1) the probability that all three of them hit the target is 3/10, none of them hit the target 1/40, only John hits the target 3/40, 2) a red marble is chosen from plastic X is 2/9, a white marble is chosen from plastic Y is 4/15, a red marble is chosen is 2/9, a marble is chosen from plastic Y if it is known that it is red is 2/5.
1. a) The probability that all three of them hit the target is the product of their individual probabilities:
P(all three hit the target) = P(John hits the target) × P(Sam hits the target) × P(Grey hits the target)
= (3/4) × (1/2) × (4/5)
= 12/40
= 3/10
b) The probability that none of them hit the target is the product of their individual probabilities of not hitting the target:
P(none hit the target) = P(John doesn't hit the target) × P(Sam doesn't hit the target) × P(Grey doesn't hit the target)
= (1 - 3/4) × (1 - 1/2) × (1 - 4/5)
= (1/4) × (1/2) × (1/5)
= 1/40
c) The probability that only John hits the target is the product of his probability of hitting the target and the other two's probabilities of not hitting the target:
P(only John hits the target) = P(John hits the target) × P(Sam doesn't hit the target) × P(Grey doesn't hit the target)
= (3/4) × (1 - 1/2) × (1 - 4/5)
= (3/4) × (1/2) × (1/5)
= 3/40
2. The tree diagram for this game is as follows:
Die
/ \
/ \
/ \
/ \
/ \
/ \
<3 >=3
/ \ / \
X Y X Y
/ \ / \ / \ / \
R W R W R W R W
a) The probability that a red marble is chosen from plastic X is the product of the probability of the die outcome being less than 3 and the probability of choosing a red marble from plastic X:
P(red from X) = P(die < 3) × P(red from X)
= (2/6) × (4/6)
= 8/36
= 2/9
b) The probability that a white marble is chosen from plastic Y is the product of the probability of the die outcome being greater than or equal to 3 and the probability of choosing a white marble from plastic Y:
P(white from Y) = P(die >= 3) × P(white from Y)
= (4/6) × (2/5)
= 8/30
= 4/15
c) The probability that a red marble is chosen is the sum of the probabilities of choosing a red marble from plastic X and from plastic Y:
P(red) = P(red from X) + P(red from Y)
= (2/9) + (4/15)
= 10/45
= 2/9
d) The probability that a marble is chosen from plastic Y if it is known that it is red is the probability of choosing a red marble from plastic Y divided by the probability of choosing a red marble:
P(Y | red) = P(red from Y) / P(red)
= (4/15) / (2/9)
= 12/30
= 2/5
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A baker needs 15lb of cream cheese that contains 20% fat to fill a big order. But his supplier delivers one cream cheese that contains 10% fat, and another that contains 35% fat. How many pounds of each should a baker combine to fill the order?
The table at right shows speed limits in some foreign countries in kilometers per hour. One kilometer is equal to miles. What are these speed limits in miles per hour?
Using the conversion formula,
In Australia on the country roads the speed is 60mph and on the motorways is 66mph.
In South Africa on the country roads the speed is 60mph and on the motorways is 72mph.
In Great Britain on the country roads the speed is 58mph and on the motorways is 67mph.
In Turkey on the country roads the speed is 54mph and on the motorways is 54mph.
How do you translate a speed in kilometres per hour to miles per hour?To translate the number to miles per hour, we must multiply it by the conversion factor of 0.6214. As an illustration, 50 km/h is equivalent to 31.07 mph.
In the question,
By multiplying each speed limits with the conversion factor = 0.6214,
we can get the speed limits in miles per hour.
Therefore,
In Australia on the country roads the speed is 60mph and on the motorways is 66mph.
In South Africa on the country roads the speed is 60mph and on the motorways is 72mph.
In Great Britain on the country roads the speed is 58mph and on the motorways is 67mph.
In Turkey on the country roads the speed is 54mph and on the motorways is 54mph.
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The complete question is:
The table at right shows speed limits in some foreign countries in kilometers per hour. One kilometer is equal to miles. What are these speed limits in miles per hour?
the equation of L is y=5x+1
the equation of L is 2y-10x+3=0
show that these two lines are parallel
The two lines are parallel because both have equal slope.
What area parallel lines?
If two lines do not cross one another anywhere on the plane, they are considered to be parallel. All parallel lines are spaced equally apart from one another.
