The collection of 4 in, 5 in, and 10in of three side lengths will not form a triangle.
What is a triangle?A triangle is a three-sided closed-plane figure formed by joining three noncolinear points.
Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.
We know, A triangle has a property that the sum of any two sides is always greater than the third side and the difference between any two sides is smaller than the third side.
From the given options if we observe the third option,
4 + 5 should be greater than 10 but it is not therefore, It is the correct option.
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1 Aiden paints an unassembled gift box to use for
his sister's birthday gift.
The base of the box measures 2 inches by5 inches. When assembled the box measures7 inches tall. If Aiden only paints the outside of the unassembled box, how many square inches will he paint?
A 64 in.²
B 70 in.²
C 118 in.²
D 126 in.²
Jade decided to rent movies for a movie marathon over the weekend. The
function g(x) represents the amount of money spent in dollars where x is the
number of movies. Does a possible solution of (6.5, $17.50) make sense for this
function? Explain your answer.
A solution (6.5, 17.50) does not make sense in the context of the problem, as the number of movies watched must be a non-negative integer.
How to interpret the ordered pair?The general format of an ordered pair is given as follows:
(x,y).
The meaning of an ordered pair (x,y) is that y = f(x), meaning that the numeric value of the function at the value of x is of y.
The variables for this problem are given as follows:
x: number of movies watched.g(x): amount of money spent.You can watch 1, 2, 3, ... and so on movies, a countable amount, hence 6.5 movies watched does not make sense in the context of this problem as decimal numbers are outside the function's domain.
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a cylinder with radius of 5ft has a volume of 314ft. Find its height
3.9979(depends on what place you need to round to
The equation is V=πr2h. First plug in what you know, 314=π5^2h. 5x5=25 and its multiplied with π. It would look like 314=25πh. Divide the 25π by both sides, leaving your h. Plug it into your calculator.
Check answer:
(I rounded the 4 to the nearest whole number) 4*25*π is approximately 314
Answer:
4 ft
---------------
Volume of cylinder formula:
V = πr²hGiven:
r = 5 ft,V = 314 ft³Find h:
314 = 3.14*5²h314 = 3.14*25hh = 314/(3.14*25)h = 4 ftA field in the shape of a rectangle has an area of 1/3 square mile. The length of the field is 2/5 mile. What is the width of the field?
If the area of a rectangle 1/3 square mile and length of the field is 2/5 mile then the width of the field is
What is Area of a Rectangle?
The quantity of unit squares that can fit inside a rectangle called its area. In other terms, the area of a rectangle is the area that it occupys. The flat surfaces of laptop monitors, blackboards, painting canvases, etc. are a few instances of rectangular shapes.
Given in the question,
Area of rectangle [tex]= \frac{1}{3} square mile[/tex]
The length of the field [tex]= \frac{2}{5} mile[/tex]
Let breadth be x.
As we know that,
Area of rectangle = [tex]length * breadth[/tex]
Now,
[tex]\frac{1}{3} = \frac{2}{5} *x[/tex]
[tex]x=\frac{1}{3} * \frac{5}{2}[/tex]
[tex]x=\frac{5}{6}[/tex]
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10. What is the length of
side a?
2
2.9.
36 cm
15cm
39 cm
a
Answer:
the answer is a=15cm
Step-by-step explanation:
c²=a²+b²
where c=Hypothenuse,a=adjacent, b=opposite
a²=c²-b²
a²=39²-36²
a²=1521-1296
a²=225
take Square root of both sides
a=15cm
trinomio cuadrado perfecto
[tex]\alpha^{3} -10\alpha ^{\frac{3}{2} } +25[/tex]
Which of the following statements about f(x)= -(x-4)² -1 are true? Select all that apply.
a) Vertex: (4, -1)
b) y-intercept: (0, -17)
c) x-intercept: (5, 0)
d) Axis of Symmetry: x = 4
e) x-intercept: (3, 0)
f) no x-intercepts
The following statements are true about the function f(x) = -( x-4) ² -1
1. The y intercept is (0,-17)
2. The axis of symmetry x = 4
3. The x intercept is (3,0)
What are quadratic function?A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.
The equation f(x) = -(x-4)² -1 can be expanded as
f(x) = -x²+8x-16-1 = -x²+8x-17
therefore when x = 0
the y-intercept = -17
therefore the y intercept = (0,-17)
The x intercept of the function,
when y = f(x) and y = 0
-(x-4)² = 1
-(x-4) = 1
-x+4 = 1
-x = -3
x = 3
the x intercept of the function is (3,0)
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Rich is buying a home for $244,800. He is making a 15% down payment and financing the rest with a 25-year loan at 5.25% interest. What will his monthly mortgage payment be?
