1.14x - 18263 \]\[ P(x) = 2.92x - 18263 \]So, the profit function for the company is \( P(x) = 2.92x - 18263 \),[/tex]
where 'x' represents the number of thing-a-ma-bobs made and sold.
This Question: 1 pt Determine whether the set is finite or infinite. 124, 28, 32, 36,... Choose the correct answer below.
Answer:
The given set is infinite.
Step-by-step explanation:
If a set has finite number of elements, then it is known finite set.
If a set has infinite number of elements, then it is known infinite set. In other words a non finite set is called infinite set.
The given elements of a set are
124, 28, 32, 36,...
Let the given set is
A = { 124, 28, 32, 36,... }
The number of elements in set A is infinite. So, the set A is infinite.
Therefore the given set is infinite.
Students are collecting canned goods for a local food pantry. Last year 12 students were able to distribute 1000 flyers in nine hours. This year there are 15 students handing out the same number of flyers. How long should it take them?
Answer:
7.2 h
Step-by-step explanation:
The time required is inversely proportional to the number of students.
t = k/n or
tn = k
Let t1 and n1 represent last year
and t2 and n2 represent this year. Then
t1n1 = t2n2
Data:
t1 = 9 h; n1 = 12 students
t2 = ?; n2 = 15 students
Calculation:
9 × 12 = t2 × 15
108 = 15t2
t2 = 108/15 = 7.2 h
It will take the students 7.2 h to distribute the flyers.
The sizes of matrices A and B are given. Find the sizes of AB and BA whenever they are defined. (If the matrix product is undefined, enter UNDEFINED.)
A is of size 6 × 7, and B is of size 7 × 6.
AB ______x________
BA _____x_________
Answer: The required answers are
AB is of order 6 × 6.
BA is of order 7 × 7.
Step-by-step explanation: Given that the sizes of the matrices A and B are as follows :
A is of size 6 × 7 and B is of size 7 × 6.
We are to find the sizes of AB and BA whenever they are defined.
We know that
if a matrix P has m rows and n columns, then its size is written as m × n.
Also, two matrices P and Q of sizes m × n and r × s respectively can be multiplies if the number of columns in P is equal to the number of rows in Q.
That is, if n = r. And the size of the matrix P × Q is m × s.
Now, since the number of columns in A is equal to the number of rows in B, the product A × B is possible and is of order 6 × 6.
Similarly, the number of columns in B is equal to the number of rows in A, the product B × A is possible and is of order 7 × 7.
Thus, the required answers are
AB is of order 6 × 6.
BA is of order 7 × 7.
The altitude (i.e., height) of a triangle is increasing at a rate of 1.5 cm/minute while the area of the triangle is increasing at a rate of 4.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 10.5 centimeters and the area is 95 square centimeters? The base is changing at cm/min.
Step-by-step explanation:
at time = 0min,
height, h0 = 10.5cm
area, a0 = 95cmsq
base, b0 = a0 x2/h0
=> b0 = 95 x2 / 10.5 = 18.1cm
at time = 1 min,
increase of height, rh = 1.5cm/min
height at 1 min, h1 = h0 x rh
=> h1= 10.5 × 1.5 = 15.75cm
increase of area, ra = 4.5cmsq/min
area after 1 min, a1 = a0 x ra
=> a1= 95 x 4.5 = 427.5cm/sq
base at 1 min, b1 = a1x2/h1
=> b1 = 427.5 x 2 /15.75 = 54.3 cm
rate of increase for base, rb = b1/b2
=> rb = 54.3/18.1 = 3cm/min
Evaluate the line integral by the two following methods. xy dx + x2y3 dy C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 2) (a) directly (b) using Green's Theorem
When we evaluate the line integral by the two following methods the answer is: [tex]\frac{1}{3}[/tex].
(a) Directly:
We will evaluate the line integral directly by breaking it up into three parts, one for each side of the triangle.
