Answer:
For all n ≥ 4, 2n < n!
Step-by-step explanation:
Let's use the induction method to prove this statement.
In the induction method, first we prove the statement for n=4
1) If n = 4 ⇒2(4) < 4! ⇒2(4) < 24 ⇒8 < 24.
Therefore the statement holds for n=4
2) Now we assume that the statement is valid for n = k
⇒2k < k!
3) Now we will prove the statement holds for n = k +1
We will prove that 2(k + 1) < (k +1)!
(k + 1)! = (k+1) (k) (k-1) .... (3) (2) (1)
If the statement is valid for k + 1, then it would mean that
2 (k + 1) < (k+1) (k) (k-1) ... (3) (2) (1)
2 < (k) (k-1).... (3) (2) (1)
which is clearly true since k ≥4
Therefore the statement n ≥4, 2n < n! is true.
Using proof by induction, we establish the base case for n = 4 and then assume 2k < k! to show 2(k+1) < (k+1)! holding true for k ≥ 4, which proves the inequality for all n ≥ 4.
Explanation:To prove that 2n < n! for all n ≥ 4, we will use proof by induction.
We start by establishing the base case for n = 4, where we compute 2^4 = 16 and 4! = 24 which confirms that 16 < 24.
Now, assume the inequality 2k < k! is true for some k ≥ 4.
For our inductive step, we need to show that if 2k < k! then 2(k+1) < (k+1)!.
Multiplying both sides of our assumption by 2 gives us 2*2k < 2*k!, and since k ≥ 4, we know that 2 < k + 1.
Multiplying our original inductive assumption by (k + 1) gives us (k+1)*2k < (k+1)*k!, which simplifies to 2(k+1) < (k+1)! as required.
(b) What's the largest product possible from two numbers adding up to 100?
Answer:
2500
Step-by-step explanation:
We have to find the largest product of two numbers whose sum is 100.
Let the two numbers be x and y.
Thus, we can write x+y=100
We can calculate the value of y as:
y = 100 - x
The product of these number can be written as: (x)(y) = (x)(100-x) = 100x - x²
Let f(x) = 100x - x²
Now, the first derivative of this function with respect to x is
[tex]\frac{df(x)}{dx}[/tex] = 100-2x
Equating [tex]\frac{df(x)}{dx}[/tex] = 0, we get,
100-2x = 0
⇒ x = 50
Now, we find the second derivative of the the function f(x) with respect to x
[tex]\frac{d^2f(x)}{dx^2}[/tex] = -2
Since, [tex]\frac{d^2f(x)}{dx^2}[/tex] < 0, then by double derivative test the function have a local maxima at x = 50
This, x = 50 and y = 100-50 =50
Largest product = (50)(50) = 2500
SOLVE this plz!!! 300 Points!!!
Explain the distance formula. Than use it to calculate the distance between A(1, 1) and B(7, -7).
Graph each point on a coordinate plane. Name the quadrant in which each is located
14. D(1, -5)
13. C(-4,3)
12. B(-2, -3)
11. A(3,5)
Given f(x) = 4x - 1, evaluate each of the following:
15. f(-4)
16. f(0)
17. f(2)
Answer:
14. Quadrant IV
13. Quadrant II
12. Quadrant III
11. Quadrant I
15. -17
16. -1
17. 7
Step-by-step explanation:
Y=∛x -8 inverse of the function
Answer:
The inverse function of [tex]\sqrt{3]{x} - 8[/tex] is [tex](x+8)^{3}[/tex]
Step-by-step explanation:
Inverse of a function:
To find the inverse of a function [tex]y = f(x)[/tex], basically, we have to reverse r. We exchange y and x in their positions, and then we have to isolate y.
In your exercise:
[tex]y = \sqrt[3]{x} - 8[/tex]
Exchanging x and y, we have:
[tex]x = \sqrt[3]{y} - 8[/tex]
[tex]x + 8 = \sqrt[3]{y}[/tex]
Now we have to write y in function of x
[tex](x+8)^{3} = (\sqrt[3]{y})^{3}[/tex]
[tex]y = (x+8)^{3}[/tex]
So, the inverse function of [tex]\sqrt{3]{x} - 8[/tex] is [tex](x+8)^{3}[/tex]
Answer the following true or false. Justify your answer.
(a) If A is a subset of B, and x∈B, then x∈A.