The required, Rich's monthly mortgage payment will be $1,246.91.
what is a mortgage payment?A mortgage payment is the amount of money that a borrower pays to a lender in order to repay a home loan.
Here,
The amount that Rich is financing with the mortgage is the difference between the total cost of the home and the down payment:
Amount financed = $244,800 - 0.15($244,800) = $208,080
To calculate the monthly mortgage payment, we can use the formula for the present value of an annuity,
[tex]PMT = (PV * r) / (1 - (1 + r)^{-n}[/tex]
First, we need to calculate the monthly interest rate. Since the annual interest rate is 5.25%, the monthly interest rate is:
r = 0.0525 / 12 = 0.004375
Next, we need to calculate the total number of payments. Since Rich is taking out a 25-year loan, he will be making 25 x 12 = 300 monthly payments.
Now we can plug these values into the formula and solve for PMT:
PMT = ($208,080 × 0.004375) / (1 - (1 + 0.004375)^(-300))
= $1,246.91
Therefore, Rich's monthly mortgage payment will be $1,246.91.
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Explain how to calculate 5% of 260 by first calculating 1% of 260
The value of 5% of 260 is 13 as it is 5 times of 1% of 260 and the value of 1% of 260 is 2.6.
Percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.
so, according to the question
Let 1% of 2600 = X
1/100 * 260 = X
X = 2.6
5% is 5 times of 1% so we can just multiply the value we got after 1% of 260
i.e., 2.6
so 5% of 260 is 5 * 2.6
= 13.0
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Monica has 2. 2 litres of milk 1. 5kg of each other ingredient. Calculate the greatest number of people she can serve with pasta bake
It is not possible to determine the greatest number of people Monica can serve with pasta bake based solely on the information provided.
The amount of milk and other ingredients she has does not necessarily indicate how many portions she can make. Other factors such as the recipe she is using, the size of each serving, and the equipment she has available will also impact how many people she can serve. In order to determine the number of portions she can make, more information is needed about the recipe and the serving size she is aiming for.It is not possible to determine the greatest number of people Monica can serve with pasta bake based solely on the information provided.
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Reece works at fries and more and the relationship between x the number of hours and y reece's total earnings can be represented by y=8. 25x who earns more per hour
Another person named Sarah earns $7.50 per hour, then Reece earns more per hour than Sarah.
If the relationship between the number of hours worked and total earnings for Reece at Fries n More is given by y = 8.25x, where y is total earnings and x is the number of hours worked, then Reece earns $8.25 per hour worked.
To compare Reece's earnings per hour to another person, we would need to know the hourly wage of the other person. For example, if another person named John earns $9.00 per hour, then John earns more per hour than Reece. Conversely, if another person named Sarah earns $7.50 per hour, then Reece earns more per hour than Sarah.
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What is true about a system of two linear equations that has no solution?
Answer: The two lines never intersect
Step-by-step explanation:
An algebraic solution for a system of equations means that the graphs of the two equations intersect at some point. If no such point can be found, it means the graphs never intersect.
In one trip to the hardware store you buy 15 boards, 3 boxes of nails, and a saw. One a second trip you buy 9 more boards and a box of nails. Define and use variables to represent the total cost.
The variables representing the total cost of the items bought are x and y.
Where X is the total cost of the items bought during the first trip and Y is the total cost of the items bought during the second trip.
Items bought on the first trip:
Boards = 15
Boxes of nails = 3
Total cost = x
Items bought on the second trip:
Boards = 9
Boxes of nails = 1
Total cost = y
Variables:Let a piece of board = b
Let a box of nails = n
Let the total cost for the first trip = x
Let the total cost for the second trip = y
Equations:x = 15b + 3n
y = 9b + n
LCM of x^2 -25 and x-5
Answer:10
Step-by-step explanation:
5.5 equals 25.then you subtract 25 by 25 and get zero.
Jerome runs a clothing store. To mark their anniversary, they have a promotional offer for their customers. He has kept a box with five 10% discount coupons, eight 15% discount coupons, seven 20% discount coupons, six 25% discount coupons, and four 40% discount coupons. Every 30 minutes, one customer gets to draw a discount coupon from the box, use it on their purchase, and then place the coupon back into the box. Jerome predicts that when a customer draws a discount coupon from box without looking, the customer will draw a 10% discount coupon one-sixth of the time.
