Answer:
(1) [tex]y=(\frac{2}{a+bx})^3[/tex]
By differentiating w.r.t. x,
[tex]\frac{dy}{dx}=3(\frac{2}{a+bx})^2\times \frac{d}{dt}(\frac{2}{a+bx})[/tex]
[tex]=3(\frac{2}{a+bx})^2\times (-\frac{2}{(a+bx)^2})[/tex]
[tex]=-\frac{24}{(a+bx)^4}[/tex]
(2) [tex]y=(at^3-3bt)^3[/tex]
By differentiating w.r.t. t,
[tex]\frac{dy}{dt}=3(at^3-3bt)^2\times \frac{d}{dt}(at^3-3bt)[/tex]
[tex]=3(at^3-3bt)^2 (3at^2-3b)[/tex]
[tex]=9t^2(at^2-3b)^2(at^2-b)[/tex]
(3) [tex]y=(t^b)(e^\frac{b}{t})[/tex]
Differentiating w.r.t. t,
[tex]\frac{dy}{dt}=t^b\times \frac{d}{dt}(e^\frac{b}{t})+\frac{d}{dt}(t^b)\times e^\frac{b}{t}[/tex]
[tex]=t^b(e^\frac{b}{t})\times \frac{d}{dt}(\frac{b}{t}) + bt^{b-1}(e^\frac{b}{t})[/tex]
[tex]=t^be^\frac{b}{t}(-\frac{b}{t^2})+bt^{b-1}e^{\frac{b}{t}}[/tex]
(4) [tex]z = ax^2.sin (4x)[/tex]
Differentiating w.r.t. x,
[tex]\frac{dz}{dt}=ax^2\times \frac{d}{dx}(sin (4x))+sin (4x)\times \frac{d}{dx}(ax^2)[/tex]
[tex]=ax^2\times cos(4x).4+sin (4x)(2ax)[/tex]
[tex]=4ax^2cos (4x)+2ax sin (4x)[/tex]
Calculate:
3 pounds (lbs) =——grams (g)
Find the area of the triangle with the vertices (2,1), (10,-1),
and(-1,8).
Answer:
The area of triangle is 25 square units.
Step-by-step explanation:
Given information: Vertices of the triangle are (2,1), (10,-1), and (-1,8).
Formula for area of a triangle:
[tex]A=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]
The given vertices are (2,1), (10,-1), and (-1,8).
Using the above formula the area of triangle is
[tex]A=\frac{1}{2}|2(-1-8)+10(8-1)+(-1)(1-(-1))|[/tex]
[tex]A=\frac{1}{2}|2(-9)+10(7)+(-1)(1+1)|[/tex]
[tex]A=\frac{1}{2}|-18+70-2|[/tex]
On further simplification we get
[tex]A=\frac{1}{2}|50|[/tex]
[tex]A=\frac{1}{2}(50)[/tex]
[tex]A=25[/tex]
Therefore the area of triangle is 25 square units.
Prove or disprove that the intersection of any collection
ofclosed sets is closed.
Answer:
Intersection of collection of any closed set is a closed set.
Step-by-step explanation:
Let F be a collection of arbitrary closed sets and let [tex]B_i[/tex] be closed set belonging to F.
We define a closed set as the set that contains its limit point or in other words it can be described that the complement or not of a closed set is an open set.
Thus, we can write R as
[tex]R =\bigcap\limits_{B_i \in F }^{} B_i[/tex]
Now, applying De-Morgan's Theorem, we have
[tex]R^c = (\bigcap\limits_{B_i \in F }^{} B_i)^c[/tex]
[tex]R^c = \bigcup\limits_{B_i \in F }^{} B_i^c[/tex]
Since we knew[tex]B_i[/tex] are closed set, thus, [tex]B_i^c[/tex] is an open set.
We also know that union of all open set is an open set.
Thus, [tex]R^c[/tex] is an open set.
Thus, R is a closed set.
Hence, the theorem.
The intersection of any collection of closed sets is indeed closed. This can be proven using the properties of complements in topology and by employing a contradiction approach where the assumption that the intersection is not closed leads to a contradiction, hence proving it must be closed.
The question addresses whether the intersection of any collection of closed sets is closed. In topology, a closed set is one where its complement (the elements not in the set) is open. One way to approach the proof is considering De Morgan's Laws, which in topology state that the intersection of closed sets is the complement of the union of their complements, which are open sets. Since the union of open sets is open, it follows that the complement (the intersection of our original sets) is closed.