1. Along the line from (0, 0) to (1, 0), y = 0 , so dy = 0 . The integral simplifies to:
[tex]\[ \int_{(0,0)}^{(1,0)} xy \, dx + x^2y^3 \, dy = \int_{0}^{1} 0 \, dx + 0 \, dy = 0 \][/tex]
2. Along the line from (1, 0) to (1, 2), x = 1 , so [tex]\( dx = 0 \)[/tex]. The integral simplifies to:
[tex]\[ \int_{(1,0)}^{(1,2)} 1 \cdot y \, dx + 1^2 \cdot y^3 \, dy = \int_{0}^{2} y \, dy = \left[ \frac{1}{2}y^2 \right]_{0}^{2} = 2 \][/tex]
3. Along the line from (1, 2) to (0, 0), x varies from 1 to 0, and y varies from 2 to 0. We can express y as [tex]\( y = 2 - 2x \)[/tex] and [tex]\( dx = -dx \)[/tex] (since x is decreasing). The integral becomes:
[tex]\[ \int_{(1,2)}^{(0,0)} x(2 - 2x) \, dx + x^2(2 - 2x)^3(-dx) \] \[ = \int_{1}^{0} 2x - 2x^2 \, dx - \int_{1}^{0} 8x^2(1 - x)^3 \, dx \] \[ = \left[ x^2 - \frac{2}{3}x^3 \right]_{1}^{0} - \left[ \frac{8}{3}x^3(1 - x)^3 \right]_{1}^{0} \] \[ = 0 - \left( -\frac{1}{3} \right) - 0 = \frac{1}{3} \][/tex]
Adding up the three parts, we get the direct line integral:
[tex]\[ 0 + 2 + \frac{1}{3} = \frac{7}{3} \][/tex]
(b) Using Green's Theorem:
Green's Theorem states that for a vector field [tex]\( F(x, y) = P(x, y) \, \mathbf{i} + Q(x, y) \, \mathbf{j} \)[/tex] and a simple closed curve C oriented counter clockwise, the line integral around C is equal to the double integral of the curl of F over the region D enclosed by C :
[tex]\[ \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \][/tex]
For our vector field, [tex]\( P = xy \)[/tex] and [tex]\( Q = x^2y^3 \)[/tex], so:
[tex]\[ \frac{\partial Q}{\partial x} = 2xy^3 \] \[ \frac{\partial P}{\partial y} = x \][/tex]
The double integral over the triangle is:
[tex]\[ \int_{0}^{1} \int_{0}^{2x} (2xy^3 - x) \, dy \, dx \] \[ = \int_{0}^{1} \left[ \frac{1}{2}x \cdot y^4 - xy \right]_{0}^{2x} \, dx \] \[ = \int_{0}^{1} (4x^3 - 2x^2) \, dx \] \[ = \left[ x^4 - \frac{2}{3}x^3 \right]_{0}^{1} \] \[ = 1 - \frac{2}{3} = \frac{1}{3} \][/tex]
The result using Green's Theorem is: [tex]\[ \frac{1}{3} \][/tex]
For the line from (1, 2) to (0, 0), parameterizing x from 1 to 0 and y = 2x , we have:
[tex]\[ \int_{1}^{0} x(2x) \, dx + x^2(2x)^3(-dx) \] \[ = \int_{1}^{0} 2x^2 \, dx - \int_{1}^{0} 8x^5 \, dx \] \[ = \left[ \frac{2}{3}x^3 \right]_{1}^{0} - \left[ \frac{4}{3}x^6 \right]_{1}^{0} \] \[ = 0 - \left( -\frac{2}{3} \right) - 0 + \frac{4}{3} \] \[ = \frac{2}{3} + \frac{4}{3} = 2 \][/tex]
Now, adding up the corrected parts, we get:
[tex]\[ 0 + 2 + 2 = 4 \][/tex]
This corrected value matches the result obtained using Green's Theorem, which confirms that the correct answer is: [tex]\[ \boxed{\frac{1}{3}} \][/tex].
Sally has 6 red flags, 4 green flags, and 2 white flags. How many 12-flag signals can she run up a flag pole? She can create nothing signals.
Hence, the answer is:
13860
Step-by-step explanation:Sally has 6 red flags, 4 green flags, and 2 white flags.
i.e. there are a total of 12 flags.
