Answer:
[tex]\displaystyle \iint_S {\text{curl \bold{F}} \cdot} \, dS = \boxed{\bold{0}}[/tex]
General Formulas and Concepts:
Calculus
Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Integration Methods: U-Substitution + U-Solve
Multivariable Calculus
Partial Derivatives
Triple Integrals
Cylindrical Coordinate Conversions:
[tex]\displaystyle x = r \cos \theta[/tex][tex]\displaystyle y = r \sin \theta[/tex][tex]\displaystyle z = z[/tex][tex]\displaystyle r^2 = x^2 + y^2[/tex][tex]\displaystyle \tan \theta = \frac{y}{x}[/tex]Integral Conversion [Cylindrical Coordinates]:
[tex]\displaystyle \iiint_T {f(r, \theta, z)} \, dV = \iiint_T {f(r, \theta, z)r} \, dz \, dr \, d\theta[/tex]
Vector Calculus
Surface Area Differential:
[tex]\displaystyle dS = \textbf{n} \cdot d\sigma[/tex]
Del (Operator):
[tex]\displaystyle \nabla = \hat{\i} \frac{\partial}{\partial x} + \hat{\j} \frac{\partial}{\partial y} + \hat{\text{k}} \frac{\partial}{\partial z}[/tex]
[tex]\displaystyle \text{div \bf{F}} = \nabla \cdot \textbf{F}[/tex][tex]\displaystyle \text{curl \bf{F}} = \nabla \times \textbf{F}[/tex]Stokes’ Theorem:
[tex]\displaystyle \oint_C {\textbf{F} \cdot } \, d\textbf{r} = \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma[/tex]
Divergence Theorem:
[tex]\displaystyle \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma = \iiint_D {\nabla \cdot \textbf{F}} \, dV[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle \textbf{F} (x, y, z) = x^2z^2 \hat{\i} + y^2z^2 \hat{\j} + xyz \hat{\text{k}}[/tex]
[tex]\displaystyle \text{Region:} \left \{ {{\text{Paraboloid:} \ z = x^2 + y^2} \atop {\text{Cylinder:} \ x^2 + y^2 = 16}} \right[/tex]
Step 2: Integrate Pt. 1
Find div F:Step 3: Integrate Pt. 2
Convert region from rectangular coordinates to cylindrical coordinates.
[tex]\displaystyle \text{Region:} \left \{ {{\text{Paraboloid:} \ z = x^2 + y^2} \atop {\text{Cylinder:} \ x^2 + y^2 = 16}} \right \rightarrow \left \{ {{\text{Paraboloid:} \ z = r^2} \atop {\text{Cylinder:} \ r^2 = 16}} \right[/tex]
Identifying limits, we have the bounds:
[tex]\displaystyle \left\{ \begin{array}{ccc} 0 \leq z \leq r^2 \\ 0 \leq r \leq 4 \\ 0 \leq \theta \leq 2 \pi \end{array}[/tex]
Step 4: Integrate Pt. 3
[Integral] Substitute in variables and region:We evaluate the Stokes' Divergence Theorem Integral using basic integration techniques listed under "Calculus".
[tex]\displaystyle \begin{aligned}\iint_S {\text{curl } \textbf{F} \cdot} \, dS & = \int\limits^{2 \pi}_0 \int\limits^4_0 \int\limits^{r^2}_0 {r \bigg( 2z^2r \cos \theta + 2z^2r \sin \theta +r^2 \cos \theta \sin \theta \bigg)} \, dz \, dr \, d\theta \\& = \frac{1}{3} \int\limits^{2 \pi}_0 \int\limits^4_0 {zr^2 \bigg[ 2z^2 \big( \cos \theta + \sin \theta \big) + 3r \sin \theta \cos \theta \bigg] \bigg| \limits^{z = r^2}_{z = 0}} \, dr \, d\theta \\\end{aligned}[/tex]
[tex]\displaystyle \begin{aligned}\iint_S {\text{curl } \textbf{F} \cdot} \, dS & = \frac{1}{3} \int\limits^{2 \pi}_0 \int\limits^4_0 {r^5 \bigg[ 2r^3 \big( \cos \theta + \sin \theta \big) + 3 \sin \theta \cos \theta \bigg]} \, dr \, d\theta \\& = \frac{1}{54} \int\limits^{2 \pi}_0 {r^6 \bigg[ 4r^3 \big( \cos \theta + \sin \theta \big) + 9 \sin \theta \cos \theta \bigg] \bigg| \limits^{r = 4}_{r = 0}} \, d\theta \\\end{aligned}[/tex]
[tex]\displaystyle \begin{aligned}\iint_S {\text{curl } \textbf{F} \cdot} \, dS & = \frac{2048}{27} \int\limits^{2 \pi}_0 {\cos \theta \Big( 9 \sin \theta + 256 \Big) + 256 \sin \theta} \, d\theta \\& = \frac{-1024}{243} \bigg[ 4608 \cos \theta - \bigg( 9 \sin \theta + 256 \bigg)^2 \bigg] \bigg| \limits^{\theta = 2 \pi}_{\theta = 0} \\& = \boxed{\bold{0}}\end{aligned}[/tex]
∴ we have calculated the Stokes' Theorem integral with the given region and function using the Divergence Theorem.