Yes, Jerome's prediction is true that customers will draw a 10% discount coupon one-sixth of the time.
What is probability?The possibility of the result of any random event is known as probability. This phrase refers to determining the likelihood that any given occurrence will occur.
Yes, Jeremore is correct.
It is given that a box has
Number of 10% discount coupons = 5
Number of 15% discount coupons = 8
Number of 20% discount coupons = 7
Number of 25% discount coupons = 6
Number of 40% discount coupons = 4.
The total coupons in the box = 5+8+7+6+4 = 30.
So, the probability that the customer will get a 10% coupon is
Number of 10% coupons/ Total coupons = 5/30= 1/6.
So, Jerome's prediction is true that customers will draw a 10% discount coupon one-sixth of the time.
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F.E.O.L. Perpendicular to a Given Line
Write the slope-intercept form of the equation of the line described.
1) through: (-2,-3), perp. to y=-x
5
3) through: (2,-2), perp. to y=-
2) through: (5,-2), perp. to y = 5x +4
4) through: (-4,-3), perp. to y=-x+2
The equation of the line are;
a. y = 5/2x + 2
b. y = -1/5x - 1
c. y = -5/2x - 3
d. y = -x - 7
What is the slope intercept form of the linea.
The line passes through point (-2, -3) and it is perpendicular to y = -2/5x
The equation of the line is y = 5/2x + 2
b.
The line passes through the point (5, -2) and it is perpendicular to y = 5x + 4
The equation of the line is y = -1/5x - 1
c.
The line passes through (-2, 2) and it is perpendicular to y = 2/5x - 4
Th equation of the line is y = -5/2x - 3
d.
The line passes through (-4, -3) and it is perpendicular to y = -x + 2
The equation of line is y = -x - 7
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a researcher randomly selects 2 fish from among 8 fish in a tank and puts each of the 2 selected fish into different containers. how many ways can this be done? a) 112 b) 28 c) 56 d) 168 e) 32
There are 28 ways to randomly select 2 fish from 8 fish and put each of the 2 selected fish into different containers. Hence, the accurate answer is option (b) 28.
To determine the number of ways to randomly select 2 fish from 8 fish and put each of the 2 selected fish into different containers, we will use the formula for combinations, which is given by -
nCk = n! / (k!(n-k)!)
where n is the total number of items, k is the number of items to be selected, and ! denotes the factorial function (the product of all positive integers up to a given number).
Here, we are given n = 8 fish and k = 2 fish.
Thus, the number of ways to select and place 2 fish from 8 fish will be -
8C2 = 8! / (2!(8-2)!) = (8 x 7) / (2 x 1) = 28
Therefore, there are 28 ways to randomly select 2 fish from 8 fish and put each of the 2 selected fish into different containers.
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question 6 the area of the rectangle is 48x3y5 square inches. its width is 6xy2 inches. what is the length of the rectangle?
The length of the rectangle is [tex]8x^2y^3[/tex] inches when the area of the rectangle is [tex]48x^3y^5[/tex] square inches.
To find the length of the rectangle, we can use the formula for the area of a rectangle, which is:
Area = length x width
We are given that the area of the rectangle is [tex]48x^3y^5[/tex] square inches, and its width is [tex]6xy^2[/tex] inches. Therefore, we can substitute these values into the formula to get:
[tex]48x^3y^5 = length * 6xy^2[/tex]
Simplifying this equation, we can cancel out the common factors of 6x and [tex]y^2[/tex] on both sides to get:
[tex]8x^2y^3 = length[/tex]
Therefore, the length of the rectangle is [tex]8x^2y^3[/tex] inches.
To verify our answer, we can substitute the length and width back into the formula for the area of a rectangle and check if it matches the given area:
Area = length x width = [tex](8x^2y^3) * (6xy^2) = 48x^3y^5[/tex]
Since this matches the given area, we can be confident that our answer is correct. Therefore, the length of the rectangle is [tex]8x^2y^3[/tex] inches.
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The frequency of people compared to the number of monthly text messages sent and received by a class are shown in the histogram.