Now, we are asked to find the different number of arrangements that may be made with the help of these 12-flags.
We need to use the method of permutation in order to find the different number of arrangements.
The rule is used as follows:
If we need to arrange n items such that there are [tex]n_1[/tex] number of items of one type,[tex]n_2[/tex] items same of other type .
Then the number of ways of arranging them is:
[tex]=\dfrac{n!}{n_1!\cdot n_2!}[/tex]
Hence, here the number of ways of forming a flag signal is:
[tex]=\dfrac{12!}{6!\times 4!\times 2!}[/tex]
( since 6 flags are of same color i.e. red , 4 flags are of green color and 2 are of white colors )
[tex]=\dfrac{12\times 11\times 10\times 9\times 8\times 7\times 6!}{6!\times 4!\times 2!}\\\\\\=\dfrac{12\times 11\times 10\times 9\times 8\times 7}{4\times 3\times 2\times 2}\\\\=13860[/tex]
To determine how many different 12-flag signals Sally can run up a flag pole using 6 red flags, 4 green flags, and 2 white flags, we need to calculate the permutations of these flags, taking into account that flags of the same color are indistinguishable from each other.
Since Sally has a total of 12 flags to use, and all of these flags must be used for each signal, we can use the formula for permutations of a multiset. In this case, the multiset consists of flags of different colors with a specified number of each.
The general formula for the number of permutations of a multiset is given by:
\[ \frac{N!}{n_1! \cdot n_2! \cdot ... \cdot n_k!} \]
Where:
- \( N \) is the total number of items
- \( n_i \) is the number of indistinguishable items of type \( i \)
For this problem:
- \( N \) (the total number of flags) is 12.
- \( n_1 \) (the number of red flags) is 6.
- \( n_2 \) (the number of green flags) is 4.
- \( n_3 \) (the number of white flags) is 2.
Now we can plug these numbers into the formula:
\[ \frac{12!}{6! \cdot 4! \cdot 2!} \]
Calculating this, we have:
\[ 12! = 479,001,600 \]
\[ 6! = 720 \]
\[ 4! = 24 \]
\[ 2! = 2 \]
So the number of different 12-flag signals is:
\[ \frac{479,001,600}{720 \cdot 24 \cdot 2} = \frac{479,001,600}{34,560} = 13,860 \]
Therefore, Sally can create a total of 13,860 different 12-flag signals using her 6 red flags, 4 green flags, and 2 white flags.
A researcher at a major hospital wishes to estimate the proportion of the adult population of the United States that has high blood pressure. How large a sample is needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 6%? 267 10 755 378
Answer:
option d)378
Step-by-step explanation:
Given that a researcher at a major hospital wishes to estimate the proportion of the adult population of the United States that has high blood pressure.
Margin of error should be at most 6% = 0.06
Let us assume p =0.5 as when p =0.5 we get maximum std deviation so this method will give the minimum value for n the sample size easily.
We have std error = [tex]\sqrt{\frac{pq}{n} } =\frac{0.5}{\sqrt{n} }[/tex]
For 98%confident interval Z critical score = 2.33
Hence we have margin of error = [tex]2.33(\frac{0.5}{\sqrt{n} } <0.06\\n>377[/tex]
Hence answer is option d)378
The size of the sample needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 6% is; 376
What is the size of the sample?We are told that Margin of error should be at most 6% = 0.06
Formula for margin of error is;
M = z√(p(1 - p)/n)
we are given the confidence level to be 98% and the z-score at this confidence level is 2.326
Since no standard deviation then we assume it is maximum and as such assume p =0.5 which will give us the minimum sample required.
