Evaluate the given integral by making an appropriate change of variables, where r is the rectangle enclosed by the lines x - y = 0, x - y = 7, x + y = 0, and x + y = 6.
A team of 10 players is to be selected from a class of 6 girls and 7 boys. Match each scenario to its probability. You have to drag the sentences to match probabilities.
Possible probabilities are:
0.07
0.12
0.44
0.49
Options for answers are:
The probability that a randomly chosen team includes all 6 girls in the class.
The probability that a randomly chosen team has 3 girls and 7 boys.
The probability that a randomly chosen team has either 4 or 6 boys.
The probability that a randomly chosen team has 5 girls and 5 boys.
Answer:
0.07 -The probability that a randomly chosen team has 3 girls and 7 boys.
0.12 -The probability that a randomly chosen team includes all 6 girls in the class.
0.44 -The probability that a randomly chosen team has 5 girls and 5 boys.
0.49-The probability that a randomly chosen team has either 4 or 6 boys.
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When a team of 10 players is to be selected from a class of 6 girls and 7 boys the probability of each scenario occurring is:
0.07 = The probability that a randomly chosen team has 3 girls and 7 boys.
0.12 = The probability that a randomly chosen team includes all 6 girls in the class.
0.44 = The probability that a randomly chosen team has 5 girls and 5 boys.
0.49 = The probability that a randomly chosen team has either 4 or 6 boys.
What is probability?
In simple words, it refers to a numerical guess of how likely an event is to occur. Hence, when it comes to probability we believe there is no guarantee the event would occur.
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How many times greater is 700 than 70
Answer:
10 times greater
Step-by-step explanation:
700÷70=10
Use cylindrical coordinates. find the volume of the solid that is enclosed by the cone z = x2 + y2 and the sphere x2 + y2 + z2 = 72.
Step 1: The intersection curve is only at the origin.
Step 2: Set up the volume integral over the sphere region.
Step 3:Evaluate the integral to find the volume: [tex]\(V = 216\pi\sqrt{2}\).[/tex]
To find the volume of the solid enclosed by the cone [tex]\(z = x^2 + y^2\)[/tex] and the sphere[tex]\(x^2 + y^2 + z^2 = 72\),[/tex] we'll first find the intersection curve of the cone and the sphere in cylindrical coordinates, then set up the triple integral to find the volume.
Step 1: Finding the intersection curve:
The cone equation in cylindrical coordinates becomes [tex]\(z = r^2\)[/tex] and the sphere equation remains the same as [tex]\(r^2 + z^2 = 72\).[/tex]
To find the intersection, we set these two equations equal to each other:
[tex]\[r^2 = r^2 + z^2\][/tex]
Substitute [tex]\(z = r^2\)[/tex] from the cone equation:
[tex]\[r^2 = r^2 + (r^2)^2\][/tex]
[tex]\[r^2 = r^2 + r^4\][/tex]
[tex]\[r^4 = 0\][/tex]
From this, we see that the only solution is (r = 0), which corresponds to the point at the origin.
Step 2: Setting up the integral for volume:
We'll integrate over the region where the cone lies within the sphere, which is the entire sphere. In cylindrical coordinates, the limits of integration are [tex]\(0 \leq r \leq 6\)[/tex] (since the sphere has radius [tex]\(\sqrt{72} = 6\)) and \(0 \leq \theta \leq 2\pi\).[/tex]
The limits for \(z\) are from the cone to the sphere, so it's from [tex]\(r^2\) to \(\sqrt{72-r^2}\).[/tex]
Thus, the volume integral is:
[tex]\[V = \iiint_{E} r \, dz \, dr \, d\theta\][/tex]
Where (E) is the region enclosed by the cone and the sphere.
Step 3: Evaluate the integral:
[tex]\[V = \int_{0}^{2\pi} \int_{0}^{6} \int_{r^2}^{\sqrt{72-r^2}} r \, dz \, dr \, d\theta\][/tex]
Let's evaluate this integral step by step:
[tex]\[V = \int_{0}^{2\pi} \int_{0}^{6} (r\sqrt{72-r^2} - r^3) \, dr \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[-\frac{1}{4}(72-r^2)^{3/2} - \frac{1}{4}r^4\right]_{0}^{6} \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[-\frac{1}{4}(0)^{3/2} - \frac{1}{4}(6^4) - \left(-\frac{1}{4}(72)^{3/2} - \frac{1}{4}(0)^4\right)\right] \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[\frac{1}{4}(72)^{3/2}\right] \, d\theta\][/tex]
[tex]\[V = \frac{1}{4}(72)^{3/2} \int_{0}^{2\pi} 1 \, d\theta\][/tex]
[tex]\[V = \frac{1}{4}(72)^{3/2} \cdot 2\pi\][/tex]
[tex]\[V = 36\pi \sqrt{72}\][/tex]
[tex]\[V = 36\pi \times 6\sqrt{2}\][/tex]
[tex]\[V = 216\pi\sqrt{2}\][/tex]
So, the volume of the solid enclosed by the cone and the sphere is [tex]\(216\pi\sqrt{2}\).[/tex]
PLEASE HELP ME!!! NOT GOOD AT MATH.