A histogram titled Daily Text Messages. The x-axis is labeled Number of Sent Texts and has intervals of 1 to 15, 16 to 30, 31 to 45, 46 to 60, 61 to 75, 76 to 90, and 91 to 105. The y-axis is labeled Frequency and starts at 0, with tick marks every 2 units up to 12. There is a shaded bar for 1 to 15 that stops at 1, for 31 to 45 that stops at 3, for 46 to 60 that stops at 4, for 61 to 75 that stops at 4, for 76 to 90 that stops at 5, and for 91 to 105 that stops at 8. There is no shaded bar for the 16 to 30 interval.
Which of the following best describes the shape of the data, and why?
The data is symmetric because the majority of students sent fewer than 45 text messages.
The data is symmetric because some students have phones and some do not.
The data is skewed because most text messages are replied to and cause the number to be at least twice what it was originally.
The data is bimodal because the popular number of text messages to send is between 91 and 105.
Range, because it measures the distinction between the highest and lowest values in a dataset, providing a clear estimate of variability for both sets of data. It is unaffected by skewness or symmetry, making it a useful measure of variability for comparing the temperature consistency of Desert Landing and Flower Town.
What is meant by histogram?A histogram is a rough illustration of how numerical values are distributed. Karl Pearson coined the phrase at the beginning.
A histogram is a visual representation of statistical data that makes use of rectangles to illustrate the frequency of data points over a range of successive numerical intervals of equal width. The independent variable and dependent variable are plotted along the horizontal axis and the vertical axis, respectively, in the most typical type of histogram.
A common tool for graphing is the histogram. It is employed to summarise continuous or discrete data that is measured on an interval scale. It is frequently used to provide a convenient manner of illustration for the main characteristics of the data distribution.
Therefore, the correct answer is both option c) and d) Range, because Flower Town is skewed and symmetric.
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Which graph shows the solution to the system of linear equations?
y = 2x − 2
4x − 2y = 4
- Coordinate plane with one line that passes through the points 0 comma negative 2 and 1 comma 0.
- Coordinate plane with one line that passes through the points 0 comma negative 1 and 1 comma 1 and another line that passes through the points 0 comma negative 2 and 1 comma 0.
- Coordinate plane with one line that passes through the points 0 comma 1 and 1 comma 2.
- Coordinate plane with one line that passes through the points 0 comma negative 2 and 1 comma 0 and another line that passes through the points 0 comma 1 and 1 comma 2.
Answer:
A
Step-by-step explanation:
Answer: the answer is A
Step-by-step explanation:
The number of each type of ticket sold in the following situation: Tickets to a football game cost $5.oo if purchased before the day of the game. They cost $7.50 if purchased at the game. For a particular game, 600 tickets were sold and the receipts were $3500.
The number of each type of ticket sold in the following situation:
1. The tickets of $5.00 = 400
2. The tickets of $7.50 = 200
What is linear equation ?A linear equation is an algebraic equation in which the highest power of the variable is 1. In simple words, a linear equation is an equation in which the variables appear only in the first degree.
Linear equations can be written in the form:
y = mx + b
where y and x are variables, m is the slope (or rate of change), and b is the y-intercept (the point where the line crosses the y-axis).
Let x be the number of tickets sold before the day of the game and y be the number of tickets sold at the game.
Then we have:
x + y = 600 (the total number of tickets sold)
and
5x + 7.5y = 3500 (the total receipts from ticket sales)
To solve for x and y, we can use substitution.
Simplify and solve for y:
3000 - 5y + 7.5y = 3500,
2.5y = 500
y = 200
Substitute y = 200 back into the equation x + y = 600 to find x:
x + 200 = 600,
x = 400.
Therefore, 400 tickets were sold before the day of the game and 200 tickets were sold at the game.
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The complete question:
The number of each type of ticket sold in the following situation: Tickets to a football game cost $5.oo if purchased before the day of the game. They cost $7.50 if purchased at the game. For a particular game, 600 tickets were sold and the receipts were $3500. Find the number of tickets sold before the day of the game and the number of tickets sold at the day of the game?
Determine the geometric mean of 20 and 30.
The geometric mean of 20 and 30 is 24.49.
What is geometric mean?
The Geometric Mean (GM) in mathematics is the average value or mean that, by calculating the product of the values of the set of numbers, denotes the central tendency of the numbers. In essence, we multiply the numbers together and calculate their nth root, where n is the total number of data values.
We are given two numbers as 20 and 30.
