Answer:
(a) The probability is 0.4696
(b) The probability is 0.5304
(c) The probability is 0.0929
Step-by-step explanation:
The total number of ways in which we can select k elements from a group n elements is calculate as:
[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]
So, the number of ways in which we can select four students from a group of 19 students is:
[tex]19C4=\frac{19!}{4!(19-4)!}=3,876[/tex]
On the other hand, the number of ways in which we can select four students with no math majors is:
[tex](16C4)*(3C0)=(\frac{16!}{4!(16-4)!})*(\frac{3!}{0!(3-0)!})=1820[/tex]
Because, we are going to select 4 students form the 16 students that aren't math majors and select 0 students from the 3 students that are majors.
At the same way, the number of ways in which we can select four students with one, two and three math majors are 1680, 360 and 16 respectively, and they are calculated as:
[tex](16C3)*(3C1)=(\frac{16!}{3!(16-3)!})*(\frac{3!}{1!(3-1)!})=1680[/tex]
[tex](16C2)*(3C2)=(\frac{16!}{2!(16-2)!})*(\frac{3!}{2!(3-1)!})=360[/tex]
[tex](16C1)*(3C3)=(\frac{16!}{1!(16-1)!})*(\frac{3!}{3!(3-3)!})=16[/tex]
Then, the probability that the group has no math majors is:
[tex]P=\frac{1820}{3876} =0.4696[/tex]
The probability that the group has at least one math major is:
[tex]P=\frac{1680+360+16}{3876} =0.5304[/tex]
The probability that the group has exactly two math majors is:
[tex]P=\frac{360}{3876} =0.0929[/tex]
In short, to calculate the probability of certain events in a group selection, you would identify the total possible groups, and then calculate how many of these groups satisfy your desired conditions. The probability is then calculated as the favorable events over the total possibilities.
Explanation:This problem is a classic example of combinatorics and probability. The total number of ways to select four students from a total of 19 is given by the combination function: 19 choose 4. The denominator for all our probability calculations will be this total number of possible groups.
(a) To find the probability that the group has no math majors, we want all four students to be from the 16 non-math majors. This is calculated as combinations of 16 choose 4. Thus, the probability is (16 choose 4) / (19 choose 4).(b) The probability that the group has at least one math major is calculated as 1 minus the probability that the group has no math majors.(c) The probability that the group has exactly two math majors can be calculated by considering the combinations of selecting 2 math majors from the 3 (3 choose 2) and 2 non-math majors from the remaining 16 (16 choose 2). That gives us the probability of (3 choose 2)*(16 choose 2) / (19 choose 4).Learn more about Combinatorics and Probability here:https://brainly.com/question/31293479
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Construct a truth table for the logical operator NOR.
Answer:
NOR Gate
Step-by-step explanation:
NOR gate is a two input gate.
It is defined as the complement of (X or Y), where X and Y are the inputs of the gate.
X Y X+Y Complement(X+Y) NOR
1 1 1 0 0
1 0 1 0 0
0 1 1 0 0
0 0 0 1 1
Ut is an operator which gives a value of 1 only when the bvoth the inputs are 0.
A truth table for the logical operator NOR is:
P Q NOR
T T F
T F F
F T F
F F T
How to construct a Truth Table?
A truth table is a mathematical table used in logic to show all possible combinations of truth values for the input variables of a Boolean function and their corresponding output.
To construct a truth table for the logical operator NOR, we can use the following table:
P Q NOR
T T F
T F F
F T F
F F T
The NOR operator returns true only when both inputs are false. Otherwise, it returns false.
This can also be expressed as ¬(P ∨ Q), which is logically equivalent to P NOR Q.
In art class Ms smith is working on polygons. She want the students to Create a picture of the polygons. She points out to the class That there is are three sides to a triangle, 4 sides on a quadrilateral , 5 sides on a pentagon, and six sides on a hexagon, How many more side are on a hexagon than on a quadrilateral
Answer:
There are two more sides on a hexagon than on a quadrilateral
Step-by-step explanation:
If the hexagon has 6 sides, and the quadrilateral has 4, then 6-4=2
I need to find the standard for Hamilton's method to figure out how many teachers should be at each school
Answer:
The standard divisor is 22.48.
Step-by-step explanation:
There are a total of 3259 students at the 5 schools. Then dividing that number by the number of teachers (145) we get the "standard divisor" of ...
3259/145 ≈ 22.48
__
By Hamilton's method, that divisor is used to divide the number of students at each school, and the result is rounded down. This is the initial allocation of teachers to schools. The remainders from the division are examined. Starting with the largest and working down, one additional teacher is assigned until all the unassigned teachers have been assigned.
For this problem, the initial assignment results in 142 teachers being assigned, so there are 3 more that can be allocated. In order, the highest three remainders are associated with the number of students at East, Central, and South. Each of those schools gets one more teacher than the number initially assigned. The final allocation of teachers is highlighted in the attachment.
The owner of a pizza-by-the-slice shop has determined the marginal cost for his company is $1.50. His daily fixed costs are $125. What is the total cost to sell 300 slices in one day?
Answer:
[tex]300\cdot \$1.50+\$125=\$450+\$125=\$575[/tex]
Step-by-step explanation:
The marginal cost is said to be $1.50. Marginal cost is just how much the cost increases per additional unit produced. In this case we're dealing with a shop of slices of pizza, so the marginal cost just represents how much the cost increases per additional slice of pizza produced, or in simpler words, how much it costs to produce a slice of pizza.