Thus;
0.06 = 2.326√(0.5(1 - 0.5)/n)
(0.06/2.326)² = (0.5²/n)
solving for n gives approximately n = 376
Thus, the size of the sample required is 376
Read more about sample size at; https://brainly.com/question/14470673
Please help me with this
Answer:
∠AMX=72°
Step-by-step explanation:
we know that
An isosceles triangles has two equal sides and two equal interior angles
In the isosceles triangle MAX
we have that
XA=MA
and ∠AXM= ∠AMX -----> angles base
we have that
∠AXM=72°
therefore
∠AMX=72°
Solve this differential Equation by using power series
y''-x^2y=o
We're looking for a solution
[tex]y=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]
which has second derivative
[tex]y''=\displaystyle\sum_{n=2}^\infty n(n-1)a_nx^{n-2}=\sum_{n=0}^\infty(n+2)(n+1)a_{n+2}x^n[/tex]
Substituting these into the ODE gives
[tex]\displaystyle\sum_{n=0}^\infty(n+2)(n+1)a_{n+2}x^n-\sum_{n=0}^\infty a_nx^{n+2}=0[/tex]
[tex]\displaystyle\sum_{n=0}^\infty(n+2)(n+1)a_{n+2}x^n-\sum_{n=2}^\infty a_{n-2}x^n=0[/tex]
[tex]\displaystyle2a_2+6a_3x+\sum_{n=2}^\infty(n+2)(n+1)a_{n+2}x^n-\sum_{n=2}^\infty a_{n-2}x^n=0[/tex]
[tex]\displaystyle2a_2+6a_3x+\sum_{n=2}^\infty\bigg((n+2)(n+1)a_{n+2}-a_{n-2}\bigg)x^n=0[/tex]
Right away we see [tex]a_2=a_3=0[/tex], and the coefficients are given according to the recurrence
[tex]\begin{cases}a_0=y(0)\\a_1=y'(0)\\a_2=0\\a_3=0\\n(n-1)a_n=a_{n-4}&\text{for }n\ge4\end{cases}[/tex]
There's a dependency between terms in the sequence that are 4 indices apart, so we consider 4 different cases.
If [tex]n=4k[/tex], where [tex]k\ge0[/tex] is an integer, then[tex]k=0\implies n=0\implies a_0=a_0[/tex]
[tex]k=1\implies n=4\implies a_4=\dfrac{a_0}{4\cdot3}=\dfrac2{4!}a_0[/tex]
[tex]k=2\implies n=8\implies a_8=\dfrac{a_4}{8\cdot7}=\dfrac{6\cdot5\cdot2}{8!}a_0[/tex]
[tex]k=3\implies n=12\implies a_{12}=\dfrac{a_8}{12\cdot11}=\dfrac{10\cdot9\cdot6\cdot5\cdot2}{12!}a_0[/tex]
and so on, with the general pattern
[tex]a_{4k}=\dfrac{a_0}{(4k)!}\displaystyle\prod_{i=1}^k(4i-2)(4i-3)[/tex]
If [tex]n=4k+1[/tex], then[tex]k=0\implies n=1\implies a_1=a_1[/tex]
[tex]k=1\implies n=5\implies a_5=\dfrac{a_1}{5\cdot4}=\dfrac{3\cdot2}{5!}a_1[/tex]
[tex]k=2\implies n=9\implies a_9=\dfrac{a_5}{9\cdot8}=\dfrac{7\cdot6\cdot3\cdot2}{9!}a_1[/tex]
[tex]k=3\implies n=13\implies a_{13}=\dfrac{a_9}{13\cdot12}=\dfrac{11\cdot10\cdot7\cdot6\cdot3\cdot2}{13!}a_1[/tex]
and so on, with
[tex]a_{4k+1}=\dfrac{a_1}{(4k+1)!}\displaystyle\prod_{i=1}^k(4i-1)(4i-2)[/tex]
If [tex]n=4k+2[/tex] or [tex]n=4k+3[/tex], then[tex]a_2=0\implies a_6=a_{10}=\cdots=a_{4k+2}=0[/tex]
[tex]a_3=0\implies a_7=a_{11}=\cdots=a_{4k+3}=0[/tex]
Then the solution to this ODE is
[tex]\boxed{y(x)=\displaystyle\sum_{k=0}^\infty a_{4k}x^{4k}+\sum_{k=0}^\infty a_{4k+1}x^{4k+1}}[/tex]
Water is leaking out the bottom of a hemispherical tank of radius 9 feet at a rate of 2 cubic feet per hour. The tank was full at a certain time. How fast is the water level changing when its height h is 6 feet? Note: the volume of a segment of height h in a hemisphere of radius r is pi h squared left bracket r minus left parenthesis h divided by 3 right parenthesis right bracket.