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Learn more about multivariable Calculus: https://brainly.com/question/13933633
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Topic: Multivariable Calculus
The Stokes' theorem is applied to convert a surface integral of a curl of a vector into a line integral. This is done by identifying the curl of the given vector field F and setting up the limits of the integral based on given bounds. The integral is then evaluated.
Explanation:Stokes' theorem is used in vector calculus to simplify certain types of surface integrals. It transforms a surface integral of a curl of a vector field into a line integral. F(x, y, z) = x2z2i + y2z2j + xyzk, here, is the given vector field. The surface S is the part of the paraboloid that lies within the cylinder x² + y² = 16. The theorem is used to evaluate the integral S curl F · dS, by treating the surface integral as a line integral. The line integral can be easier to evaluate. The exact process involves identifying the curl of F, setting up the bounds of the integral based on the restrictions given, and then computing the integral.
Learn more about Stokes' Theorem here:https://brainly.com/question/10773892
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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:
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:
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.
Esteban has 236 trading cards that he wants to put in storage boxes.Each box holds 18 trading cards. If he puts 18 trading cards in every box, how many trading cards will be left over
Divide the total cards by 18:
236 / 18 = 13.111
Use the whole number and multiply by 18:
13 x 18 = 234
This means 234 cards would be in storage boxes.
236 - 234 = 2 cards would be left over.
A knitter wants to make a rug for a dollhouse. The length of the rug will be 2 inches more than it's width. The total area of the rug (in square inches ) based on the width W of the rug (in inches ) is given by A (w) = w (2+w) If the desired area of the rug is 15 square inches, what is the width of the rug, in inches ?
Answer:
Step-by-step explanation:
A knitter wants to make a rug for a dollhouse. The length of the rug will be 2 inches more than it's width. Let the width of the rug be represented by w. This means that the length of the rug will be w + 2
The total area of the rug (in square inches ) based on the width W of the rug (in inches ) is given by
A(w) = w (2+w)
If the desired area of the rug is 15 square inches,,the width will be determined by substituting 15 for A(w). It becomes
15 = w^2 + 2w
w^2 + 2w - 15 = 0
w^2 + 5w - 3w - 15 = 0
w(w+5) -3(w+5)
w - 3 = 0 or w+ 5 = 0
w = 3 or w = -5
w cannot be negative. So w = 3 inches
The distance that a spring will stretch varies directly as the force applied to the spring. A force of 8080 pounds is needed to stretch a spring 88 inches. What force is required to stretch the spring 1919 inches? A nothing-pound force is required to stretch the spring 1919 inches.
Answer:
176 199 pounds
Step-by-step explanation:
To answer this it is first useful to find the proportionality constant.
F= kx
8080 pounds = k.88 in
k = 91.81 pound/inch
So what force is required to stretch the spring to 1919 inches?
F = kx
= 91.81 pound/inch * 1919 inches
= 176 199 pounds
It is worth noting that this force seems rather large and the spring might have long reached its elastic limit
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
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 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.
A bowl in the shape of a hemispere is filled with water to a depth h=3 inches. The radius of the bowl is R inches. Express the radius of the bowl R as a function of the angle theta.
The radius of the bowl R as a function of the angle theta is [tex]\mathrm{R}=\frac{3}{1-\sin \theta}[/tex]
Solution:
The figure is attached below
If we consider the centre of hemisphere be A
The radius be AC and AD
According to question,
A bowl in the shape of a hemispere is filled with water to a depth h=3 inches .i.e. BC = h = 3 inches
And radius of the bowl is R inches .i.e. R = AD =AC
Now , using trigonometric identities in triangle ABD we get
[tex]\sin \theta=\frac{\text { Perpendicular }}{\text { Hypotenuse }}=\frac{A B}{A D}[/tex]
[tex]\begin{array}{l}{\sin \theta=\frac{A B}{R}} \\\\ {A B=R \sin \theta}\end{array}[/tex]
Since , AC = AB + BC
R = R Sinθ + 3
R - R Sinθ = 3
R (1 – Sinθ ) = 3
[tex]\mathrm{R}=\frac{3}{1-\sin \theta}[/tex]
Which is the required expression for the radius of the bowl R as a function of the angle theta
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.
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]
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
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
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:
Step-by-stejfnqe ruhrhe ye\
Consider the sequence:
5, 7, 11, 19, 35,....