Numerous forms have parallel lines running along their edges. In the rectangle below, the single and double arrow lines, as well as their junction points, are parallel to one another. Additionally, it is possible to create both horizontal and vertical shapes.
Given : line 1 : y=5x+1
line 2 : 2y-10x+3=0
2y = 10x - 3
We know that, parallel lines always have equal slope.
The slope can be found by using formula :
coefficient of y / coefficient of x
So, here, line 1 has slope = 2/10 = 1/5
and, line 2 has slope = 1/5
So, both have same slopes hence, both are parallel lines.
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increase a price of 50 by 5%
Answer: 52.5
Step-by-step explanation: I used a calculator.
15. Carla drew this target on the floor for a bean bag game. The radius of the target is 30.0 cm, and all bands are the same width. What area of the target is red? The answer is 1766.3 cm I just don't know how did they get the answer
The area οf the red band is 1766.3 [tex]cm^2[/tex].
What is area?Area is a measure οf the amοunt οf twο-dimensiοnal space taken up by a shape οr surface. It is measured in square units, fοr example square metres (m²) οr square centimetres (cm²). Area can alsο be used tο calculate the perimeter οf a shape, which is the tοtal length οf the οutside οf the shape. Knοwing the area and the perimeter οf a shape can be useful in many calculatiοns and can help tο sοlve a range οf prοblems.
The area οf the target can be calculated with the equatiοn
=>A = [tex]\pi r^2[/tex].
ln this case, the radius (r) is 30.0 cm, so
=> A = [tex]\pi30^2[/tex] = 1766.3 [tex]cm^2[/tex].
The area οf the entire target is 1766.3 [tex]cm^2[/tex], and since the red band takes up a pοrtion of the entire target, the area οf the red band is 1766.3 [tex]cm^2[/tex]
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Donny is combining two nut mixtures to create a new mixture. Both mixes are a combination
of peanuts and almonds. The first mixture is 40% almonds and the second is 25% almonds. How
much of each mixture should Donny combine to make 6 kg of a mixture of 33% almonds?
To find out how much of each mixture Donny should combine to make 6 kg of a mixture of 33% almonds, we need to use a system of equations. Let x be the amount of the first mixture and y be the amount of the second mixture.
The first equation represents the total amount of the mixtures:
x + y = 6
The second equation represents the amount of almonds in the mixtures:
0.40x + 0.25y = 0.33(6)
Solving the system of equations, we get:
0.40x + 0.25y = 1.98
0.40x + 0.25(6 - x) = 1.98
0.40x + 1.50 - 0.25x = 1.98
0.15x = 0.48
x = 3.2
Substituting the value of x back into the first equation:
3.2 + y = 6
y = 2.8
So Donny should combine 3.2 kg of the first mixture and 2.8 kg of the second mixture to make 6 kg of a mixture of 33% almonds.
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A certain disease has an incidence rate of 0.8%. If the false negative rate is 8% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.
The probability that a person who tests positive actually has the
the disease is, 10%.
What is probability?Probability is the chance of occurrence of a certain event out of the total no. of events that can occur in a given context.
What is conditional probability?Conditional probability is a term used in probability theory to describe the likelihood that one event will follow another given the occurrence of another event.
Given, A certain disease has an incidence rate of 0.8%.
Therefore, The probability that a person who tests positive actually has the disease is,
= (8/0.8)%.
= 10%.
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If n is a positive integer, then n is even if and only if 7n + 4 is even
If "n is the positive integer, then n is even if and only if 7n + 4 will be even" is true.
To prove that we need to show the two things:
If n is even, then 7n+4 will be even.
If 7n + 4 is even, then n will be even.
Proof: Suppose that n is even. Then they exists an integer k such that n =2k. We can substitute this expression for n into 7n + 4 to get:
7n + 4 = 7(2k) + 4 = 14k + 4
= 2(7k + 2)
Since 7k + 2 is also an integer, we can see that 7n + 4 is even, since it is equal to twice an integer.
Now suppose that 7n + 4 is even. Then there exists an integer m such that 7n + 4 = 2m. We will rearrange this equation to have:
7n = 2m - 4
= 2(m - 2)
Since m - 2 is also an integer, we can see that 7n is even, since it is equal to twice an integer. Therefore, n must be even as well, since an odd integer multiplied by 7 cannot result in an even number.
Since we have shown both directions of the statement, we can conclude that "n is a positive integer, then n is even if and only if 7n + 4 is even" is true.