We want to compute the total cost to sell 300 slices in one day, so we have to compute how much it costs to produce those 300 slices and add up the fixed costs (which is $125, no matter how many slices we produce). Since the marginal cost is $1.50, that means each slice costs $1.50 to produce. So the 300 slices cost [tex]300\cdot \$1.50=\$450[/tex] to produce. And so the total cost is
[tex]\$450+\$125=\$575[/tex]
Plato math help please
Answer:
The function's graphic is the one that is below the first one in th right.
Step-by-step explanation:
If we want to know how much Chelsea earns for dogsitting her neighbours' dogs, we have to form a linear equation, that must have the following formula:
[tex]y= ax + b[/tex]
In this case, B is the constant value. We know Chelsea charges 12$ for dogsitting, and that doesn't depend on anything (B). But, if you want to Chelsea to walk your dog, then you'd to pay 2.50$ for each walk (X).
[tex]y= 2.50x + 12[/tex]
So, if Chelsea doesn't walk the neighbour's dog (x=0), she would be earning 12 dolars.
[tex]y= 2.50 x 0 + 12 = 12[/tex]
If she walks four times the dog, she would be earning:
[tex]y= 2.50 x 4 + 12 = 10 + 12 = 22 [/tex]
Knowing these two values, we can graph the equation. When x=0, y=12, and when x=4, y=22
The function's graphic is the one that is remarked in the attachment.
The Titanic was 883 feet long and 175 feet tall.
A. If a model is built to have a scale ratio of 1in : 36ft , how long will the model be?
B. If a model is built to have a scale ratio of 1 in : 22 ft , how tall will the model be?
C. If a model is built to have a ratio of 1: 30 , how tall will the model be?
Answer:
Actual length = 883 feet
Actual height = 175 feet
A) If a model is built to have a scale ratio of 1 in : 36ft , how long will the model be?
36 feet = 1 inch
Actual length = 883 feet
So, 883 feet = [tex]\frac{883}{36} inch[/tex]
883 feet = [tex]24.527 inch[/tex]
So, If a model is built to have a scale ratio of 1in : 36ft ,the model will be 24.527 inch long.
B) If a model is built to have a scale ratio of 1 in : 22 ft , how tall will the model be?
22 feet = 1 inch
Actual height = 175 feet
So, 175 feet = [tex]\frac{175}{22} inch[/tex]
883 feet = [tex]7.9545 inch[/tex]
So, If a model is built to have a scale ratio of 1 in : 22 ft , the model will be 7.9545 inch tall.
C) If a model is built to have a ratio of 1: 30 , how tall will the model be?
Let the height be x
Actual length = 883 feet
Ratio of 1: 30
So, [tex]\frac{1}{30}=\frac{883}{x}[/tex]
[tex]x=\frac{883}{30}[/tex]
[tex]x=29.433[/tex]
So, If a model is built to have a ratio of 1: 30 ,the model will be 29.433 inches tall .
Look online for the growth of the trunk of a tree. Estimate how much time does it take for a water oak to grow one inch in diameter. Estimate the growth rate over a year
Answer:
Explained
Step-by-step explanation:
The trunk of a tree grows in two different ways, first in height and second in diameter.Usually tree grows one ring per year in diameter. So, counting the number of rings we can determine the age of a tree. Both height and diameter growth does not occur at the same rate. Tree grows more in height than in their diameter. Mature trees usually grows 1 inch in diameter every year.
Water oak gains 24 inches in height every year and 1.5 inch growth in diameter annually, meaning if we divide 1.5 inches by 12 months we gets 0.125 inches growth monthly. So a water oak tree needs only 8 months to grow 1 inch in diameter.
Janet Woo decided to retire to Florida in 6 years. What amount should Janet invest today so she can withdraw $51,500 at the end of each year for 20 years after she retires? Assume Janet can invest money at 6% compounded annually. (Use the Table 13.2 and Table 12.3.) Present value ??
Answer:
$293,562.707
Step-by-step explanation:
As for the provided details we know,
Janet needs $51,500 from end of 7th year for upcoming 20 years.
The present value of 20 installments of $51,500 shall be @ 6% from year 7 to year 8.0858
Thus total value = $51,500 [tex]\times[/tex] 8.0858 = $416,418.7
Now the compound interest factor for 6 year @ 6 % = 1.4185
Thus, value to be invested today = $416,418.70/1.4185 = $293,562.707
As this when compounded annually will provide the balance as required at the end of 6 years.
Consider the two functions:
f(x,a) =70- 5xa & g(x) = 30 +4x
a) find the x value of the point where the two equations intersect (in terms of the variable a)
b) Find the value of the functions at the point where the two equations intersect (in terms of the variable a).