For example, consider two closed intervals on the real line, A and B. The intersection, A B, would be the set of all points that are in both A and B. The fact that both A and B contain their boundary points ensures that their intersection also contains boundary points, maintaining closedness.
Proof strategy involves contradiction: assume the intersection is not closed. This would imply that its complement is not open, violating the definition that complements of closed sets are open, thus leading to a contradiction and proving the proposition is true.
A research company desires to know the mean consumption of meat per week among males over age 25. They believe that the meat consumption has a mean of 3.8 pounds, and want to construct a 85% confidence interval with a maximum error of 0.06 pounds. Assuming a standard deviation of 1.3 pounds, what is the minimum number of males over age 25 they must include in their sample? Round your answer up to the next integer.
To construct an 85% confidence interval with a maximum error of 0.06 pounds (with a standard deviation of 1.3 pounds), a sample size of 275 males over age 25 would be required.
Explanation:The subject of this question revolves around the mathematical concept of
confidence intervals
by utilizing the formula to find the appropriate sample size. As per the question about meat consumption among males over age 25, the company wish to construct an
85% confidence interval
with a maximum error of 0.06 pounds, with the standard deviation being 1.3 pounds. To find the minimum sample size necessary for the research, we'll have to use the following formula:
n=z²*σ²/E²
, where z is the z-score representative of the desired confidence level, σ is the known standard deviation, and E is the maximum acceptable error. Looking up the Z table or Z score table, 85% confidence corresponds to a z score of approximately 1.44. Substituting these values into the formula gives us n=1.44² * 1.3² / 0.06². This results to roughly 274.94, but since we can't have a fraction of a person, we round up to the nearest whole number, giving us a minimum sample size of 275.
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A classroom has ten students. Three students are freshman, two are sophomores, and five are juniors. Three students are randomly selected (without replacement) to participate in a survey. Consider the following events: A = Exactly 1 of the three selected is a freshman B = Exactly 2 of the three selected are juniors Find the following probability. If needed, round to FOUR decimal places. Pr(A∩B) = ___________
Answer:
0.25
Step-by-step explanation:
We have a total of ten student, and three students are randomly selected (without replacement) to participate in a survey. So, the total number of subsets of size 3 is given by 10C3=120.
On the other hand A=Exactly 1 of the three selected is a freshman. We have that three students are freshman in the classroom, we can form 3C1 different subsets of size 1 with the three freshman; besides B=Exactly 2 of the three selected are juniors, and five are juniors in the classroom. We can form 5C2 different subsets of size 2 with the five juniors. By the multiplication rule the number of different subsets of size 3 with exactly 1 freshman and 2 juniors is given by
(3C1)(5C2)=(3)(10)=30 and
Pr(A∩B)=30/120=0.25
Final answer:
To find the probability that exactly one freshman and exactly two juniors are selected, we calculate the combination of selecting one from three freshmen and two from five juniors, then divide by the total combinations of selecting three students from ten. The probability is 0.25.
Explanation:
We need to find the probability Pr(A∩B) where:
A is the event that exactly 1 of the three selected is a freshman.B is the event that exactly 2 of the three selected are juniors.For both events A and B to occur simultaneously, we must select one freshman and two juniors in our three student picks. The number of ways to choose one freshman out of three is C(3,1), and the number of ways to choose two juniors out of five is C(5,2). The total number of ways to choose any three students out of ten is C(10,3). Hence, the probability is:
Pr(A∩B) = (C(3,1) × C(5,2)) / C(10,3)
Calculating this gives:
Pr(A∩B) = (3 × 10) / 120 = 30 / 120
Pr(A∩B) = 0.25
Mathematics with applications in the management, Natural, and Social Sciences Twelfth edition
Chapter 6: Systems of Linear Equations and Matrices
6.1 Exercises
23.) According to Google Trends, popular interest in LED lightbulbs has been soaring. while interest in CFLs has been dropping. The following equations approximate the Google Trends rating (on a scale from 0-100) in year x, where x = 10 corresponds to the year 2010. ( Data from www.google.com/trends.)
LED: -25x + 6y = 20
CFL: 15x + 2y = 322
When did interest in ED lighting Surpass CFL lighting? Round your answer to the nearest year.
Answer: 2014
Step-by-step explanation:
LED: -25x + 6y = 20
CFL: 15x + 2y = 322
we need to find which year it was the same to know where LED lighting surpassed CFL lighting
-25x + 6y = 20
15x + 2y = 322 (-3)
-25x + 6y = 20
-45x - 6y = -966
-70x = -946
x = 13.51
The nearest year would be 14 which is 2014
By solving the system of equations, we find that the interest in ED lighting surpassed CFL lighting around the year 2014.