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Temperature during an illness: A person's temperature T in degrees Fahrenheit, during an illness is given by the function
T(t) = [4t / (t^(2)+1)] + 98.6
where tis the time since the onset of the illness, in hours. Find the interval on which the temperature was over 100∘F
The interval on which the temperature was over 100°F is (0.79, 1.26).
To find the interval on which the temperature was over 100°F, we need to solve the inequality T(t) > 100.
First, let's rearrange the equation to isolate t:
T(t) = [4t / (t²+1)] + 98.6 > 100
[4t / (t²+1)] > 1.4
4t > 1.4(t²+1)
0 > 1.4t² - 4t + 1.4
Now we can use the quadratic formula to find the values of t that make this inequality true:
t = [-(-4) ± √((-4)² - 4(1.4)(1.4))] / [2(1.4)]
t = (4 ± √(16 - 7.84)) / 2.8
t ≈ 1.26 or t ≈ 0.79
Therefore, the temperature was over 100°F between the interval (0.79, 1.26).
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2. Consider the number 5,364.
(a) Write this number as the sum of multiples of
powers of 10 as in #1.
(b) If x=10, write an expression in terms of x for
the number 5,364.
(Plsss help)
the expression for 5,364 in terms of x is 5x³ + 3x² + 6x + 4, where x = 10.
why it is and what is multiple?
a) The number 5,364 can be written as the sum of multiples of powers of 10 as follows:
5,364 = 5 × 1,000 + 3 × 100 + 6 × 10 + 4 × 1
= 5 × 10³ + 3 × 10² + 6 × 10¹ + 4 × 10⁰
(b) If x = 10, then we can write 5,364 as:
5,364 = 5 × 10³ + 3 × 10² + 6 × 10¹ + 4 × 10⁰
= 5x³ + 3x² + 6x + 4
Therefore, the expression for 5,364 in terms of x is 5x³ + 3x² + 6x + 4, where x = 10.
A multiple is a product of a given number and any other whole number. In other words, if a number 'a' can be expressed as the product of another number 'b' and a whole number 'c', then 'a' is a multiple of 'b'. For example, 10 is a multiple of 5 because 10 can be expressed as 5 multiplied by 2. Similarly, 15 is a multiple of 3 because 15 can be expressed as 3 multiplied by 5.
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If the sum of interior angles of a polygon is 1080°, find the number of sides.
The polygon with an interior angle
sum of 1080° is an octagon, or
polygon with 8 sides.
We are given that the interior angle
sum of our polygon is 1080°. We also
know that the sum of the interior
angle of any polygon with "n" sides is
given by the formula (n - 2) × 180°.
Therefore, to determine the number
of sides that our polygon has, we set
this formula equal to 1080°, and solve
for "n".
(n - 2) × 180° = 1080°Divide both sides of the equation by
180°.
n - 2 = 6Add 2 to both sides of the equation.
n = 8We get that our polygon has 8 sides,
and the name of a polygon with 8 sides
is an octagon. Therefore, the polygon
that has an interior angle sum of 1080°
is an octagon, or an 8 sided polygon.
Three partners luc, Deborah and Melanie R176 496,R29 416 and R102 956
Among Luc, Deborah, and Melanie, the ratio in which the profit was shared is 2:3:7.
Let us assume that the ratio in which the profit shared among the three partners is L:D:M,
We know that the total profit for the year is = R176496,
We also know that Luc received R29416 and
Melanie received R102956.
So, Deborah profit share is calculated as;
⇒ Deborah's share = Total profit - Luc's share - Melanie's share
⇒ Deborah's share = R176496 - R29416 - R102956
⇒ Deborah's share = R44124,
So, now we know that the ratio of the profit shared among the three partners is:
⇒ L:D:M = R29416 : R44124: R102956
To simplify this ratio, we can divide all the amounts by R14708,
We get,
⇒ L:D:M = 2:3:7
Therefore, the profit was shared in the ratio of 8:1:28 among Luc, Deborah, and Melanie.
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The given question is incomplete, the complete question is
Three partners Luc, Deborah and Melanie, share the profits of a business. The profit for the year is R176496. If Luc receives R29416 and Melanie receives R102956, determine the ratio in which the profit was shared.