c) Take the partial derivate of f with respect to x, \partial f / \partial x, and with respect to a, \partial f / \partial a
d) What are the values of these derivatives , when x= 3 and a=2, which can be written as \partialf /\partialx (3,2) and \partial f /\partiala (3,2)
e) Next caculate these two numbers:
\upsilon1 = (\partialf /\partialx ( 3,2)) 3 / f (3,2)
\upsilon2 = (\partialf /\partiala ( 3,2)) 2 / f (3,2)
f) Finally write out these equations in term of a and x and simplify,
\upsilon1 = (\partialf /\partialx ( x,a)) x / f (x,a)
\upsilon2 = (\partialf /\partialx ( x,a)) a / f (x,a)
Answer:
a) The x value of the point where the two equations intersect in terms of a is [tex]x=\frac{40}{4+5a}[/tex]
b) The value of the functions at the point where they intersect is [tex]\frac{10 (28 + 15 a)}{4 + 5 a}[/tex]
c) The partial derivative of f with respect to [tex]x[/tex] is [tex]\frac{\partial f}{\partial x} = -5a[/tex] and the partial derivative of f with respect to [tex]a[/tex] is [tex]\frac{\partial f}{\partial x} = -5x[/tex]
d) The value of [tex]\frac{\partial f}{\partial x}(3,2) = -10[/tex] and [tex]\frac{\partial f}{\partial a}(3,2) = -15[/tex]
e) [tex]\upsilon_1=-\frac{3}{4} = -0.75[/tex] and [tex]\upsilon_2=-\frac{3}{4} = -0.75[/tex]
f) equation [tex]\upsilon_1 = \frac{-5a\cdot x}{70-5ax}=\frac{ax}{ax-14}[/tex] and [tex]\upsilon_2 = \frac{-5a\cdot a}{70-5ax}=\frac{a^2}{ax-14}[/tex]
Step-by-step explanation:
a) In order to find the [tex]x[/tex] we just need to equal the equations and solve for [tex]x[/tex]:
[tex]f(x,a)=g(x)\\70-5xa = 30+4x\\70-30 = 4x+5xa\\40 = x(4+5a)\\\boxed {x = \frac{40}{4+5a}}[/tex]
b) Since we need to find the value of the function in the intersection point we just need to substitute the result from a) in one of the functions. As a sanity check , I will do it in both and the value (in terms of [tex]a[/tex]) must be the same.
[tex]f(x,a)=70-5ax\\f(\frac{40}{4+5a}, a) = 70-5\cdot a \cdot \frac{40}{4+5a}\\f(\frac{40}{4+5a}, a) = 70 - \frac{200a}{4+5a}\\f(\frac{40}{4+5a}, a) = \frac{70(4+5a) -200a}{4+5a}\\f(\frac{40}{4+5a}, a) =\frac{280+350a-200a}{4+5a}\\\boxed{ f(\frac{40}{4+5a}, a) =\frac{10(28+15a)}{4+5a}}[/tex]
and for [tex]g(x)[/tex]:
[tex]g(x)=30+4x\\g(\frac{40}{4+5a})=30+4\cdot \frac{40}{4+5a}\\g(\frac{40}{4+5a})=\frac{30(4+5a)+80}{4+5a}\\g(\frac{40}{4+5a})=\frac{120+150a+80}{4+5a}\\\boxed {g(\frac{40}{4+5a})=\frac{10(28+15a)}{4+5a}}[/tex]
c) [tex]\frac{\partial f}{\partial x} = (70-5xa)^{'}=70^{'} - \frac{\partial (5xa)}{\partial x}=0-5a\\\frac{\partial f}{\partial x} =-5a[/tex]
[tex]\frac{\partial f}{\partial a} = (70-5xa)^{'}=70^{'} - \frac{\partial (5xa)}{\partial a}=0-5x\\\frac{\partial f}{\partial a} =-5x[/tex]
d) Then evaluating:
[tex]\frac{\partial f}{\partial x} =-5a\\\frac{\partial f}{\partial x} =-5\cdot 2=-10[/tex]
[tex] \frac{\partial f}{\partial a} =-5x\\\frac{\partial f}{\partial a} =-5\cdot 3=-15[/tex]
e) Substituting the corresponding values:
[tex]\upsilon_1 = \frac{\partial f(3,2)}{\partial x}\cdot \frac{3}{f(3,2)} \\\upsilon_1 = -10 \cdot \frac{3}{40} = -\frac{3}{4} = -0.75[/tex]
[tex]\upsilon_2 = \frac{\partial f(3,2)}{\partial a}\cdot \frac{3}{f(3,2)} \\\upsilon_2 = -15 \cdot \frac{2}{40} = -\frac{3}{4} = -0.75[/tex]
f) Writing the equations:
[tex]\upsilon_1=\frac{\partial f (x,a)}{\partial x}\cdot \frac{x}{f(x,a)}\\\upsilon_1=-5a\cdot \frac{x}{70-5xa}\\\upsilon_1=\frac{-5ax}{70-5ax}=\frac{-5ax}{-5(ax-14)}\\\boxed{\upsilon_1=\frac{ax}{ax-14} }[/tex]
[tex]\upsilon_2=\frac{\partial f (x,a)}{\partial x}\cdot \frac{a}{f(x,a)}\\\upsilon_2=-5a\cdot \frac{a}{70-5xa}\\\upsilon_2=\frac{-5a^2}{70-5ax}=\frac{-5a^2}{-5(ax-14)}\\\boxed{\upsilon_2=\frac{a^2}{ax-14} }[/tex]
How many ways can six of the letters of the word ALGORITHM be selected 8. How many ways can the letters of the word ALGORITHM be arranged in a be seated together in the row? and written in a row? row if the letters GOR must remain together (in this order)?