Explanation:To find out when the interest in ED lighting surpassed CFL lighting, we'll need to solve the system of equations given by:
LED: -25x + 6y = 20
CFL: 15x + 2y = 322
We can use substitution or elimination methods to solve this system of equations. For the elimination method, multiply the first equation by 2 and the second equation by 6 to make the y-coefficients equal:
-50x + 12y = 40
90x + 12y = 1932
Now, subtract the first equation from the second:
140x = 1892. Hence, x = 1892/140 = 13.51
So, the interest in ED lighting surpassed CFL lighting in the year 2013.51. Since we round to the nearest year, we can say that this happened around the year 2014.
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Find a general solution of y" + 4y = 0.
Answer:
[tex]y(x)=C_{1}cos2x+C_{2}sin(2x)[/tex]
Step-by-step explanation:
It is a linear homogeneous differential equation with constant coefficients:
y" + 4y = 0
Its characteristic equation:
r^2+4=0
r1=2i
r2=-2i
We use these roots in order to find the general solution:
[tex]y(x)=C_{1}cos2x+C_{2}sin(2x)[/tex]
the number of employees at a certain company is described by the
function P(t)= 300 (1.5)2t where t is the time in years.
how long does it take for the number of employess at this company
to increase by 10%.
Answer:
It will take 0.1175 years or the number of employees at this company to increase by 10%.
Step-by-step explanation:
We are given that the number of employees at a certain company is described by the function [tex]P(t)= 300 (1.5)^{2t}[/tex]
Initial no. of employees = 300
Increase% = 10%
So, New no. of employees = [tex]300+\frac{10}{100} \times 300[/tex]
= [tex]330[/tex]
Now we are supposed find how long does it take for the number of employees at this company to increase by 10%.
So, [tex]330= 300 (1.5)^{2t}[/tex]
[tex]\frac{330}{300}= (1.5)^{2t}[/tex]
[tex]1.1= (1.5)^{2t}[/tex]
[tex]t=0.1175[/tex]
So, it will take 0.1175 years or the number of employees at this company to increase by 10%.
For each function below, determine whether or not the function is injective and whether| or not the function is surjective. Be sure to justify your answers. (a) f : N -> N given by f(n) =n+ 2
(b) f P({1,2, 3}) -» N given by f(A) = |A| (Note: P(S) denotes the power set of a set S.)
Answer:
a) injective but not surjective. b) neither injective nor surjective.
Step-by-step explanation:
A function is injective if there aren't repeated images. To check if a function is injective we are going to suppose that for two values in the domain the image is equal, then we need to find that the two values are equal.
a) f : N → N with f(n) = n+2.
suppose that for n and m natural numbers, f(n)=f(m). Then
n+2 = m+2
n+2-2 = m
n = m.
Then, f(n) is injective.
Now, a function is surjective if every term m in the codomain there exists a pre-image of that element, that is to say, there exists an n such that f(n) = m. That is, the range of the function is equal to the codomain.
In this case, f(n) is not surjective. For example, if we would have that
n+2 = 1
n = 1-2
n = -1
but -1 is not a natural number, then for m=1 we don't have a pre image in f.
b) f: P({1, 2, 3}) → N with f(A) = |A| (amount of elements in the subset A).
Now, for {1,2,3} we can have subsets of 0, 1, 2 or 3 elements. Then, the range of the function f is {1, 2, 3} (0 is not included because 0 is not natural). Then, the range is not all the natural numbers and therefore the function is not surjective.
Now, let's check if f is injective. Let {1} and {2} subsets of {1, 2, 3}. Then
f({1}) = |{1}| = 1.
f ({2}) = |{2}| = 1.
We have two different subsets with the same image, then f is not injective.
The USS Enterprise was 1,123 feet in length.
A. What is the scale for a model that is 30 inches long?
B. What is the scale for a model that is 2 feet long?
Answer:
The scale for a model that is 30 inches long is 449.2 inches.
The scale for a model that is 2 feet long is 561.5 feet.
Step-by-step explanation:
Consider the provided information.
The USS Enterprise was 1,123 feet in length.
Part (A) What is the scale for a model that is 30 inches long?
1 feet = 12 inches.
First convert the length of USS enterprise in inches.
1×1,123 feet = 12×1,123 inches
1,123 feet = 13476 inches
Divide original length with the length of model.
The scale for the model is:
[tex]\frac{13476}{30}=449.2\ inches[/tex]
Hence, the scale for a model that is 30 inches long is 449.2 inches.