(b) The set {(x,y) ∈ R2 | x > 0 and x < 0} is empty.
(c) If A and B are square matrices, then AB is also square.
(d) A and B are subsets of a set S, then A∩B and A∪B are also subsets of S.
(e) For a matrix A, we define A^2 = AA.
Answer:
a) False b) False c) True d) True e) True
Step-by-step explanation:
a) If A is a subset of B, and x∈B, then x∈A. False
Suppose A is Z (Set of Integers), and B is R (Set of Real Numbers). Then A is a subset of B. x ∈ B, let's say x equals π. If x ∈ B (B = Real Numbers) and x=π then x ∉ A (Z).
We could call A, any other subset of Real numbers (Q, I,..) and we both would come up to the same conclusion when it comes to Real numbers the Set.
So this conclusion is False. Not always an element of a subset is an element of a set.
b) False
For this one, I've drawn some lines, and it is useful to work with them.
Given the set {(x,y) ∈ R²| 0<x<0} then all negative and positive numbers but zero belong to this set.
c) If A and B are square matrices, then AB is also square. True
Taking into account the rules for multiplying matrices. The number of columns of A must be the same number of lines of B to there can be a matrix product.
Whenever we multiply square matrices, we'll always get square matrices. Then this conclusion is true.
d) A and B are subsets of a set S, then A∩B and A∪B are also subsets of S. True
Suppose A= Z (Integer Numbers) and B=Q (Rational Numbers) and S= R (Real Numbers)
A∩B = Z∩Q=∅ and A∪B =Z∪Q = subset
Since the ∅ empty set ⊂ in every set and ZUQ is another Subset of R this is a True conclusion.
e) True. For a matrix A, we define A²= AA
For any Power of Matrices, all we have to do is multiply any given matrix by itself for a given number of times.
M²=M*M
M³=M*M*M
Solve the equation for x. cx+b=3(x-c) XFIİ (Simplify yo nswer.)
Answer:
The value of x is [tex]\frac{(3c+b)}{3-c}[/tex].
Step-by-step explanation:
The given equation is
[tex]cx+b=3(x-c)[/tex]
Using distributive property we get
[tex]cx+b=3(x)+3(-c)[/tex]
[tex]cx+b=3x-3c[/tex]
To solve the above equation isolate variable terms.
Subtract 3x and b from both sides.
[tex]cx-3x=-3c-b[/tex]
Taking out common factors.
[tex]x(c-3)=-(3c+b)[/tex]
Divide both sides by (c-3).
[tex]x=-\frac{(3c+b)}{c-3}[/tex]
[tex]x=\frac{(3c+b)}{3-c}[/tex]
Therefore the value of x is [tex]\frac{(3c+b)}{3-c}[/tex].
In a manufacturing operation, a part is produced by machining, polishing, and painting. If there are three machine tools, four polishing tools, and three painting tools, how many different routings (consisting of machining, followed by polishing, and followed by painting) for a part are possible?
Answer:
36Step-by-step explanation:
As per the given question,
In a manufacturing operation, a part is produced by machining, polishing, and painting.
Number of machine tools = 3
Number of polishing tools = 4
Number of painting tools = 3
Now,
For finding the different routing consisting of machining, followed by polishing, and followed by painting, we have to simply multiply the number of machine tools, polishing tools and painting tools.
Therefore,
The different routing (consisting of machining, followed by polishing, and followed by painting) for a part are possible = 3 × 4 × 3 = 36.
Hence, the required answer is 36.
Number of ways a process can be done is the count of total distinguished ways that process can be done. The count of different routings possible for a part is 36
In how many ways, two things with a and b choices be done sequentially?Suppose that a process A can be done in 'a' different ways.
And there is a process following A, call it B, can be done in 'b' different ways. Then, the process A then B can be done in a×b different ways.
This is called rule of product in combinatorics.
Since there are 3 processes to be done subsequently(machining, polishing, then painting), and each of them can be done in 3, 4, and 3 ways respectively, thus,
Total number of routings possible for a part is [tex]3 \times 4 \times 3 = 36[/tex] ways.