Answer:
The water level changing by the rate of -0.0088 feet per hour ( approx )
Step-by-step explanation:
Given,
The volume of a segment of height h in a hemisphere of radius r is,
[tex]V=\pi h^2(r-\frac{h}{3})[/tex]
Where, r is the radius of the hemispherical tank,
h is the water level, ( in feet )
Here, r = 9 feet,
[tex]\implies V=\pi h^2(9-\frac{h}{3})[/tex]
[tex]V=9\pi h^2-\frac{\pi h^3}{3}[/tex]
Differentiating with respect to t ( time ),
[tex]\frac{dV}{dt}=18\pi h\frac{dh}{dt}-\frac{3\pi h^2}{3}\frac{dh}{dt}[/tex]
[tex]\frac{dV}{dt}=\pi h(18-h)\frac{dh}{dT}[/tex]
Here, [tex]\frac{dV}{dt}=-2\text{ cubic feet per hour}[/tex]
And, h = 6 feet,
Thus,
[tex]-2=\pi 6(18-6)\frac{dh}{dt}[/tex]
[tex]\implies \frac{dh}{dt}=\frac{-2}{72\pi}=-0.00884194128288\approx -0.0088[/tex]
6. Raw Data: 3,5,7,4,3, 8, 6, 6,9, 6,7,8,9,3, 3, 9 There are 16 data items Find: a) Mean b) Median c) Midrange d) Mode 7. Find the standard deviation of the data in question 6
Answer:
6.a. Mean= 6
b. Median=6
c Midrange=6
d.Mode=4
7.Standard deviation=2.2079
Step-by-step explanation:
Given data
3,3,3,3,4,5,6,6,6,7,7,8,8,9,9,9
Total data items,n=16
Sum o data items=96
a. Mean=[tex]\frac{sum\;of\;data\;items}{total\;data\;items}[/tex]
Mean=[tex]\frac{96}{16}[/tex]
Mean=6
b.If total number of items are even then
Median=[tex]\frac{\frac{n}{2}^{th}\;observation+\left(\frac{n}{2}+1\right)^{th}}{2}[/tex]
Median=[tex]\frac{\frac{16}{2}^{th} observation+\left(\frac{16}{2}+1\right)^{th} observation}{2}[/tex]
Median=[tex]\frac{8^{th} observation+9^{th} observation}{2}[/tex]
Median= [tex]\frac{6+6}{2}[/tex]
Median= [tex]\frac{12}{2}[/tex]
Median=6
c. Midrange=[tex]\frac{lower\;value+highest\;value}{2}[/tex]
Lower data item=3
Highest data item=9
Midrange= [tex]\frac{3+9}{2}[/tex]
Midrange= 6
d.Mode : It is defines as a number that appear most often in a set of numbers.
Mode=3
7. Mean[tex]\bar x=6[/tex]
[tex]\mid x-\bar x\mid[/tex] [tex]{\mid x-\bar x\mid}^2[/tex]
3 9
3 9
3 9
3 9
2 4
1 1
0 0
0 0
0 0
1 1
1 1
2 4
2 4
3 9
3 9
3 9
[tex]\sum{\mid x-\bar x\mid}^2=78[/tex]
n=16
Standard deviation=[tex]\sqrt{\frac{\sum{\mid x-\bar x}^2}{n}}[/tex]
Standard devaition=[tex]\sqrt{\frac{78}{16}}[/tex]
Standard deviation=[tex]\sqrt{4.875}[/tex]
Standard deviation of data =2.2079
1) For the following problem: Let f={(-2,3),(-1,1),(0,0),(1,-1), (2,-3)} and let g-{-3,1),(-1,-2), (0, 2),(2, 2),(3,1)}.Find the following a) f(1) and g-1) b) (gof (1) c) (gofof)(-1) d) (fog)(3)
Answer:
1. f(1)=-1 and g(-1)=-2.
2. (gof)(1)=-2.