Part (B) What is the scale for a model that is 2 feet long?
Since, both the units are in feet, so simply divide original length with the length of model.
The scale for the model is:
[tex]\frac{1123}{2}=561.5\ feet[/tex]
Hence, the scale for a model that is 2 feet long is 561.5 feet.
Norgestrel and ethinyl estradiol tablets are available containing 0.5 mg of norgestrel and 50 μg of ethinyl estradiol. How many grams of each ingredient would be used in making 10,000 tablets?
Answer:
5g of norgestel are used in making 10,000 tablets.
0.5g of ethinyl estradiol are used in making 10,000 tablets.
Step-by-step explanation:
This problem can be solved as a rule of three problem.
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.
Unit conversion problems, like this one, is an example of a direct relationship between measures.
First step: Grams of norgestrel
Each tablet contais 0.5mg of norgestrel. How many miligrams are in 10,000 tablets?
1 tablet - 0.5 mg
10,000 tablets - x mg
x = 10,000*0.5
x = 5,000 mg
Now we have to convert 5,000 mg to g. Each g has 1,000 mg. So:
1g - 1,000 mg
xg - 5,000 mg
1,000x = 5,000
[tex]x = \frac{5,000}{1,000}[/tex]
x = 5g,
5g of norgestel are used in making 10,000 tablets.
Final step: Grams of ethinyl estradiol
50ug = 0.05 mg.
So
1 tablet - 0.05mg
10,000 tablets - xg
x = 10,000*0.05
x = 500 g
Now we have to convert 500 mg to g. Each g has 1,000 mg. So:
1g - 1,000 mg
xg - 500 mg
1,000x = 500
[tex]x = \frac{500}{1,000}[/tex]
x = 0.5g,
0.5g of ethinyl estradiol are used in making 10,000 tablets.
a. Every set has an element. b. The smallest perfect number is 28. c. Is 1.5 an irrational number? d. Please find the popular approximation of pi. e. Can you extract the root of -25?
Answer:
a) False. b) False. c) No, it's Rational. d) pi=355/113 e) 5i (for Complex Set of Numbers)
Step-by-step explanation:
a) Since there is the empty set. And an axiom assures us the existence of this Set. "There is a set such no element belongs to it"
∅ has no elements.
b) A perfect number is a positive integer equals to the sum of proper divisions
The smallest is 6. Since the proper divisors of 6={3,2,1}. 6=1+2+3
28 is a perfect number, but not the smallest. It is perfect since
28 proper divisors={14,7,4,2} 28=1+2+3+4+5+6+7
c) No, An Irrational number cannot be written as a fraction a/b where "a" and "b" are integers. 1.5 is a rational one, since 3/2 =1.5
d)
[tex]\pi[/tex]=22/7 -1/791= 355/113
e) Not for Real Numbers, since it is not defined for Real numbers. But for the set of Complex 5i
A restaurant sold 6 hamburgers every day for a week. How many hamburgers were sold during the week
Answer:
42 hamburgers
Step-by-step explanation:
Your answer is 42.
6 x 7 = 42
Find a formula for Y(t) with Y(0)=1 and draw its graph. What is Y\infty?
a. Y'+2Y=6
b. Y'+2Y=-6
Answer:
[tex](a)\ y(t)\ =\ -2e^{-2t}+3[/tex]
[tex](b)\ y(t)\ =\ 4e^{-2t}-3[/tex]
Step-by-step explanation:
(a) Given differential equation is
Y'+2Y=6
=>(D+2)y = 6
To find the complementary function, we will write
D+2=0
=> D = -2
So, the complementary function can be given by
[tex]y_c(t)\ =\ C.e^{-2t}[/tex]
To find the particular integral, we will write
[tex]y_p(t)\ =\ \dfrac{6}{D+2}[/tex]
[tex]=\ \dfrac{6.e^{0.t}}{D+2}[/tex]
[tex]=\ \dfrac{6}{0+2}[/tex]
= 3
so, the total solution can be given by
[tex]y_(t)\ =\ C.F+P.I[/tex]
[tex]=\ C.e^{-2t}\ +\ 3[/tex]
[tex]y_(0)=C.e^{-2.0}\ +\ 3[/tex]
but according to question
1 = C +3
=> C = -2
So, the complete solution can be given by
[tex]y_(t)\ =\ -2.e^{-2.t}\ +\ 3[/tex]
(b) Given differential equation is
Y'+2Y=-6
=>(D+2)y = -6
To find the complementary function, we will write
D+2=0
=> D = -2
So, the complementary function can be given by
[tex]y_c(t)\ =\ C.e^{-2t}[/tex]
To find the particular integral, we will write
[tex]y_p(t)\ =\ \dfrac{-6}{D+2}[/tex]
[tex]=\ \dfrac{-6.e^{0.t}}{D+2}[/tex]
[tex]=\ \dfrac{-6}{0+2}[/tex]
= -3
so, the total solution can be given by
[tex]y_(t)\ =\ C.F+P.I[/tex]
[tex]=\ C.e^{-2t}\ -\ 3[/tex]
[tex]y_(0)\ =C.e^{-2.0}\ -\ 3[/tex]
but according to question
1 = C -3
=> C = 4
So, the complete solution can be given by
[tex]y_(t)\ =\ 4.e^{-2.t}\ -3[/tex]
Which number is not the square of a whole number?