Thus, The count of different routings possible for a part is 36
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Use the variation of parameters method to solve the DE y"+y'- 2y=1
Answer:
[tex]y(t)\ =\ C_1.e^{-2t}+C_2e^t-\ t.\dfrac{e^{-2t}}{3}-\dfrac{1}{3}[/tex]
Step-by-step explanation:
Given differential equation is,
y"+y'-2y=1
[tex]=>\ (D^2+D-2D)y\ =\ 1[/tex]
To find the complementary function we will write,
[tex]D^2+D-2=0[/tex]
[tex]=>\ D\ =\ \dfrac{-1+\sqrt{1^2+4\times 2\times 1}}{2\times 1}\ or\ \dfrac{-1-\sqrt{1^2+4\times 2\times 1}}{2\times 1}[/tex]
[tex]=>\ D\ =\ -2\ or\ 1[/tex]
Hence, the complementary function can be given by
[tex]y(t)\ =\ C_1e^{-2t}\ +\ C_2e^t[/tex]
Let's say,
[tex]y_1(t)\ =\ e^{-2t}\ \ =>y'_1(t)\ =\ -2e^{-2t}[/tex]
[tex]y_2(t)\ =\ e^{t}\ \ =>y'_2(t)\ =\ e^{t}[/tex]
[tex]g(t)\ =\ 1[/tex]
Wronskian can be given by,
[tex]W\ =\ y_1(t).y'_2(t)\ -\ y_2(t).y'_1(t)[/tex]
[tex]=\ e^{-2t}.e^{t}\ -\ e^{t}.(-2e^{-2t})[/tex]
[tex]=\ e^{-t}\ +\ 2e^{-t}[/tex]
[tex]=\ 3.e^{-t}[/tex]
Now, the particular integral can be given by
[tex]y_p(t)=\ -y_1(t)\int\dfrac{y_2(t).g(t)}{W}dt\ +\ y_2(t)\int\dfrac{y_1(t).g(t)}{W}dt[/tex]
[tex]=\ -e^{-2t}\int\dfrac{e^t.1}{3.e^{-t}}+e^t\int\dfrac{e^{-2t}.1}{3.e^{-t}}dt[/tex]
[tex]=\ -e^{-2t}\int\dfrac{1}{3}dt+\dfrac{e^t}{3}\int e^{-t}dt[/tex]
[tex]=\ \dfrac{-e^{-2t}}{3}.t\ -\ \dfrac{e^t}{3}.e^{-t}[/tex]
[tex]=\ -t.\dfrac{e^{-2t}}{3}-\dfrac{1}{3}[/tex]
Hence, the complete solution can be given by
[tex]y(t)\ =\ C_1.e^{-2t}+C_2e^t-\ t.\dfrac{e^{-2t}}{3}-\dfrac{1}{3}[/tex]
Select the best answer that defines the radius of a circle.
The radius is a line segment joining two distinct points on the circle.
The radius is a line segment that starts at the center of the circle and ends at a point on the circle.
The radius is the boundary of a circle.
The radius is a line segment that starts at one point on the circle, passes through the center of the circle, and ends at another point on the circle.
Answer:
The radius is a line segment that starts at the center of the circle and ends at a point on the circle.
Step-by-step explanation:
The radius is half of the diameter. The diameter is one line going across the whole, through the midpoint. The radius starts at the midpoint and that's why it's only half of the diameter.
Answer: The radius is a line segment that starts at the center of the circle and ends at a point on the circle.
Step-by-step explanation:
The radius of a circle is the distance from the center of the circle to any point on it.Let's check all the options.
The radius is a line segment joining two distinct points on the circle. → Wrong.
Reason :- Radius joins center and any point on circle not any two points.
The radius is the boundary of a circle. → Wrong.
Reason :- Circumference is the boundary of circle ,
Formula for circumference C= 2π r , where r is radius .
The radius is a line segment that starts at one point on the circle, passes through the center of the circle, and ends at another point on the circle.→ Wrong.
Reason :- Its diameter that starts at one point on the circle, passes through the center of the circle, and ends at another point on the circle and it is twice of radius.
So , the best answer that defines the radius of a circle is The radius is a line segment that starts at the center of the circle and ends at a point on the circle.
1000.0 cm3 of a metallic cylinder has a mass of 556 gram. Calculate the density of the cylinder.
Answer:
[tex]\text{Density}=\frac{0.556\text{ Grams}}{\text{ cm}^3}[/tex]
Step-by-step explanation:
We are asked to find the density of a cylinder whose volume is 1000.0 cubic cm and mass is 556 grams.