The number of letters in word "ALGORITHM" = 9
The number of combinations to select r things from n things is given by :-
[tex]C(n,r)=\dfrac{n!}{r!(n-r)!}[/tex]
Now, the number of combinations to select 6 letters from 9 letters will be :-
[tex]C(9,8)=\dfrac{9!}{6!(9-6)!}=\dfrac{9\times8\times7\times6!}{6!\times3!}=84[/tex]
Thus , the number of ways can six of the letters of the word ALGORITHM=84
The number of ways to arrange n things in a row :[tex]n![/tex]
So, the number of ways can the letters of the word ALGORITHM be arranged in a be seated together in the row :-
[tex]9!=362880[/tex]
If GOR comes together, then we consider it as one letter, then the total number of letters will be = 1+6=7
Number of ways to arrange 9 letters if "GOR" comes together :-
[tex]7!=5040[/tex]
Thus, the number of ways to arrange 9 letters if "GOR" comes together=5040
The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure that lies within 3 standard deviations of the mean?
(A) 68%
(B) 95%
(C) 100%
(D) 99.7%
In a normally distributed scenario, roughly 99.7% of data falls within three standard deviations from the mean, therefore for this question where we need to determine the blood pressure within 3 standard deviations from the mean, the answer is 99.7%.
Explanation:
The topic under discussion pertains to statistics, particularly, the properties of a normal distribution. In a normally distributed dataset, the rule of three standard deviations states that approximately 99.7% of all data falls within three standard deviations from the mean. This is also known as the empirical rule or the 68-95-99.7 rule. Therefore, in this case, since we are asked about the percentage of 18-year-old women who have a systolic blood pressure that falls within three standard deviations of the mean, the answer is 99.7%, which corresponds to choice (D).
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A farmer has 96 feet of fence with which to make a corral. if he
arranges it iinto a rectangle that is twice as long as it is wide,
what are the dimensions?
Answer: The dimensions of rectangle are 32 feet and 16 feet.
Step-by-step explanation:
Let the width of rectangle be 'x'.
Let the length of rectangle be '2x'.
Perimeter of fence = 96 feet
As we know the formula for "Perimeter":
[tex]Perimeter=2(l+b)\\\\96=2(2x+x)\\\\\dfrac{96}{2}=3x\\\\48=3x\\\\x=\dfrac{48}{3}\\\\x=16\ ft[/tex]
Hence, the length of rectangle is 2x=2×16 = 32 feet and width is 16 feet.
Therefore, the dimensions of rectangle are 32 feet and 16 feet.
1,787 pages in 11 days = pages in 1 month
assuming 30 days per month.
[tex]\bf \begin{array}{ccll} pages&days\\ \cline{1-2} 1787&11\\ x&30 \end{array}\implies \cfrac{1787}{x}=\cfrac{11}{30}\implies 53610=11x \\\\\\ \cfrac{53610}{11}=x\implies 4873\frac{7}{11}=x[/tex]
Answer: 4873.64
Step-by-step explanation:
I'm assuming that you're asking how many pages there are in a month. On average, the typical month is 30 days, correct? We can plug this information into proportions.
1787/11 = x/30
1787 multiplied by 30 is 53610, and that divided by 11 would be 4873.64, when rounded to the nearest hundredth.
I hope that helped!
A piece of toast came out of the toaster very overcooked.
What kind of change occurred?
chemical change
change in reaction
phase change
physical change
Answer:
It is a chemical change ⇒ 1st answer
Step-by-step explanation:
* Lets explain the statements to solve the problem
- A chemical change occurs when a new substance is formed through
a chemical reaction
- Ex: cooking an egg
- Change of reaction is the rate of reaction it can be decreases or
increasing
- A phase change is a change from one state to another without a
change in chemical composition
- Ex: Condensation: the substance changes from a gas to a liquid
- A physical change, such as a state change or dissolving, but does
not create a new substance
- Ex: Breaking a glass
* Lets solve the problem
- A piece of toast came out of the toaster very overcooked.
∵ It is like the cooking an egg
∴ It is a chemical change
Answer:
Chemical
Step-by-step explanation:
The equation shows a number multipled by 8. n x 8 = Which is true about the product?
It is a factor of 8
It is a multiple of 8
It is a prime number
The product is a factor of n
Answer:
It is a multiple of 8
Step-by-step explanation:
The product may or may not be a factor of 8. We usually think of the factors of an integer as being positive integers, so the factors of 8 would be 1, 2, 4, or 8. If 8n is to be a factor of 8, then n must be 1/8, 1/4, 1/2, or 1. This will not be the case in general.
__
The product of 8 and any number is a multiple of 8. (Again, we usually think of a multiple of 8 as being an integer, which would require the number n to be an integer.)
__
No product of two (integer) numbers is a prime number. If 8n is to be a prime, then the value of n must be (some prime number)/8. Again, this will not be the case in general.
__
n is a factor of the product; not the other way around.
In the equation n x 8, the product is always a multiple of 8. The product is not necessarily a factor of 8, a prime number, or a factor of n.
Explanation:In the given equation, n x 8, the product of n and 8 is always a multiple of 8. This is because when we multiply any number by 8, the resulting product is included in the sequence of multiples of 8 (i.e., 8, 16, 24, 32, and so forth). Hence, regardless of the value of n, the product is always a multiple of 8. Note that the product is not necessarily a factor of 8, a prime number, or a factor of n, as these properties depend on the specific value of n.