Select one:
a. 100
b. 400
c. 800
d. 144
Answer:
C. 800
Step-by-step explanation:
[tex]\sqrt{100} =10; \ \sqrt{400}=20; \ \sqrt{144}=12; \ \sqrt{800}=28.284271247461900976033774484194.[/tex]
At a newsstand, out of 46 customers, 27 bought the Daily News, 18 bought the Tribune, and 6 bought both papers.
Use a Venn diagram to answer the following questions:
How many customers bought only one paper?
How many customers bought something other than either of the two papers?
Answer:
[tex]\text{Customer that bought only one paper}=33[/tex]
[tex]\text{Customer that bought something other than either of the two papers}=7[/tex]
Step-by-step explanation:
We have been given that at a newsstand, out of 46 customers, 27 bought the Daily News, 18 bought the Tribune, and 6 bought both papers.
[tex]\text{Customer that bought only Daily news}=27-6[/tex]
[tex]\text{Customer that bought only Daily news}=21[/tex]
[tex]\text{Customer that bought only Tribune}=18-6[/tex]
[tex]\text{Customer that bought only Tribune}=12[/tex]
The customer that bought only one paper would be the sum of customers, who bought only Daily news or Tribune.
[tex]\text{Customer that bought only one paper}=21+12[/tex]
[tex]\text{Customer that bought only one paper}=33[/tex]
Therefore, 33 customers bought only one paper.
[tex]\text{Customer that bought something other than either of the two papers}=46-(27+18-6)[/tex]
[tex]\text{Customer that bought something other than either of the two papers}=46-(45-6)[/tex]
[tex]\text{Customer that bought something other than either of the two papers}=46-39[/tex]
[tex]\text{Customer that bought something other than either of the two papers}=7[/tex]
Therefore, 7 customers bought something other than either of the two papers.
An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A) = 0.5 and P(B) = 0.9..(a) If the Asian project is not successful, what is the probability that the European project is also not successful?______Explain your reasoning.(A) Since the events are independent, then A' and B' are independent, too.(B) Since the events are independent, then A' and B' are mutually exclusive.(C) Since the events are not independent, then A' and B' are mutually exclusive.(D) Since the events are independent, then A' and B' are not independent.
Answer:
The probability that the European proyect is not succesfull is 0.1
Step-by-step explanation:
Since A and B are independent, A' and B' are independent too.
The probability of B' is P(B')=1-P(B) = 1 - 0.9 = 0.1
Prove that if n or m is an odd integer, then n*m is an even integer. Proposed proof: Suppose that n*m is odd. Then n*m = 2k + 1 for some integer k. Therefore, n or m must be odd.
Answer: Ok, we have two numbers, and one of them is an odd integer, and the other is even.
Lets call M to the odd integer and N the even.
We know that a even integer can be written as 2k, where k is a random integer, and a odd integer can be written as 2j + 1, where j is also a random integer.
then M = 2k, N= 2j+1
then the product of M and N is: M*N = 2*k*(2*j + 1) = 2*(k*2*j + k)
is obvious to see that (k*2*J + k) is a integer, because k and j are integers.
then if we call g = ( k*2*J + k), we can write M*N=2g, and we already know that this is an even number. So M*N is a even integer.
Use Gauss's approach to find the following sums (do not use formulas) a 1+2+3+4 998 b. 1+3+5 7+ 1001 a The sum of the sequence is
Answer:
(a) 498501
(b) 251001
Step-by-step explanation:
According Gauss's approach, the sum of a series is
[tex]sum=\frac{n(a_1+a_n)}{2}[/tex] .... (1)
where, n is number of terms.