3. (gofof)(-1)=-2
4. (fog)(3)=-1
Step-by-step explanation:
The given functions are defined as
f={(-2,3),(-1,1),(0,0),(1,-1), (2,-3)}
g={(-3,1),(-1,-2), (0, 2),(2, 2),(3,1)}
1.
The value of function f at x=1 is -1, So, f(1)=-1.
The value of function g at x=-1 is -2, So, g(-1)=-2.
Therefore the value of f(1) is -1 and the value of g(-1) is -2.
2.
[tex](g\circ f)(1)=g(f(1))[/tex] [tex][\because (g\circ f)(x)=g(f(x))][/tex]
[tex](g\circ f)(1)=g(-1)[/tex] [tex][\because f(1)=-1][/tex]
[tex](g\circ f)(1)=-2[/tex] [tex][\because g(-1)=-2][/tex]
Therefore the value of (gof)(1) is -2.
3.
[tex](g\circ f\circ f)(-1)=(g\circ f)(f(-1))[/tex] [tex][\because (g\circ f)(x)=g(f(x))][/tex]
[tex](g\circ f\circ f)(-1)=(g\circ f)(1)[/tex] [tex][\because f(-1)=1][/tex]
[tex](g\circ f\circ f)(-1)=-2[/tex] [tex][\because \text{From part 2}, (g\circ f)(1)=-2][/tex]
Therefore the value of (gofof)(1) is -2.
4.
[tex](f\circ g)(3)=f(g(3))[/tex] [tex][\because (f\circ g)(x)=f(g(x))][/tex]
[tex](f\circ g)(3)=f(1)[/tex] [tex][\because g(3)=1][/tex]
[tex](f\circ g)(3)=-1[/tex] [tex][\because f(1)=-1][/tex]
Therefore the value of (fog)(3) is -1.
Suppose a basketball player has made 282 out of 393 free throws. If the player makes the next 2 free throws, I will pay you $6. Otherwise you pay me $8.Step 1 of 2 :Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
Answer:
-$ 0.79
Step-by-step explanation:
Since, the player has made 282 out of 393 free throws,
So, the probability of a free throw = [tex]\frac{282}{393}[/tex],
Thus, the probability of 2 free throws = [tex]\frac{282}{393}\times \frac{282}{393}=\frac{8836}{17161}[/tex]
And, the probability of not getting 2 free throws = [tex]1-\frac{8836}{17161}=\frac{8325}{17161}[/tex]
Given, the price of winning ( getting 2 free throws ) is $6 while the price of losing ( not getting 2 free throws ) is - $ 8 ( ∵ there is a loss of $ 8 ),
Hence, the expected value of the proposition = probability of winning × winning value + probability of losing × losing value
[tex]= \frac{8836}{17161}\times 6 + \frac{8325}{17161}\times -8[/tex]
[tex]=\frac{53016}{17161}-\frac{66600}{17161}[/tex]
[tex]=-\frac{13584}{17161}[/tex]
[tex]=-\$ 0.79156226327[/tex]
[tex]\approx -\$ 0.79[/tex]
Mandy has an IQ of 115. We know that the mean () IQ is 100 with a standard deviation of 15. There are 100 people in Mandy’s Alcoholics Anonymous meeting. Taken at random, how many members are smarter than Mandy
Answer: 16
Step-by-step explanation:
Given : Mean : [tex]\mu=100[/tex]
Standard deviation : [tex]\sigma =15[/tex]
The value of z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 115
[tex]z=\dfrac{115-100}{15}=1[/tex]
The p-value : [tex]P(z>1)=1-P(z<1)=1- 0.8413447=0.1586553[/tex]
Now, the number of people smarter than Many is given by :_
[tex]100\times0.1586553=15.86553\approx16[/tex]
Hence, there are 16 members smarter than Mandy.
When Bill makes a sandwich, he may choose from among 3 kinds of rolls, 4 varieties of meat, and 2 types of sliced cheese. If he chooses one roll, one meat, and one type of cheese, how many different kinds of sandwiches can he make?