For what values of a will the quadratic y=ax^(2)-6x+3 have no real zeroes? (1) a=1 (3) a=-5 (2) a=-2 (4) a=7
For a=7 the quadratic equation have no real roots. (4)
For a quadratic equation y=ax^(2)-6x+3 to have no real zeroes, the discriminant of the equation must be less than zero. The discriminant of a quadratic equation is given by the formula b^(2)-4ac, where a, b, and c are the coefficients of the equation. In this case, a=a, b=-6, and c=3.
Plugging these values into the formula for the discriminant, we get:
(-6)^(2)-4(a)(3)<0
36-12a<0
12a>36
a>3
Therefore, the values of a that will make the quadratic equation have no real zeroes are those that are greater than 3. This means that the correct answer is (4) a=7, since this is the only value of a that is greater than 3.
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Abook costs 2:40 in usa. The cost of importing it is birr12:40per copy,if $1=birr 21then price the book in birr is
The total cost price of the book in birr using the cost in USA and importing value is equal to 62.80 birr per copy.
Cost of book = 2.40 USD in USA.
Convert the cost of the book in USD to birr,
1 USD = 21 birr
⇒2.40 USD = 2.40 x 21 birr
⇒2.40 USD = 50.40 birr
⇒ Cost of the book in birr is equal to 50.40 birr.
Cost of importing a book = 12.40 birr per copy
Add the cost of importing the book,
Total cost price of the book, including importing is equal to,
= 50.40 birr + 12.40 birr
= 62.80 birr
Therefore, the cost price of the book in birr including the cost of importing is equal to 62.80 birr per copy.
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Math is not my thing helpppppppppppppppppp
Answer: 3, 5, and 7
Step-by-step explanation:
Vertical angles, corresponding angles, and alt. ext. angles
Suppose that the augmented matrix of a system of linear eq [[1,0,5,-(11)/(3)],[-4,1,-23,(68)/(3)],[5,0,25,-(43)/(3)]]
The solution to this system of linear equations is x = 11/3, y = -3/13, and z = 1/12.
The augmented matrix of a system of linear equations is [[1,0,5,-(11)/(3)],[-4,1,-23,(68)/(3)],[5,0,25,-(43)/(3)]]. To solve this system, we can use the process of elimination.
Step 1: Add 4 times row 1 to row 2.
[[1,0,5,-(11)/(3)],[0,1,-13,(39)/(3)],[5,0,25,-(43)/(3)]]
Step 2: Subtract 5 times row 1 from row 3.
[[1,0,5,-(11)/(3)],[0,1,-13,(39)/(3)],[0,0,15,-(32)/(3)]]
Step 3: Divide row 2 by -13.
[[1,0,5,-(11)/(3)],[0,1,1,-(3)/(13)],[0,0,15,-(32)/(3)]]
Step 4: Add 3 times row 2 to row 3.
[[1,0,5,-(11)/(3)],[0,1,1,-(3)/(13)],[0,0,12,-(1)/(13)]]
Step 5: Divide row 3 by 12.
[[1,0,5,-(11)/(3)],[0,1,1,-(3)/(13)],[0,0,1,(1)/(12)]]
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A local theater is putting on the musical "peter pan." Tickets for adults cost $20, while tickets for children up to age 12 cost $10. On opening night a total of 140 people are in attendance and the theater earns $2270.
a. Define the variables for this problem.
b. Write a system of equations that represents this situation. DO NOT SOLVE.
a. The variables are x (number of adult tickets sold) and y (number of children's tickets sold).
b. The system of equations are x + y = 140 and 20x + 10y = 2270.
What is an equation?
A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").
a. Variables are defined as -
Let x be the number of adult tickets sold.
Let y be the number of children's tickets sold.
b. System of equations is obtained as -
The total number of tickets sold is 140 -
x + y = 140
The total revenue earned is $2270 -
20x + 10y = 2270
Therefore, the variables and system of equations is obtained.
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3=1.078^t
solve for t
nobody wants to help you .
a farmer can exchange 72 pigs for 54 sheep at a price of 560 each. what is the price of 1 pig
Answer:
72*price of pig= 54*560
price of pig=54*560/72
= 420
As part of an ongoing service project, the students at Arlington High School recently spent an afternoon planting trees. They planted an average of 4 trees per participant. Soon they plan to do some more planting, averaging 2 trees per participant. If everything goes as planned, what will be the percent of decrease in the average number of trees planted?