(a)
The given series is
1+2+3+4+...+998
here,
[tex]a_1=1[/tex]
[tex]a_n=998[/tex]
[tex]n=998[/tex]
Substitute [tex]a_1=1[/tex], [tex]a_n=998[/tex] and [tex]n=998[/tex] in equation (1).
[tex]sum=\frac{998(1+998)}{2}[/tex]
[tex]sum=499(999)[/tex]
[tex]sum=498501[/tex]
Therefore the sum of series is 498501.
(b)
The given series is
1+3+5+7+...+ 1001
The given series is the sum of dd natural numbers.
In 1001 natural numbers 500 are even numbers and 501 are odd number because alternative numbers are even.
[tex]a_1=1[/tex]
[tex]a_n=1001[/tex]
[tex]n=501[/tex]
Substitute [tex]a_1=1[/tex], [tex]a_n=1001[/tex] and [tex]n=501[/tex] in equation (1).
[tex]sum=\frac{501(1+1001)}{2}[/tex]
[tex]sum=\frac{501(1002)}{2}[/tex]
[tex]sum=501(501)[/tex]
[tex]sum=251001[/tex]
Therefore the sum of series is 251001.
To find the sum of the sequences using Gauss's approach, we create pairs from the sequence that each have the same sum and then multiply the number of pairs by this common sum. For 1 to 998, this results in 499 pairs each summing to 999. For 1, 3, 5, ... to 1001, there are 501 pairs each summing to 1002.
The student is asking how to find the sum of two sequences using Gauss's approach, which does not involve the use of formulas. This approach, also known as Gauss's trick, involves pairing numbers from opposite ends of a sequence and then multiplying the number of pairs by the common sum of each pair to find the total sum.
Let's illustrate this for the sequences given:
For the sequence 1, 2, 3, ..., 998, we pair the first and last numbers (1 and 998), the second and second-to-last numbers (2 and 997), and so on until we reach the middle of the list. Each pair sums up to 999. Since there are 998 numbers in total, there will be 998/2 = 499 pairs. The sum of the sequence is 499 * 999.
For the sequence 1, 3, 5, ..., 1001, we recognize that this is an arithmetic series with a common difference of 2. We can pair the first and last terms (1 and 1001) to get a sum of 1002. Since the sequence has (1001-1)/2 + 1 terms, we will have (1000/2) + 1 = 501 pairs. Thus, the sum of the sequence is 501 * 1002.
Gauss's approach to summing an arithmetic series can be visualized by considering the example of summing the first n natural numbers, which results in the formula (n² + n)/2.
fill in the missing number 8×____=4×8
Answer:
4
Step-by-step explanation:
8 *4 is the same as 4*8; commutative property
Answer:
[tex]4[/tex]
Step-by-step explanation:
8 x 4 is exactly the same as 4 x 8, both equaling 32.
No matter how insert 4 and 8, it will always be the same
[tex]x \times y = y \times x [/tex]
^^^
Arithmetic Modular Composite Numbers (4 marks). Carry out the following calcula- tions by hand by using the Chinese Remainder Theorem to split each operation into two operations modulo smaller numbers. You must show your work to receive full credit (a) 23 x 36 mod 55 (b) 29 x 51 mod 91
Answer:
a) 23 x 36 (mod 55) = 3 (mod 55)
b) 23 x 36 (mod 55) = 23 (mod 91)
Step-by-step explanation:
The Chinese Remainder Theorem lets us split a composite modulo into its prime components and solve for smaller numbers.
a) Using the Chinese Remainder Theorem, we have that 55 = 11 x 5
Since 11 and 5 are relatively prime numbers, we can use the Theorem and rewrite 23 x 36 mod 55 as: 23 x 36 (mod 11) and 23 x 36 (mod 5).