Formula for geometric mean is
G.M = √x₁ × ... × xₐ
So, using this we get
⇒G.M = √20 × 30
⇒G.M = √600
⇒G.M = 24.49
Hence, the geometric mean of 20 and 30 is 24.49.
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Which table of values satisfies the linear equation y=−34x −6?
The table of values satisfies the linear equation y = −3x/4 − 6 is: C. table C.
How to determine the solution?In order to determine which ordered pairs are valid and true solutions based on the given linear equation, we would have to test the given ordered pairs by substituting their values into the linear equation as follows;
For ordered pair (-4, -9), we have:
y = −3x/4 − 6
-9 = −3(-4)/4 − 6
-9 = 3 − 6
-9 = -3 (False).
For ordered pair (-8, -12), we have:
y = −3x/4 − 6
-12 = −3(-8)/4 − 6
-12 = 8 − 6
-12 = 2 (False).
For ordered pair (-4, -3), we have:
y = −3x/4 − 6
-3 = −3(-4)/4 − 6
-3 = 3 − 6
-3 = -3 (True).
For ordered pair (0, -6), we have:
y = −3x/4 − 6
-6 = −3(0)/4 − 6
-6 = 0 − 6
-6 = -6 (True).
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Complete Question:
Which table of values satisfies the linear equation y = −3x/4 − 6?
Shelly incorrectly solves the equation 4−2x=−2.
Answer:
x=3
Step-by-step explanation:
x=3
4-2x=-2
-2x=-6
x=3
Answer:
Positive
Step-by-step explanation:
She incorrectly solved it because 4-2x = 2 ( Positive )
Select all the angle relationships in △
△
ABC that are correct.
In ΔABC, we can conclude that m∠A < m∠C and m∠B < m∠A < m∠C. Hence, options a. and d. are the accurate solutions.
The angle opposite to the longest side of the triangle is the largest angle of the triangle, similarly, the angle opposite to the smallest side of the triangle is the smallest angle of the triangle.
In the given ΔABC, we have,
The longest side = AB = 3.1
The angle opposite to side AB = m∠C
The smallest side = AC = 2.9
The angle opposite to side AC = m∠B
side CB = 3
The angle opposite to side CB = m∠A
We have,
side AC < side CB < side AB
m∠B < m∠A < m∠C
From the given relations in the options provided, we get,
a. m∠A < m∠C
d. m∠B < m∠A < m∠C
are correct interpretations.
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The complete question is -
Select all the angle relationships in △ABC that are correct.
Mr. Taylor has been looking at buying a house, and he is even thinking about having a pool put into his backyard!
Your Task: Design a pool that will fit inside Mr. Taylor's backyard and find out how much water it will take to fill the pool.
Rules!
1. Your pool design must include at least two different solids that are connected
2. Your pool must fit inside Mr. Taylor's backyard and leave at least 3 feet of space on each side of it for construction purposes
3. Your pool must be accessible and realistic
*Look at the photo for the blueprint of the house.
*You must clearly describe the pool design. Write this in a word-written response
To design a pool that would fit inside Mr. Taylor's backyard, you would need to know the dimensions of the available space.
You would also need to take into account any landscaping, slopes, and other features of the backyard that could impact the design and construction of the pool.
Once you have the necessary information about the backyard, you can start designing a pool that would be accessible and realistic. Here are some tips and guidelines for designing a pool.
Choose a shape and size that fits the available space and is functional. Rectangular or oval shapes are typically the most common and practical, but other shapes can be used as well.
Consider the depth of the pool. For safety reasons, the depth should be at least 4 feet at the shallow end and 8 feet at the deep end. You may also want to consider adding a gradual slope in the pool to make it easier for people to enter and exit.
Choose the material for the pool. Concrete, fiberglass, and vinyl are common materials used for pools. Each material has its own advantages and disadvantages, so research the options and choose the one that is best for your design and budget.
Consider adding features such as a diving board, slide, or waterfall to make the pool more enjoyable and attractive.
Calculate the amount of water it will take to fill the pool. This will depend on the size and depth of the pool. To get an estimate, you can use online pool volume calculators or consult with a pool contractor.
Once you have a design for the pool and know how much water it will take to fill, you can determine if it is feasible and within Mr. Taylor's budget.
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PLEASE HELP ME AS SOON AS POSSIBLE!!
A university purchased two different paintings in one year. The value of Painting A over time is modeled by f(x)=30,000(1.068)^x. The value of Painting B is represented by the graph at the right. Find the average rate of change of the value of each artwork over a -year time period. Which art work's value is increasing more quickly?