Let x be a random variable giving the number of aces in a random draw of 4 cards from an ordinary deck of 52 cards. construct a table showing the probability distribution of x
The provided table outlines the probability distribution for drawing varying numbers of aces in a random draw of 4 cards from a standard 52-card deck, using combinatorial probability calculations.
Explanation:To answer your question, we need to consider the different possibilities for pulling aces in a draw of four cards. There are 4 aces in a standard deck of 52 cards, and here's a table showing the probability distribution for each outcome:
x = 0 - this represents no aces drawn. The number of ways to choose no aces from 4 aces and 4 non-aces from 48 non-aces is (4 choose 0)*(48 choose 4). So the probability P(X=0) = (comb(4, 0)*comb(48, 4))/comb(52,4)x = 1 - one ace is drawn. The probability P(X=1) = (comb(4, 1)*comb(48, 3))/comb(52,4).x = 2 - two aces are drawn. The probability P(X=2) = (comb(4, 2)*comb(48, 2))/comb(52,4).x = 3 - three aces are drawn. The probability P(X=3) = (comb(4, 3)*comb(48, 1))/comb(52,4).x = 4 - all four aces are drawn, and the probability P(X=4) = (comb(4, 4)*comb(48, 0))/comb(52,4).Please note that 'comb(a, b)' in this case represents 'a choose b', which is a way to compute combinations in probability.
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in the problem 10-4=6, whats the correct term for the number 4
What is the probability that a randomly selected person will have a birthday in march? assume that this person was not born in a leap year. express your answer as a simplified fraction or a decimal rounded to four decimal places?
there are 31 days in August
365 days in a year
31/365 = 0.0849 probability
Answer:
0.0849
Step-by-step explanation:
Let's consider the event: a birthday takes place in March. The probability (P) of such event is:
[tex]P = \frac{favorable\ cases }{possible\ cases}[/tex]
The favorable cases are 31 because March has 31 days.
The possible cases are 365 because there are 365 days in a year.
The probability of a birthday being in March is:
[tex]P=\frac{31}{365} =0.0849[/tex]
What is the least common multiple of 2, 10 and 6
The least common multiple of 2, 10 and 6 by using the definition of multiple is 30.
The lowest possible number that can be divisible by all the given numbers is called as least common multiple (LCM). It is the smallest multiple which is common in all the numbers.
The least common multiple of 2, 10 and 6 can be calculated as:
The multiples of 2 are: 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34.........
The multiples of 10 are: 10,20,30,40,50,60,70.............
The multiples of 6 are: 6,12,18,24,39,36,42,48,54,60,66...........
The lowest common multiple among 2,10,6 is 30.
Thus, the least common multiple of 2, 10 and 6 is 30.
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How do you work out a percentile rank of a score of 57
What is the future value of $20 a week for 10 years at 6 percent interest? assume the first payment occurs at the end of this week?
Juan will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of
$55
and costs an additional
$0.30
per mile driven. The second plan has no initial fee but costs
$0.80
per mile driven.
For what amount of driving do the two plans cost the same?
What is the cost when the two plans cost the same?
Let x and y represent two real numbers. Write an algebraic expression to denote the quotient obtained when the product of the two numbers is divided by their sum.
An algebraic expression to denote the quotient obtained when the product of the two numbers is divided by their sum is (xy)/(x+y).
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that let x and y represent two real numbers. The expression to denote the quotient obtained when the product of the two numbers is divided by their sum is written as,
E = (xy)/(x+y).
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What is the value of x to the nearest tenth?
The quotient of 8.4x10^9 and a number n results in 5.6 x 10^27 What is the value of n?
Answer:
[tex]1.5\times 10^{-18}[/tex]
Step-by-step explanation:
We have been given that the quotient of [tex]8.4\times 10^9[/tex] and a number n results in [tex]5.6\times 10^{27}[/tex]
To find the value of n we will write our given information in an equation as:
[tex]\frac{8.4\times 10^9}{n}=5.6\times 10^{27}[/tex]
[tex]\frac{8.4\times 10^9}{5.6\times 10^{27}}=n[/tex]
Using quotient rule of exponents [tex]\frac{a^m}{a^n}=a^{m-n}[/tex] we will get,
[tex]\frac{8.4\times 10^{9-27}}{5.6}=n[/tex]
[tex]\frac{8.4\times 10^{-18}}{5.6}=n[/tex]
[tex]1.5\times 10^{-18}=n[/tex]
Therefore, the value of n is [tex]1.5\times 10^{-18}[/tex].
A new car dealer offered to sell her demonstrator model for 90% of the retail price. If the sale price was $9,750, what was the retail price?
90% = 9,750
9750/90 = 1%
1% = 108.3
100% = 10,833.3
original retail price = $10,833.30
EASY 5 POINTS!!! This box has the same length, width, and height as the 2 basketballs. Which expression gives the volume of the empty space encircling the basketballs inside the box?
the upper left on TTM Trust
Use cylindrical coordinates. find the volume of the solid that lies between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z2 = 2
To find the volume of the solid that lies between the paraboloid z = x^2 + y^2 and the sphere x^2 + y^2 + z^2 = 2 using cylindrical coordinates, set up the integral for the volume by rewriting the sphere equation in cylindrical coordinates, determining the limits for r and z, and evaluating the integral.