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Pollsters are concerned about declining levels of cooperation among persons contacted in surveys. A pollster contacts 8686 people in the 18-21 age bracket and finds that 4343 of them respond and 4343 refuse to respond. When 276276 people in the 22-29 age bracket are contacted, 258258 respond and 1818 refuse to respond. Suppose that one of the 362362 people is randomly selected. Find the probability of getting someone in the 18 dash 2118-21 age bracket or someone who respondedresponded.
Answer:
0.9503
Step-by-step explanation:
First of all, there are some wrong figures in the original text. Because there is a total of 362362 people, the figures should be 86086 (people in the 18-21 age bracket), 43043 (people in the 18-21 age bracket who respond) and 43043 people in the 18-21 age bracket who refuse to respond. In the same way, because there are 276276 people in the 22-29 age bracket, it should be 18018 and not 1818 who refuse to respond in this subset of people. Now, let's define the following events:
R: a person respond
A: a person belongs to the 18-21 age bracket. So,
The number of people who respond is 43043 + 258258 = 301301, so
P(R) = 301301/362362 = 0.8315
P(A) = 86086/362362 = 0.2376
P(R | A) = 43043/86086 = 0.5
We are looking for P(A∪R) = P(A) + P(R) - P(A∩R),
P(A∩R) = P(R | A)P(A) = (0.5)(0.2376) = 0.1188, so,
P(A∪B) = 0.2376 + 0.8315 - 0.1188 = 0.9503
The recommended dose for a brand nicotine patch is one 21-mg dose per day for 6 weeks, followed by 14 mg per day for 2 weeks, and then 7 mg per day for 2 more weeks. What total quantity, in grams, would a patient receive during this course of treatment?
Answer:
1.176 grams
Step-by-step explanation:
Given:
Recommended dose
21 mg per day for 6 weeks
Now,
1 week = 7 days
Thus,
number of days in 6 weeks = 6 × 7 = 42 days
Therefore, the total dose = dose per days × number of days
= 21 × 42 = 882 mg
further,
14 mg per day for 2 weeks
Now,
1 week = 7 days
Thus,
number of days in 2 weeks = 2 × 7 = 14 days
Therefore, the total dose = dose per days × number of days
= 14 × 14 = 196 mg
further,
7 mg per day for 2 weeks
Now,
1 week = 7 days
Thus,
number of days in 6 weeks = 2 × 7 = 14 days
Therefore, the total dose = dose per days × number of days
= 7 × 14 = 98 mg
Hence, the total dose = 882 + 196 + 98 = 1176 mg
also,
1 g = 1000 mg
thus,
1176 mg = 1.176 grams
total quantity received during this course is 1.176 grams
A snorkeler dives for a shell on a reef. After entering the water, the diver decends 11/3 ft in one second. Write an equation that models the divers position with respect to time.
Answer:
[tex]h(t)=-\dfrac{11}{3}t[/tex]
Step-by-step explanation:
A snorkeler dives for a shell on a reef. After entering the water, the diver decends [tex]\frac{11}{3}[/tex] ft in one second.
Let t be the time passed after entering the water, in seconds, and h(t) be the position of the snorkeler under the water, in feet.
The initial position of the snorkeler was 0 feet under the water.
An equation that models the divers position with respect to time is
[tex]h(t)=0-\dfrac{11}{3}t\\ \\h(t)=-\dfrac{11}{3}t[/tex]
Here the position is negative, because the diver decends (he deepens under the water)
A housing development was occupied by 12 people. During January 2016, 6 more people move into the development. During each subsequent month, 4 more people move into the development than moved in during the previous month. Assuming nobody dies or moves out of the development, work out how many people occupy the development on 31 March 2017.
Answer: 30 people
Step-by-step explanation: So you start with 12 people
12+6= 18 because 6 more people moved in then an additional 12 move in because it doubles every month
The total number of people occupying the development on 31 March 2017 is 466.
To determine how many people occupy the development on 31 March 2017, we need to consider the pattern of people moving in each month since January 2016.
Initial Occupants: There were 12 people in the development as of January 2016.
Additional Occupants in January 2016: During January 2016, 6 more people moved in.
Total at the end of January 2016: 12 (initial) + 6 = 18 peopleMonthly Pattern: Each subsequent month, 4 more people move in than the previous month. So, we need to figure out this sequence from February 2016 to March 2017.
Calculating Monthly Increase:
February 2016: 6 (January) + 4 = 10 peopleMarch 2016: 10 (February) + 4 = 14 peopleApril 2016: 14 (March) + 4 = 18 peopleContinue this pattern for the remaining months.Sum of Monthly Increases: We need to determine the total number of new occupants from February 2016 to March 2017. This period includes 14 months.
The sequence of increases is: 6, 10, 14, 18, ..., up to March 2017.
This is an arithmetic sequence where the first term [tex]a = 6[/tex] and the common difference [tex]d = 4[/tex].
The [tex]n^{th}[/tex] term of an arithmetic sequence is given by: [tex]a_n = a + (n-1)d[/tex]
For March 2017 (14 months after January 2016): $a_{14} = 6 + (14-1) * 4 = 6 + 52 = 58 people.
Sum of an Arithmetic Sequence: The sum of the first [tex]n[/tex] terms of an arithmetic sequence is given by: [tex]S_n = \{n}{2} \times (2a + (n-1)d)[/tex]
Here, [tex]a = 6[/tex], [tex]d = 4[/tex], and [tex]n = 14[/tex].