Rate of change of Panting A over a year period
2000Rate of change of Panting B over a year period
2500Painting B increases more quickly
How to find the rate of changeRate of change of Panting A over a year period, using the function f(x)=30,000(1.068)^x
at x = 0, f(x) = 30,000
at x = 1, f(x) = 32,040
rate of change = (30 000 - 32 000) / (0 - 1) = -2000 / -1 = 2000
Rate of change of painting B over a year period
= (25 - 27.5) * 1000 / (0 - 1)
= -2.5 * 1000 / -1
= 2500
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how would you describe the output of the graph for:
inputs from 0 to 1
inputs from 3 to 4
We can describe the output of the graph for:
inputs from 0 to 1 - slow output.inputs from 3 to 4 - rapid output.What is a Graph?This is referred to as a pictorial representation or a diagram that represents data or values in an organized manner.
From the graph given, we can deduce that the variation at 0-1 curve is small while the variation at 3-4 curve is large which is therefore the reason why slow and rapid output were chosen respectiviely.
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twenty five people, consisting of women and men are lined up in a random order. find the probability that the ninth woman to appear is in position 17. that is, find the probability there are women in positions thru and a woman in position 17
The probability that the ninth woman to appear is in position 17 is about 5.59%
We can approach this problem by using the binomial probability distribution. Let X be the number of women among the first 16 people in the line. Then X follows a binomial distribution with parameters n = 16 and p, the probability that any given person among the first 16 is a woman. We want to find the probability that X = 8, since this means that the ninth woman to appear is in position 17.
The probability of any given person being a woman is not specified in the problem, but we can assume that it is 0.5 for simplicity. Therefore, p = 0.5, and we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where (n choose k) is the binomial coefficient, which gives the number of ways to choose k items from a set of n items. In this case, it gives the number of ways to choose k women from the first 16 people in the line.
Using this formula with n = 16 and k = 8, we get:
P(X = 8) = (16 choose 8) * 0.5^8 * 0.5^8
= 12870 * 0.00390625
= 50.27
This means that the probability of exactly 8 women appearing among the first 16 people in the line is about 50.27%.
Given that there are 8 women among the first 16 people, the probability that the ninth woman appears in position 17 is 1/9, since there are 9 possible positions for the ninth woman to appear, and they are all equally likely.
Therefore, the overall probability that the ninth woman to appear is in position 17 is:
P(X = 8) * (1/9) = 50.27% * (1/9) = 5.59%
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if the first one to arrive will wait only 20 min before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [hint: the event of interest is a
The probability that they have dinner at the health food restaurant is 0.643.
To calculate the probability that the first one to arrive has dinner at the health-food restaurant given that they will wait only 20 minutes before leaving to eat elsewhere, we need to use conditional probability.
Let A be the event that the first one to arrive has dinner at the health-food restaurant, and let B be the event that the first one to arrive waits only 20 minutes before leaving to eat elsewhere.
We want to find P(A|B), the probability that A occurs given that B occurs.
We know that P(A) = 0.3, the probability that the first one to arrive has dinner at the health-food restaurant. We also know that P(B|A) = 0.6, the probability that the first one to arrive waits only 20 minutes before leaving to eat elsewhere given that they have dinner at the health-food restaurant.
To calculate P(B), the probability that the first one to arrive waits only 20 minutes before leaving to eat elsewhere, we need to use the law of total probability:
P(B) = P(B|A) x P(A) + P(B|A') x P(A')
where A' is the complement of A, i.e., the event that the first one to arrive does not have dinner at the health-food restaurant. We know that P(A') = 1 - P(A) = 1 - 0.3 = 0.7, and we are given that P(B|A') = 0.1, the probability that the first one to arrive waits only 20 minutes before leaving to eat elsewhere given that they do not have dinner at the health-food restaurant.
Therefore, we can calculate P(B) as:
P(B) = P(B|A) x P(A) + P(B|A') x P(A')
= 0.6 x 0.3 + 0.1 x 0.7
= 0.21 + 0.07
= 0.28
Finally, we can calculate P(A|B) using Bayes' theorem:
P(A|B) = P(B|A) x P(A) / P(B)
= 0.6 x 0.3 / 0.28
= 0.643
Therefore, the probability that the first one to arrive has dinner at the health-food restaurant given that they will wait only 20 minutes before leaving to eat elsewhere is approximately 0.643 or 64.3%.
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