Explanation:To find the volume of the solid that lies between the paraboloid z = x^2 + y^2 and the sphere x^2 + y^2 + z^2 = 2
We can rewrite the sphere equation in cylindrical coordinates as r^2 + z^2 = 2. The limits for r are from 0 to √(2-z^2), and for z, they are from 0 to √(2-r^2).
The volume can be found by integrating the constant 1 over the limits of r and z: V = ∭1 dz dr dθ. Evaluate this integral to find the volume of the solid.
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Find the x and y intercepts of 10x-15y=30
If each dimension of the rectangular prism is doubled, how will its total surface area change?
mc006-1.jpg
[Not drawn to scale]
The surface area doubles.
The surface area triples.
The surface area increases by a factor of four.
The surface area increases by a factor of eight.
Answer:
inceases by factor four
Step-by-step explanation:
A car rents for $180 per week plus $0.75 per mile. Find the rental cost for a two-week trip of 500 miles for a group of three people.
Answer:
$735
Step-by-step explanation:
$180/week * 2 weeks = $360 for 2 weeks
$0.75/mile * 500 miles = $375
$360 + $375 = $735 for 2 week trip of 50 miles
A photograph has a length that is 6 inches longer than its width, x. So its area is given by the expression x(x+6) square inches. If the area of the photograph is 112 square inches, what is the width of the photograph?
The width of the photograph is 8 inches
let
width of the photograph = x
length of the photograph = 6 + x
Area = x(6 + x)
112 = x(6 + x)
112 = 6x + x²
x² + 6x - 112 = 0
x² - 8x + 14x - 112 = 0
x(x - 8) + 14(x - 8) = 0
(x + 14)(x - 8)
x = -14 or 8
x = 8
x cannot be negative so we use only 8.
width of the photograph = 8 inches
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If Ying Zyiyu covers the first 200 miles of a trip going 50mph. and the last hundred going 40mph, what was his average speed for the entire trip?
Answer:
46.15
Step-by-step explanation:
I do RSM :)
March 21, the 80th day of the year, is the spring equinox. find the number of hours of daylight in fairbanks on this day
A ball of radius 0.200 m rolls with a constant linear speed of 3.00 m/s along a horizontal table. the ball rolls off the edge and falls a vertical distance of 2.08 m before hitting the floor. what is the angular displacement of the ball while the ball is in the air
The angular displacement of the ball while it's in the air is 9.77 radians.
Here's how we can find it:
Time in the air: First, we need to find the time the ball spends in the air. We can use the following kinematic equation for vertical motion:
h = 1/2 * g * t^2
where:
h is the vertical distance (2.08 m)
g is the acceleration due to gravity (9.81 m/s²)
t is the time in the air
Solving for t, we get:
t = sqrt(2 * h / g) = sqrt(2 * 2.08 m / 9.81 m/s²) ≈ 1.43 s
Angular velocity: The angular velocity (ω) of the ball is related to its linear velocity (v) and radius (r) by the following equation:
ω = v / r
where:
v is the linear velocity (3.00 m/s)
r is the radius (0.200 m)
Plugging in the values, we get:
ω = 3.00 m/s / 0.200 m = 15 rad/s
Angular displacement: Finally, the angular displacement (θ) of the ball is the product of its angular velocity (ω) and the time (t) it spends in the air:
θ = ω * t = 15 rad/s * 1.43 s ≈ 21.45 rad ≈ 9.77 rad (rounded to three significant figures)
Therefore, the angular displacement of the ball while in the air is approximately 9.77 radians.
What is the slope of a line that is perpendicular to the line whose equation is 0.5x−5y=9
To find the slope of a line perpendicular to the line defined by the equation 0.5x−5y=9, first convert the equation into slope-intercept form to determine the slope of the original line. The slope of the line perpendicular to this is its negative reciprocal. For this specific problem, the slope of the line perpendicular to the given line is -10.
Explanation:Firstly, rearrange the given equation, 0.5x−5y=9, into the slope-intercept format, i.e. y=mx+b. Here, 'm' is the slope of the line. To illustrate this rearrangement: 5y = 0.5x - 9, then divide through by 5 to get y = 0.1x - 1.8. Thus, the slope of the given line is 0.1.
Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. In this case, the slope of the given line is 0.1, so the negative reciprocal (which represents the slope of the line perpendicular to this line) is -1/0.1, which equals -10. This is the slope of the line perpendicular to the given line.
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Solve the following system of equations.
7x -8y= -19
-2x +5y =0
x=
y=
Harry rolls a number cube what is the probability that he will roll an even number or a number greater than four
An artisan is creating a circular street mural for an art festival. The mural is going to be 50 feet wide. One sector of the mural spans 38 degrees. What is the area of the sector to the nearest square foot?
Children of age 3-6 years require 25% less sleep per day than infants, who need 16hours. How much slee do the 3-6 year olds need?