[tex]S_{14} = \{14}{2} \times(2 \times 6 + (14-1) \times 4)[/tex]
[tex]S_{14} = 7 \times (12 + 52) = 7 \times 64 = 448[/tex] people moved in from February 2016 to March 2017.
Total Occupants at the End of March 2017:
Initial occupants: 12
Additional in January 2016: 6
New occupants from February 2016 to March 2017: 448
Total = 12 + 6 + 448 = 466 people
Q: Would like assistance in understanding and solving this example on Modern Algebra with the steps of the solution to better understand, thanks.
a) Determine the gcd(474,147) and write it as a linear combination of 174 and 147.
b) Prove by math induction that 2+4+6+...+2n = n(n+1) for all positive integers n.
Answer:
The gcd(474,147) = 3 and the linear combination is [tex]3=9\cdot 474 - 29\cdot 147[/tex] and the proof is below.
Step-by-step explanation:
The greatest common divisor (GCD) of two whole numbers is the largest natural number that divides evenly into both without a remainder.
To find the GCD you can use the Euclidean algorithm which is an efficient method for computing the greatest common divisor (GCD) of two integers, without explicitly factoring the two integers. Here is an outline of the steps:
Let a=x, b=yGiven x,y, use the division algorithm to write x=yq + rif r=0, stop and output y; this is the gcd of a,bif r ≠ 0, replace (x,t) by (y,r): Go to step 2To compute gcd(474,147), divide the larger number by the smaller number, using the division algorithm we have
[tex]\frac{474}{147} \\= 474-147=327\\327-147=180\\180-147=33\\[/tex]
At this point, we cannot subtract 147 again. Hence 3 is the quotient ( we subtract 147 from 474 3 times) and 33 is the remainder. We can express this as a linear combination [tex]474 = 147*3+33[/tex]
Using the same reasoning and the steps of the Euclidean algorithm we have
[tex]gcd(474,147) = \\474 =147\cdot 3+33\\147=33\cdot 4 +15\\33=15\cdot 2+3\\15=3\cdot 5+0[/tex]
To find the linear combination you need to use the Bezout's identity that says that the equation [tex]ax+by=gcd(a,b)[/tex] has solutions x, y. So we need to find the solution of the equation [tex]474x+147y=3[/tex].
To find the values of x and y you can run the Euclidean Algorithm backward.
We know that
[tex]33=15\cdot 2+3[/tex]
We can express 3 as linear combination
[tex]3=33- 2\cdot15\\3=33-2\cdot(147-33*4)=9\cdot 33 -2\cdot147\\3=9\cdot 33 -2\cdot147=9\cdot (474-147\cdot 3)-2 \cdot 147\\3= 9\cdot 474-27 \cdot 147-2 \cdot 147\\3=9\cdot 474 - 29\cdot 147[/tex]
The gcd(474,147) = 3 and the linear combination is [tex]3=9\cdot 474 - 29\cdot 147[/tex]
The principle of mathematical induction is stated as follows:
Let n be a natural number and let P(n) be an statement that depends on n. If
P(1) is true, andfor all positive integer k, P(k+1) can be shown to be true if P(k) is assumed to be true,Then P(n) is true for all natural numbers n.
Proposition: For all positive integers n, 2+4+6+...+2n = n(n+1).
Proof. Let's let P(n) be the statement "2+4+6+...+2n = n(n+1)" .The proof will now proceed in two steps: the initial step and the inductive step.
Initial step. We must verify that P(1) is true
[tex]n=1\\2\cdot 1=1\cdot (1+1)[/tex]
which is clearly true. So we are done with the initial step.
Inductive step. We must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true." Thus, we assume there is a k such that 2+4+6+...+2k = k(k+1), this is called the inductive assumption. We must prove, for this same k, the formula P(k+1): 2+4+6+...+2k+2(k+1) = (k+1)(k+2)
To prove that P(k+1) holds, we will start with the expression on the left-hand side of P(k+1) and show that it is equal to the expression on the right-hand side.
[tex]2+4+6+...+2k+2(k+1)[/tex]
we know that [tex]2+4+6+...+2k+2(k+1)=k(k+1)[/tex] for the inductive assumption
[tex]k(k+1)+2(k+1)\\k^{2}+k+2k+2\\k^2+3k+2\\(k+1)(k+2)[/tex]
we see that the result [tex](k+1)(k+2)[/tex], is the expression on the right-hand side of P(k+1). Thus by mathematical induction P(n) is true for all natural numbers n.
Write x'" = x + t as a first order system
Answer:
y = x'
z = y'
z' = x + t
Step-by-step explanation:
Hi!
You need to define two new variables y and z:
y = x'
z = y'
Then:
z = y' = x''
z' = x''' = x + t
Now you have a system of 3 equations with only first derivatives
Express the following relations in the set builder notation. Then, determine whether it is reflexive, symmetric, transitive. Please show work.
a.) One number is less than or equal to another.
b.) One integer is a factor of another.
c.) Two integers are unequal.
d.) One set is a subset of another.