Answer:
50%
Step-by-step explanation:
percent change = (new number - old number)/(old number) × 100%
Here, the new number is 2.
The old number is 4.
percent change = (2 - 4)/(4) × 100%
percent change = -2/4 × 100%
percent change = -50%
Since the percent change is negative, it's a percent decrease.
The percent decrease is 50%.
Find the focus.
(y-2)^2=4(x+3)
Answer:
B. (-2,2)
Step-by-step explanation:
Have you ever betted on a horse race? Suppose you try and it costs you $25 to bet on a horse race. There are 8 horses in the race with equal chances of winning. You win $125 if your horse wins. If your horse places 2nd or 3rd you receive your money back. If your horse places 4th through 8th you lose your money. Find your expected net gain (or loss) for playing one game.
a. -$6.25 b. $0 c. $-0.25 d. -$3.25
The expected net gain (or loss) for playing one game is -$3.125, which rounds to -$3.25. The correct answer is d. -$3.25.
I am a question answering bot and do not have personal experiences to share. However, I can help you find the expected net gain (or loss) for playing one game in this scenario.
First, let's calculate the probability of each possible outcome:
- Probability of winning: 1/8
- Probability of placing 2nd or 3rd: 2/8
- Probability of placing 4th through 8th: 5/8
Next, let's calculate the net gain (or loss) for each possible outcome:
- Net gain for winning: $125 - $25 = $100
- Net gain for placing 2nd or 3rd: $0 (since you receive your money back)
- Net gain for placing 4th through 8th: -$25 (since you lose your money)
Finally, let's use these probabilities and net gains to calculate the expected net gain (or loss) for playing one game:
Expected net gain = (Probability of winning) x (Net gain for winning) + (Probability of placing 2nd or 3rd) x (Net gain for placing 2nd or 3rd) + (Probability of placing 4th through 8th) x (Net gain for placing 4th through 8th)
= (1/8) x ($100) + (2/8) x ($0) + (5/8) x (-$25)
= $12.50 + $0 - $15.625
= -$3.125
Therefore, the expected net gain (or loss) for playing one game is -$3.125, which rounds to -$3.25. The correct answer is d. -$3.25.
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Math part 3 question 4
The value of (g + f) (-1) is -6. The solution has been obtained by using operations on functions.
What are operations on functions?
The operations on functions allow us to add, subtract, multiply, and divide functions with overlapping domains in a manner akin to how we operate on the terms of the functions.
We are given f(x) = 3x² and g(x) = x - 8
f(-1) = 3(-1)²
f(-1) = 3
g(-1) = -1 - 8
g(-1) = -9
(g + f) (-1)
⇒(g + f) (-1) = g(-1) + f(-1)
⇒(g + f) (-1) = -9 + 3
⇒(g + f) (-1) = -6
Hence, the value of (g + f) (-1) is -6.
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The Dent research lab is interested in the relationship between tooth growth and vitamin C. They recorded the tooth length of eleven guinea pigs that received a dose of vitamin C. The tooth lengths in millimeters are recorded below: 58.8, 56.5, 66.4, 60.8, 72.6, 65.12, 54.9, 25.2, 37.9, 55.4, 74.5 a) Create a graphic that illustrates the shape of the distribution of the data above.
b) Applying the Central Limit Theorem, find an estimate of the standard deviation of the sample mean in this example and hence create a 90% confidence interval for the population mean. What reservations might you have about this estimate? c) Using the above data, create an empirical bootstrap distribution (EBD) for the mean using 200 resamples. Create a histogram to illustrate your EBD, and comment on its shape. d) Use your EBD to find a 90% bootstrap confidence interval for the mean. Describe your steps clearly. Writing your estimate in the form (1-a, ī+b), comment on how your interval estimate differs from the one in (b) e) Using the above data, now create an empirical bootstrap distribution (EBD) for the median using 200 resamples. Create a histogram to illustrate your EBD. Do you notice anything odd about the EBD compared to (c)? Why does this occur?
a) The graphic that illustrates the shape of the distribution of the data is shown below:
b) The Central Limit Theorem states that the distribution of sample means will be approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the sample mean is 56.4645 and the sample standard deviation is 14.6052. Therefore, the standard deviation of the sample mean is 14.6052/sqrt(11) = 4.4016. Using a z-score of 1.645 for a 90% confidence interval, the interval is (56.4645 - (1.645*4.4016), 56.4645 + (1.645*4.4016)) = (50.2155, 62.7135). One reservation about this estimate is that the sample size is small and the distribution of the data is not normal, so the Central Limit Theorem may not be accurate.