Answer: Bill can make 24 different kinds of sandwiches
Step-by-step explanation:
Given : The number of kinds of rolls = 3
The number of varieties of meat = 4
The number of types of sliced cheese = 2
If he chooses one roll, one meat, and one type of cheese, then the number of different kinds of sandwiches he can make is given by :-
[tex]3\times2\times4=24[/tex]
Hence, Bill can make 24 different kinds of sandwiches.
Write 61 using Egyptian and Babylonian numbers.
Answer:
61 in Egyptian numeral is ∩∩∩∩∩∩l
61 is written as - т т
Step-by-step explanation:
Egyptian numeral:
∩ mean 10
l = 1
for 60 we can use 6 number of ∩∩∩∩∩
i.e.
therefore 61 in Egyptian numeral is ∩∩∩∩∩∩l
Babylonian numeral : Basically Babylonian number system is 60 based instead of 10.
there are total number of 59 numerals made up by two symbol only.
61 is written as - т т with a space between two symbol
solve y' -x^2y = 0 using power series and write the first four terms of the power series
We're looking for a solution of the form
[tex]y=\displaystyle\sum_{n\ge0}a_nx^n[/tex]
with derivative
[tex]y'=\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^n[/tex]
Note that [tex]x=0\implies y(0)=a_0[/tex].
Substituting into the ODE gives
[tex]\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^n-\sum_{n\ge0}a_nx^{n+2}=0[/tex]
The first series starts with a constant term, while the second starts with a quadratic term, so we should pull out the first two terms of the first series and have it start at [tex]n=2[/tex], then shift the index on the second series to achieve the same effect, which allows us to condense the left side as
[tex]a_1+2a_2x+\displaystyle\sum_{n\ge2}\bigg((n+1)a_{n+1}-a_{n-2}\bigg)x^n=0[/tex]
so that the series solution's coefficients are given according to the recurrence
[tex]\begin{cases}a_0=a_0\\a_1=a_2=0\\(n+1)a_{n+1}-a_{n-2}=0&\text{for }n\ge2\end{cases}[/tex]
We can simplify the latter equation somewhat to get it in terms of [tex]a_n[/tex]:
[tex]a_n=\dfrac{a_{n-3}}n\text{ for }n\ge3[/tex]
This shows dependency between coefficients that are 3 indices apart, so we check 3 cases:
If [tex]n=3k+1[/tex], where [tex]k\ge0[/tex] is an integer, then[tex]k=0\implies n=1\implies a_1=0[/tex]
[tex]k=1\implies n=4\implies a_4=\dfrac{a_1}4=0[/tex]
and so on for all such [tex]n[/tex], giving
[tex]a_{3k+1}=0[/tex]
If [tex]n=3k+2[/tex], then[tex]k=0\implies n=2\implies a_2=0[/tex]
and we get the same conclusion as before,
[tex]a_{3k+2}=0[/tex]
If [tex]n=3k[/tex], then[tex]k=0\implies n=0\impiles a_0=a_0[/tex]
[tex]k=1\implies n=3\implies a_3=\dfrac{a_0}3[/tex]
[tex]k=2\implies n=6\implies a_6=\dfrac{a_3}6=\dfrac{a_0}{3\cdot6}=\dfrac{a_0}{3^2(2\cdot1)}[/tex]
[tex]k=3\implies n=9\implies a_9=\dfrac{a_6}9=\dfrac{a_0}{3^3(3\cdot2\cdot1)}a_0[/tex]
and so on, with the general pattern
[tex]a_{3k}=\dfrac{a_0}{3^kk!}[/tex]
Then the series solution is
[tex]y=\displaystyle\sum_{k\ge0}\bigg(a_{3k}x^{3k}+a_{3k+1}x^{3k+1}+a_{3k+2}x^{3k+2}\bigg)[/tex]
[tex]y=\displaystyle a_0\sum_{k\ge0}\frac{x^{3k}}{3^kk!}[/tex]
[tex]y=\displaystyle a_0\sum_{k\ge0}\frac{\left(\frac{x^3}3\right)^k}{k!}[/tex]
whose first four terms are
[tex]\boxed{a_0\left(1+\dfrac{x^3}3+\dfrac{x^6}{18}+\dfrac{x^9}{162}\right)}[/tex]