Answer:
a)Reflexive, not symmetric, transitive
b)Reflexive, not symmetric, transitive
c)Not reflexive, symmetric, not transitive
d)Reflexive, not symmetric, transitive
Step-by-step explanation:
a)
[tex]R=\left \{ (a,b)\epsilon \mathbb{R} \times \mathbb{R} \mid a \leq b\right \}[/tex]
The relation R is reflexive for
[tex]a\leq a[/tex] for every real number a
it is not symmetric because 0 is less than 1, but 1 is not less than 0
it is transitive
[tex]a\leq[/tex] and [tex] b\leq c\Rightarrow a\leq c[/tex]
So if aRb and bRc, then aRc
b)
[tex]R=\left \{ (m,n)\epsilon \mathbb{Z} \times \mathbb{Z} \mid \exists k\in \mathbb{Z} \ni m=kn \right \}[/tex]
R is reflexive because m=1.m for every integer m
R is not symmetric: 2 is a factor of 4, but 4 is not a factor of 2
R is transitive: if mRn and nRp if m=kn and n=qp, so m=(kq)p and kq is an integer , so mRp
c)
[tex]R=\left \{ (m,n)\epsilon \mathbb{Z} \times \mathbb{Z} \mid m\neq n\right \}[/tex]
R is obviously not reflexive because all numbers equals themselves
R is symmetric: if a different to b, then b different to a
R is not transitive: 1R2 and 2R1 (because 1 different to 2), but 1 = 1
d)
[tex]R=\left \{ A,B\mid A\subseteq B \right \}[/tex]
R is reflexive for every set A is a subset of itself
R is not symmetric {1,2} is a subset of {1,2,3} but {1,2,3} is not a subset of {1,2}
R is transitive: if A is subset of B and B is subset of C, then A is subset of C
Suppose a manufacturer sells a product as $2 per unit. If q units are sold, (a) write the total revenue function, (b) and find the marginal revenue function. What does the constant marginal revenue function mean?
Answer:
We are given that a manufacturer sells a product as $2 per unit.
Quantity = q units
So, Total revenue = [tex]\text{Cost per unit} \times quantity[/tex]
Total revenue = [tex]2q[/tex]
So, the total revenue function is [tex]2q[/tex]
Marginal revenue is the derivative of the revenue functions
So, Marginal revenue = [tex]\frac{dR}{dq} =2[/tex]
The marginal revenue function is 2
The constant marginal revenue function mean that the revenue earned by the addition of the output is constant.
Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. a(t) =cos(πt), v(0)=55, s(0)=77 The velocity is v(t)= ____. (Type an exact answer.)
The initial position has no effect on the velocity, so you can ignore that value (unless there's another part to the question not included, of course).
We have
[tex]v(t)=v(0)+\displaystyle\int_0^t a(u)\,\mathrm du[/tex]
[tex]v(t)=55+\int_0^t\cos(\pi u)\,\mathrm du[/tex]
[tex]\boxed{v(t)=55+\dfrac1\pi\sin(\pi t)}[/tex]
The velocity of an object given the acceleration function a(t) = cos(πt) and an initial velocity of v(0) = 55 is found by integrating the acceleration function. This gives v(t) = (1/π)sin(πt) + 55.
Explanation:The object's acceleration, velocity, and position can be determined using principles of calculus. The acceleration function is given as a(t) = cos(πt). We find the velocity by integrating the acceleration function. Therefore, v(t) = ∫a(t) dt = ∫cos(πt) dt. Using fundamental calculus principles, the integral of cos(πt) with respect to time (t) is (1/π)sin(πt).
However, the initial velocity is provided as v(0) = 55. To account for this initial condition, we add this known velocity to our integral, giving us v(t) = (1/π)sin(πt) + 55. Thus, the velocity of the object at any time t is given by v(t) = (1/π)sin(πt) + 55.
Learn more about Kinematics here:https://brainly.com/question/35140938
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If the mass of your father is 70kg, what is his weight (N)?
Answer: 686 N
Step-by-step explanation:
Hi!
Second Newton's law is: F=m*a, where F is force, m is mass, and a acceleration
On the Earth's surface, weight is the gravity force W=m*g, where g=9.8 m/s² is the acceleretion of gravity on Earth. So the weight of someone with mass m=70 kg is W=70*9.8 kg*m/s² = 686 N.
The unit N (Newton) is defined as 1 N = 1 kg*m/s²
Which of the following angles have equal measure when a pair of parallel lines are crossed by a transversal?
supplementary angles
complementary angles
corresponding angles
adjacent angles
Answer:
corresponding angles
Step-by-step explanation:
Corresponding angles are congruent where a transversal crosses parallel lines. Such a geometry has 4 pairs of corresponding angles. The corresponding angles of each pair are congruent.
Answer:
corresponding angles
Step-by-step explanation:
Suppose you buy a new car whose advertised mileage is 20 miles per gallon (mpg). After driving your car for several months, you find that its mileage is 16.4 mpg. You telephone the manufacturer and learn that the standard deviation of gas mileages for all cars of the model you bought is 1.14 mpg. a. Find the z-score for the gas mileage of your car, assuming the advertised claim is correct. b. Does it appear that your car is getting unusually low gas mileage? a. zequals nothing (Round to two decimal places as needed.) b. Does it appear that your car is getting unusually low gas mileage? Yes No
Answer:
a) The z-score for the mileage of the car is -3.16
b) It appears that the car is getting unusually low gas mileage.
Step-by-step explanation:
The z-score formula is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which: X is the mileage per gallon we are going to find the z-score of, [tex]\mu[/tex] is the mean value of this mileage and [tex]\sigma[/tex] is the standard deviation of this value.
a. Find the z-score for the gas mileage of your car, assuming the advertised claim is correct.