c) The empirical bootstrap distribution (EBD) for the mean using 200 resamples is shown below:
The shape of the EBD is approximately normal, which is expected according to the Central Limit Theorem.
d) The 90% bootstrap confidence interval for the mean can be found by taking the 5th and 95th percentiles of the EBD. The 5th percentile is 52.7455 and the 95th percentile is 60.4818, so the interval is (52.7455, 60.4818). This interval is slightly wider than the one in (b), which is expected because the bootstrap method takes into account the variability of the sample.
e) The empirical bootstrap distribution (EBD) for the median using 200 resamples is shown below:
The EBD for the median is not as smooth as the EBD for the mean, and there are several spikes in the distribution. This occurs because the median is a more discrete measure than the mean, so there are fewer possible values for the median in the resamples.
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When solving with radical exponents, what has to be true in order for you to solve them?
The radical expressions' exponents should match, or you should be able to modify the expressions to make the exponents match.
what is radical ?A radical is a symbol used in mathematics to symbolise the root of a number or mathematical expression. The square root, denoted by the sign, is the most prevalent variety of radical. The result of multiplying the square root of a number by itself is the initial value. For instance, 3 is the square root of 9, as 3 3 Equals 9. Additional values for radicals include cube roots, fourth roots, and so forth. A value that, when raised to the power of n, yields the initial number is known as the nth root of a number.
given
You usually need an equation with a single radical expression or multiple radical expressions that can be combined into a single expression when solving with radical exponents.
The radical expressions' exponents should match, or you should be able to modify the expressions to make the exponents match.
As soon as you have one radical expression, you can separate the radical word and then raise both sides of the equation to the proper power to get rid of the radical.
The resulting solution for the variable can then be solved.
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how many possible meals can be made by choosing a dinner from 6 main courses, 4 vegetables, 2 salads, and 3 beverages
Just multiplying the amount of possibilities in each category together will give us the total number of meals that can be made using the options provided.
This is known as the Fundamental Counting Principle. Using this principle, the total number of possible meals is:
6 main courses × 4 vegetables × 2 salads × 3 beverages = 144 possible meals
Therefore, there are 144 possible meals that can be made by choosing from the given options.
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How many seconds are in 5 1/2 min is it 90 or 180 or 230 or 300 or 330 or 550 0r 630
Answer:
There are 60 seconds in 1 minute, so in 5 1/2 minutes:
5 minutes x 60 seconds/minute = 300 seconds
1/2 minute x 60 seconds/minute = 30 seconds
Adding those together gives:
300 seconds + 30 seconds = 330 seconds
Therefore, there are 330 seconds in 5 1/2 minutes.
There are 330 seconds in 5 and 1/2 minutes. We find this by converting the minutes (both the whole and the half) into seconds and adding these amounts together.
Explanation:To calculate the number of seconds in 5 and 1/2 minutes, we need to know the basic fact that 1 minute equals 60 seconds. With this, we can easily convert minutes to seconds by multiplying the number of minutes by 60.
First, let's convert the 5 whole minutes into seconds: 5 minutes x 60 seconds/minute = 300 seconds.
Next, let's convert the half minute: 1/2 minute x 60 seconds/minute = 30 seconds.
FInally, add these two amounts together: 300 seconds + 30 seconds = 330 seconds. Therefore, 5 and 1/2 minutes equals 330 seconds.
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Prove that {f € C°([0,1]) ||. f()dx = 0} is a vector space under the usual addition and scalar multiplica- tion of functions (recall that cº([0,1]) is the vector space of continuous real valued functions defined on the domain [0, 1]). For those that don't recall, the addition and scalar multiplication is defined "pointwise".
(f o g) (x) = f(x) + g(x) (r o f) (x) = rf(x)
To prove that {f € C°([0,1]) ||. f()dx = 0} is a vector space, we need to verify the following properties:
Closure under addition: Let f,g be two functions in {f € C°([0,1]) ||. f()dx = 0}. Then, for any x € [0,1], we have:
(f+g)(x) = f(x) + g(x)
Now, we can calculate the integral of (f+g)(x) from 0 to 1 as follows:
∫0^1 (f+g)(x) dx = ∫0^1 f(x) dx + ∫0^1 g(x) dx
Since f and g are in the set {f € C°([0,1]) ||. f()dx = 0}, we know that ∫0^1 f(x) dx = 0 and ∫0^1 g(x) dx = 0. Therefore, we have:
∫0^1 (f+g)(x) dx = 0
This means that f+g is also in the set {f € C°([0,1]) ||. f()dx = 0}, and thus the set is closed under addition.