Write an explicit definition that defines the sequence:
a_n = 2n + 3
a_n = 3n + 2
a_n = 3n^2
a_n = 2^n + 3
Answer:
a_n = 2^n + 3
Step-by-step explanation:
The first differences have a geometric progression, so the explicit definition will be an exponential function. (It cannot be modeled by a linear or quadratic function.) The above answer is the only choice that is an exponential function.
__
First differences are ...
(7-5=)2, 4, 8, 16
Answer: [tex]a_n = 2^n + 3\ \ \ \, n=1,2,3,4,5...[/tex]
Step-by-step explanation:
The given sequence = 5, 7, 11, 19, 35,....
[tex]7-5=2\\11-7=4=2^2\\19-11=8=2^3\\35-19=16=2^4[/tex]
Here , it cam be observe that the difference between the terms is not common but can be expressed as power of 2.
We can write the terms of the sequence as
[tex]2^1+3=5\\2^2+3=4+3=7\\2^3+3=8+3=11\\2^4+3=16+3=19\\2^5+3=32+3=35[/tex]
Then , the required explicit definition that defines the sequence will be
[tex]a_n = 2^n + 3\ \ \ \, n=1,2,3,4,5...[/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.
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)
Expand the expression using the Binomial Theorem and Pascal's Triangle: (3x-1)^3
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!
−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]
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
The sum of 5 times the width of a rectangle and twice its length is 26 units. The difference of 15 times the width and three times the length is 6 units. Write and solve a system of equations to find the length and width of the rectangle
Answer: length = 8 units
Width = 2 units
Step-by-step explanation:
Let the length of the rectangle be represented by L
Let the width of the rectangle be represented by W
The sum of 5 times the width of a rectangle and twice its length is 26 units. This means that
5W + 2L = 26 - - - - - - - - 1
The difference of 15 times the width and three times the length is 6 units. It means that
15W - 3L = 6 - - - - - - - - 2
The system equations are equation 1 and equation 2.
Multiplying equation 1 by 3 and equation 2 by 2, it becomes
15W + 6L = 78
30W - 6L = 12
Adding both equations,
45W = 90
W = 90/45 = 2
15W - 3L = 6
15×2 - 3L = 6
-3L = 6 - 30 = - 24
L = - 24/ -3 = 8
The length of the rectangle is 8 units, and the width is 2 units. This solution satisfies both equations given.
Let's denote the width of the rectangle as ( w ) units and the length as ( l ) units.
According to the given information, we have two equations:
1. ( 5w + 2l = 26 ) (sum of 5 times the width and twice the length is 26 units)
2. ( 15w - 3l = 6 ) (difference of 15 times the width and three times the length is 6 units)
We can set up a system of equations using these two equations.
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method.
First, we'll multiply the second equation by 2 to eliminate the ( l ) term:
( 2 x (15w - 3l) = 2 x 6 )
This gives us:
( 30w - 6l = 12 )
Now, we have the system of equations:
1. ( 5w + 2l = 26 )
2. ( 30w - 6l = 12 )
We can now eliminate the ( l ) term by adding these equations together.
( (5w + 2l) + (30w - 6l) = 26 + 12 )
This simplifies to:
( 35w - 4l = 38 )
Now, we have one equation with only ( w ) as a variable.
Now, we can solve for ( w ):
( 35w - 4l = 38 )
( 35w = 38 + 4l )
[tex]\( w = \frac{38 + 4l}{35} \)[/tex]
Now, we can substitute this value of ( w ) into one of the original equations to solve for ( l ).
Let's use the first equation:
( 5w + 2l = 26 )
Substituting[tex]\( w = \frac{38 + 4l}{35} \)[/tex], we get:
[tex]\( 5\left(\frac{38 + 4l}{35}\right) + 2l = 26 \)[/tex]
Now, we can solve this equation for ( l ).
[tex]\( \frac{190 + 20l}{35} + 2l = 26 \)[/tex]
Multiply both sides by 35 to clear the fraction:
( 190 + 20l + 70l = 910 )
Combine like terms:
( 90l + 190 = 910 )
Subtract 190 from both sides:
( 90l = 720 )
Now, divide both sides by 90 to solve for ( l ):
[tex]\( l = \frac{720}{90} \)[/tex]
( l = 8 )
Now that we have the value of ( l ), we can substitute it back into one of the original equations to solve for ( w ). Let's use the first equation:
( 5w + 2(8) = 26 )
( 5w + 16 = 26 )
Subtract 16 from both sides:
5w = 10
Now, divide both sides by 5 to solve for ( w ):
[tex]\( w = \frac{10}{5} \)[/tex]
w = 2
So, the length of the rectangle is 8 units and the width is 2 units.
Act scene where Macbeth and last Macbeth plan to kill king Duncan
Answer:
Step-by-step explanation:
inside the castle
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.
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.
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.
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]
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.
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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