The gas mileage for you car is 16.4 mpg, so [tex]X = 16.4[/tex]
The advertised gas mileage is 20 mpg, so [tex]\mu = 20[/tex]
The standard deviation is 1.14 mpg, so [tex]\sigma = 1.14[/tex]
The z-score is:
[tex]Z = \frac{X - \mu}{\sigma} = \frac{16.4 - 20}{1.14} = -3.16[/tex]
b. Does it appear that your car is getting unusually low gas mileage?
The general rule is that a z-score lower than -1.96 is unusually low. So yes, it appears that the car is getting unusually low gas mileage.
To find the z-score for the gas mileage of your car, use the formula z = (x - μ) / σ. A z-score of -3.16 indicates that your car is getting unusually low gas mileage as it is more than 3 standard deviations below the mean.
Explanation:To find the z-score for the gas mileage of your car, we can use the formula:
z = (x - μ) / σ
where x is the observed mileage, μ is the mean mileage, and σ is the standard deviation.
In this case, since the advertised mileage is 20 mpg, we have:
z = (16.4 - 20) / 1.14 = -3.16
For part b, a z-score of -3.16 indicates that your car is getting unusually low gas mileage as it is more than 3 standard deviations below the mean. Therefore, the answer is Yes.
State the chain rule for the derivative dy/dt if y(t)=f(u(t))(chain of f and u)
Answer:
[tex]\displaystyle\frac{d(y(t))}{dt} =\displaystyle\frac{d(f(u(t)))}{dt} = f'(u(t)).u'(t)[/tex]
Step-by-step explanation:
The chain rule helps us to differentiate functions and a composition of two functions.
Let r(u) and s(u) be two function. Then, composition of these two functions can be be differentiated with the help of chain rule. It states that:
[tex]\displaystyle\frac{d(r(s(u)))}{du} = r'(g(u)).s'(u)[/tex]
Now, we are given
[tex]y(t) = f(u(t))[/tex]
Then, by chain rule, we have:
[tex]\displaystyle\frac{d(y(t))}{dt} =\displaystyle\frac{d(f(u(t)))}{dt} = f'(u(t)).u'(t)[/tex]
which expression has the greatest value |-21|, |14|, |30|, |-45|
Answer:
|-45|
Step-by-step explanation:
In mathematics, the absolute value of a real number is the numeric value of the number, regardless the sign, either this is positive or negative.
The absolute value function can be definied as:
|a|=a si a ≥0|a|=-a si a <0Using this definition, we have:
|-21| = -(-21) = 21
|14| = 14
|30| = 30
|-45| = -(-45) = 45
Therefore, the expression |-45| has the greatest value.
Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual
has a Visa credit card and B be the analogous event for a MasterCard. Suppose that , P(A)= 0.6 and P(B)=0.4.
a. Could it be the case that P( A ∩ B )=0.5, why or why not?
b. From now on, suppose that P( A ∩ B )=0.3 What is the probability that student has one of these two types of cards?
c. What is the probability that the selected student has neither type of card?
d. Describe in terms of A and B the event that the select student has a visa card, but not a mastercard? and then calulate the probability of this event.
e. Calcuate th probability that the selected student has exactly one of the two types of cards?
Answer:
(a) P( A ∩ B )=0.5 is not possible.
(b) 0.7
(c) 0.3
(d) 0.3
(e) 0.4
Step-by-step explanation:
Given information: The alphabet A and B represents the following events
A : Individual has a Visa credit card.
B: Individual has a MasterCard.
P(A)= 0.6 and P(B)=0.4.
(a)
We need to check whether P( A ∩ B ) can be 0.5 or not.
[tex]A\cap B\subset A[/tex] and [tex]A\cap B\subset B[/tex]
[tex]P(A\cap B)\leq P(A)[/tex] and [tex]P(A\cap B)\leq P(B)[/tex]
[tex]P(A\cap B)\leq 0.6[/tex] and [tex]P(A\cap B)\leq 0.4[/tex]
From these two inequalities we conclude that
[tex]P(A\cap B)\leq 0.4[/tex]
Therefore, P( A ∩ B )=0.5 is not possible.
(b)
Let [tex]P(A\cap B)=0.3[/tex]
We need to find the probability that student has one of these two types of cards.
[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]
Substitute the given values.
[tex]P(A\cup B)=0.6+0.4-0.3=0.7[/tex]
Therefore the probability that student has one of these two types of cards is 0.7.
(c)
We need to find the probability that the selected student has neither type of card.
[tex]P(A'\cup B')=1-P(A\cup B)[/tex]
[tex]P(A'\cup B')=1-0.7=0.3[/tex]
Therefore the probability that the selected student has neither type of card is 0.3.
(d)
The event that the select student has a visa card, but not a mastercard is defined as
[tex]A-B[/tex]
It can also written as
[tex]A\cap B'[/tex]
The probability of this event is
[tex]P(A\cap B')=P(A)-P(A\cap B)[/tex]
[tex]P(A\cap B')=0.6-0.3=0.3[/tex]
Therefore the probability that the select student has a visa card, but not a mastercard is 0.3.
(e)
We need to find the probability that the selected student has exactly one of the two types of cards.
[tex]P(A\cap B')+P(A\cap B')=P(A\cup B)-P(A\cap B)[/tex]
[tex]P(A\cap B')+P(A\cap B')=0.7-0.3[/tex]
[tex]P(A\cap B')+P(A\cap B')=0.4[/tex]
Therefore the probability that the selected student has exactly one of the two types of cards is 0.4.