First we will work with 23 x 36 (mod 11)
[tex](23)(36)(mod 11) = (1)(3) (mod 11)[/tex] (Since 23 is congruent to 1 modulo 11 and 36 is congruent to 3 modulo 11)
Now we do the same with 23 x 36 (mod 5)
[tex](23)(36) (mod 5) = (3)(1) (mod5) = 3 (mod 5)[/tex]
Now we will use the Chinese Remainder Theorem to solve this pair of equations:
x = 3 (mod 11) and x = 3(mod 5)
[tex]x=5y+3\\5y+3=3(mod 11)\\5y=0(mod 11)\\y=0 (mod 11)\\y=11z\\x=5(11z)+3\\x=55z + 3\\x=3(mod 55)\\[/tex]
b) We are going to use the same procedure from a)
91 = 13 x 7
29 x 51 (mod 91) = 29 x 51 (mod 13) and 29 x 51 (mod 7)
29 x 51 (mod 13) = 3 x 12 (mod 13) = 36 (mod 13) = 10 (mod 13)
29 x 51 (mod 7) = 1 x 2 (mod 7) = 2 (mod 7)
Our pair of equations is x = 10 (mod 13) and x = 2 (mod 7)
[tex]x= 7y + 2\\7y + 2 = 10(mod13)\\7y= 8(mod13)\\y= 3 (mod 13)\\y=13y+3\\x=7(13y+3) + 2\\x=91y +21+2\\x=91y+23\\x= 23 (mod 91)[/tex]
At a large department store, the average number of years of employment for a cashier is 5.7 with a standard deviation of 1.8 years. If the number of years of employment at this department store is normally distributed, what is the probability that a cashier selected at random has worked at the store for over 10 years?
Answer: 0.0019
Step-by-step explanation:
Let x be the random variable that represents the number of years of employment at this department store.
Given : The number of years of employment at this department store is normally distributed,
Population mean : [tex]\mu=5.7[/tex]
Standard deviation : [tex]\sigma=1.8[/tex]
Z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]
Now, the z-value corresponding to 10 : [tex]z=\dfrac{10-5.7}{1.8}\approx2.39[/tex]
P-value = [tex]P(x>10)=P(Z>2.89)=1-P(z\leq2.89)[/tex]
[tex]=1-0.9980737=0.0019263\approx0.0019\text{ (Rounded to nearest ten thousandth)}[/tex]
Hence, the probability that a cashier selected at random has worked at the store for over 10 years = 0.0019
Linda enrolls for 10 credit hours for each two semsters at a cost of $600 per credit hour. in addition textbooks $400 per semster.
Linda's total cost for college for this academic year is calculated as $12,800, encompassing both tuition and textbooks. This is part of an observed trend in increasing higher education costs.
Explanation:Cost of Tuition: Linda is paying $600 per credit hour for 20 credit hours (10 each semester), so $600 * 20 = $12,000 in total for tuition.
Cost of Textbooks: She is also spending $400 per semester for textbooks, so $400 * 2 = $800 in total for textbooks.
Total Cost: Thus, Linda's total cost for the academic year would be the sum of these two costs, i.e, $12,000 + $800 = $12,800. The costs of tuition, textbooks and other expenses are part of the rising trend of higher education costs. Despite this, the value of education still remains high, as it can lead to better job prospects and higher earning potential in the future.
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Write the prime factorization of the number. 18,234
Answer:
[tex]18234=2\times 3\times 3\times 1013[/tex]
Step-by-step explanation:
We are given that a number 18234
We have to find the prime factorization of the number
Prime factorization : The number written is in the product of prime numbers is called prime factorization.
In order to find the prime factorization we will find the factors of given number
[tex]18234=2\times 3\times 3\times 1013[/tex]
Hence, the prime factorization of [tex]18234=2\times 3\times 3\times 1013[/tex]
9 + 22 = x + 1
HALPP
Hey!
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Solution:
9 + 22 = x + 1
9 + 22 - x = x + 1 - x
31 - x = 1
31 - x 31 = 31 - 1
x = 30
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Answer:
x = 30
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Hope This Helped! Good Luck!
Answer:
x = 30
Step-by-step explanation:
9 + 22 = x + 1
9 + 22 = 31
31 = x + 1
-1 -1
30 = x
x = 30
In 1970 the male incarceration rate in the U.S. was approximately 190 inmates per 100,000 population. In 2008 the rate was 960 inmates per 100,000 population. What is the percent increase in the male incarceration rate during this period?
Answer:
405.26%
Step-by-step explanation:
We have been given that in 1970 the male incarceration rate in the U.S. was approximately 190 inmates per 100,000 population. In 2008 the rate was 960 inmates per 100,000 population.
[tex]\text{Percent increase}=\frac{\text{Final amount}-\text{Initial amount}}{\text{Initial amount}}\times 100[/tex]
[tex]\text{Percent increase}=\frac{960-190}{190}\times 100[/tex]
[tex]\text{Percent increase}=\frac{770}{190}\times 100[/tex]
[tex]\text{Percent increase}=4.052631578947\times 100[/tex]
[tex]\text{Percent increase}=405.2631578947\%[/tex]
[tex]\text{Percent increase}\approx 405.26\%[/tex]
Therefore, the percent increase in the male incarceration rate during the given period is 405.26%.
Find the missing length to the nearest tenth of a meter of the
right triangle. One side is 1.4m and the other is 3.1. What is the
third side?