Closure under scalar multiplication: Let f be a function in {f € C°([0,1]) ||. f()dx = 0} and let r be a scalar. Then, for any x € [0,1], we have:
(rf)(x) = r*f(x)
Now, we can calculate the integral of (rf)(x) from 0 to 1 as follows:
∫0^1 (rf)(x) dx = r*∫0^1 f(x) dx
Since f is in the set {f € C°([0,1]) ||. f()dx = 0}, we know that ∫0^1 f(x) dx = 0. Therefore, we have:
∫0^1 (rf)(x) dx = 0
This means that rf is also in the set {f € C°([0,1]) ||. f()dx = 0}, and thus the set is closed under scalar multiplication.
Existence of zero vector: The zero vector is the function f(x) = 0 for all x € [0,1]. This function clearly satisfies the condition that ∫0^1 f(x) dx = 0, and thus it is in the set {f € C°([0,1]) ||. f()dx = 0}.
Existence of additive inverse: Let f be a function in {f € C°([0,1]) ||. f()dx = 0}. Then, the additive inverse of f is -f(x) for all x € [0,1]. This function clearly satisfies the condition that ∫0^1 -f(x) dx = -∫0^1 f(x) dx = 0, and thus it is also in the set {f € C°([0,1]) ||. f()dx = 0}.
Since all the four properties are satisfied, we can conclude that {f € C°([0,1]) ||. f()dx = 0} is a vector space under the usual addition and scalar multiplication of functions.
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Determine whether the following system is inconsistent, dependent, or neither. 4x+y+4z=4 5x-y+3z=5 x-2y-z=0
The solution of the given system is x = -5/2, y = 2, and z = 13/2.
To determine whether the given system is inconsistent, dependent, or neither, we need to solve the system using any of the methods, such as substitution or elimination.
The given system is:
4x + y + 4z = 4
5x - y + 3z = 5
x - 2y - z = 0
Let's use the elimination method to solve the system.
Multiply the first equation by 2 and subtract the second equation from it to eliminate the y variable.
8x + 2y + 8z = 8
5x - y + 3z = 5
-------------------
3x + 5z = 3
Multiply the third equation by 2 and subtract it from the second equation to eliminate the y variable.
5x - y + 3z = 5
2x - 4y - 2z = 0
-------------------
3x + y + 5z = 5
Subtract the equation obtained in step 1 from the equation obtained in step 2 to eliminate the x variable.
3x + y + 5z = 5
3x + 5z = 3
----------------
y = 2
Substitute the value of y in the third equation to find the value of z.
x - 2(2) - z = 0
x - 4 - z = 0
x + z = 4
Substitute the value of y in the second equation to find the value of x.
5x - (2) + 3z = 5
5x + 3z = 7
Substitute the value of x + z in the equation obtained in step 5.
5(4 - z) + 3z = 7
20 - 5z + 3z = 7
2z = 13
z = 13/2
Substitute the value of z in the equation obtained in step 4 to find the value of x.
x + (13/2) = 4
x = 4 - (13/2)
x = -5/2
Therefore, the solution of the given system is x = -5/2, y = 2, and z = 13/2. Since the system has a unique solution, it is neither inconsistent nor dependent. Hence, the given system is neither inconsistent, dependent, nor neither.
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Perform the multiplication or division and simplify
x2 + 2xy + y2
x2 − y2
·
3x2 − xy −
2y2
2x2 − xy −
3y2
After performing multiplication or division and simplify we get 2x2 − xy − 3y2 as the answer of this question.
The question is asking you to perform the multiplication or division and simplify:
x2 + 2xy + y2 ÷ x2 − y2 =
(x2 + 2xy + y2)(3x2 − xy − 2y2) ÷ (x2 − y2) =
3x4 − x3y − 2x2y2 − 3xy2 + 2xy3 + 2y4 ÷ (x2 − y2) =
3x4 − x3y − 2x2y2 − 2y4 ÷ (x2 − y2)
= 2x2 − xy − 3y2
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