Answer:
length of the third side may be either 3.4 m or 2.8 m.
Step-by-step explanation:
Measure of two sides of a right triangle are 1.4 m and the other side is 3.1 m.
then we have to find the measure of third side.
If the third side of the triangle is its hypotenuse then
(Third side)² = (1.4)² + (3.1)²
Third side = [tex]\sqrt{(1.4)^{2}+(3.1)^{2}}[/tex]
= [tex]\sqrt{1.96+9.61}[/tex]
= [tex]\sqrt{11.57}[/tex]
= 3.4 m
If the third side is one of the perpendicular sides of the triangle and 3.1 m is hypotenuse, then
(Third side)² + (1.4)² = (3.1)²
(Third side)²= (3.1)² - (1.4)²
Third side = [tex]\sqrt{(3.1)^{2}-(1.4)^{2}}[/tex]
= [tex]\sqrt{9.61-1.96}[/tex]
= [tex]\sqrt{7.65}[/tex]
= 2.76 m
≈ 2.8 m
Therefore, length of the third side may be either 3.4 m or 2.8 m.
At Tech High there are three math teachers, Mary, Tom, and Alex. All students at Tech High take math, with 21% taking class with Mary, 15% with Tom, and 64% taking class with Alex. Every year 4% of Mary’s students fail, 6% of Tom’s students fail, and 13% of Alex’s student fail. What is the probability that a Tech High student fails math? If needed, round to FOUR decimal places.
Answer:
The probability that a Tech High student fails math is 0.1006
Step-by-step explanation:
21% students are taking class with Mary.
Every year 4% of Mary’s students fail,
15% students are taking class with Tom
Every year 6% of Tom’s students fail.
64% students are taking class with Alex
Every year 13% of Alex's students fail.
Now we are supposed to find the probability that a Tech High student fails math.
[tex]P(\text{Tech High student fails math}) = P(\text{taking class with Mary})\times P(\text{Mary students fail})+P(\text{taking class with Tom}) \times P(\text{Tom students fail})+P(\text{taking class with Alex}) \times P(\text{Alex student fail})[/tex]
[tex]P(\text{Tech High student fails math}) = 0.21 \times 0.04+0.15 \times 0.06+0.64 \times 0.13[/tex]
[tex]P(\text{Tech High student fails math}) =0.1006[/tex]
Hence the probability that a Tech High student fails math is 0.1006
A solution for direct IV bolus injection contains 125 mg of drug in each 25 mL of injection. What is the concentration of drug in terms of μg/μL?
Answer:
The concentration of the drug is 5ug/uL
Step-by-step explanation:
The first step of the problem is the conversion of the quantities of the drug in mg and mL to ug and uL.
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.
Unit conversion problems, like this one, is an example of a direct relationship between measures.
First step: Conversion of 125mg to ug
Each mg has 1,000ug. So:
1mg - 1,000ug
125mg - xug
x = 1,000*125
x = 125,000 ug
Second step: Conversion of 25 mL to uL
Each mL has 1,000uL. So:
1mL - 1,000uL
25mL - x uL
x = 25*1,000
x = 25,000uL
Concentration:
[tex]C = \frac{125,000 ug}{25,000uL} = 5ug/uL[/tex]
The concentration of the drug is 5ug/uL
The concentration of the drug in terms of μg/μL is 5 μg/μL
The given parameters are:
125 mg of drug in each 25 mL
The concentration (k) of the drug is then calculated as:
[tex]k = \frac{125 mg}{25mL}[/tex]
Divide
[tex]k = 5\frac{mg}{mL}[/tex]
The units of the injection and the drug is in mill-.
So, the concentration can be rewritten as:
[tex]k = 5\frac{\mu g}{\mu L}[/tex]
Hence, the concentration of the drug in terms of μg/μL is 5 μg/μL
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3. 36 percent of 18 is 18 percent of what number?
Final answer:
To find the number of which 18 percent is equal to 36 percent of 18, you set up the equation 0.36 × 18 = 0.18 × x and solve for x to find that x equals 36.
Explanation:
The student has asked: "36 percent of 18 is 18 percent of what number?" To solve this equation, we need to set it up as follows:
Let's assume the number we are searching for is x. Then, according to the question,
0.36 × 18 = 0.18 × x
Now, we can solve for x by dividing both sides of the equation by 0.18:
x = (0.36 × 18) / 0.18
After doing the calculation:
x = 6.48 / 0.18
x = 36
So, 36 percent of 18 is 18 percent of 36.