[tex]\text{Density}=\frac{\text{Mass}}{\text{Volume}}[/tex]
Substitute the given values:
[tex]\text{Density}=\frac{556\text{ Grams}}{1000.0\text{ cm}^3}[/tex]
[tex]\text{Density}=\frac{0.556\text{ Grams}}{\text{ cm}^3}[/tex]
Therefore, the density of the metallic cylinder is 0.556 grams per cubic centimeter.
The solution of a certain differential equation is of the form y(t)=aexp(7t)+bexp(11t), where a and b are constants. The solution has initial conditions y(0)=1 and y′(0)=4. Find the solution by using the initial conditions to get linear equations for a and b.
Answer:
Step-by-step explanation:
Given that the solution of a certain differential equation is of the form
[tex]y(t) = ae^{7t} +be^{11t}[/tex]
Use the initial conditions
i) y(0) =1
[tex]1=a(1)+b(1)\\a+b=1[/tex] ... I
ii) y'(0) = 4
Find derivative of y first and then substitute
[tex]y'(t) = 7ae^{7t} +11be^{11t}\\y'(0) =7a+11b \\7a+11b =4 ...II[/tex]
Now using I and II we solve for a and b
Substitute b = 1-a in II
[tex]7a+11(1-a) = 4\\-4a+11 =4\\-4a =-7\\a = 1.75 \\b = -0.75[/tex]
Hence solution is
[tex]y(t) = 1.75e^{7t} -0.75e^{11t}[/tex]
Answer:
y(t) = a exp(3t) + b exp(4t) conditions, y(0) = 3 y'(0) = 3 y(0) = a exp(3 x 0) + b exp(4 x 0) = a exp(0) + b exp(0) = (a x 1) + (b x 1) = a + b y'(0) = 0 so the linear equation is, a + b = 3
Step-by-step explanation:
52.25 is ___% of 1,1050.00
Answer: 0.4729%
Step-by-step explanation:
The formula to find the percent of a part in total amount :-
[tex]\%=\dfrac{\text{Part}}{\text{Total}}\times100[/tex]
Given : Total amount = 1,1050.00
Part of total amount = 52.25
Now, substitute all the values in the formula , we get
[tex]\%=\dfrac{52.25}{11050}\times100\\\\\Rightarrow\ \%=\dfrac{5225}{1105000}\times100=\dfrac{5225}{11050}=0.472850678733\approx0.4729\%[/tex]
Hence, 52.25 is 0.4729% of 1,1050.00.
The side of a lake has a uniform angle of elevation of 15degrees
30minutes. How far up the side of the lake does the water rise if,
during the flood season, the height of the lake increases by 7.3
feet?
During the flood season, the water rises 26.4 feet up the side of the lake.
The angle of elevation is given as 15 degrees 30 minutes,
Now, it can be converted to decimal degrees as 15.5 degrees.
Let's denote the distance up the side of the lake as x feet.
Now set up a trigonometric equation using the tangent function:
tan(15.5°) = (7.3 feet) / x
We can solve for x by rearranging the equation:
x = (7.3 feet) / tan(15.5°)
Evaluating this expression gives us:
x = (7.3 feet) / 0.277
x = 26.4 feet
Therefore, the water rises 26.4 feet up the side of the lake.
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An urn contains three red balls, five white balls, and two black balls. Four balls are drawn from the urn at random without replacement. For each red ball drawn, you win $6, and for each black ball drawn, you lose $9. Let X represent your net winnings Compute E(X), your expected net winnings E(x)
Answer:
0
Step-by-step explanation:
total number of balls = 3+5+2= 10
Probability of getting red P(R) = 3/10
Probability of getting white P(W) = 5/10
Probability of getting black P(B) = 2/10
for each red ball drawn you win $6 and for each black ball drawn you loose $9 dollars
E(X)= 6×3/10 +0×5/10 -9×2/10= 0
E(X)= 0
A committee on community relations in a college town plans to survey local businesses about the importance of students as customers. From telephone book listings, the committee chooses 282 businesses at random. Of these, 69 return the questionnaire mailed by the committee. The population for this study is
A. the 282 businesses chosen.
B. all businesses in the college town.
C. the 69 businesses that returned the questionnaire.
D. None of the above.
Answer:
option C the 69 businesses that returned the questionnaire.
Step-by-step explanation:
It is given in the question that only 69 businesses out of all 282 randomly chosen businesses have returned the questionnaire mailed by the committee.
Therefore,
The data available for the study by the committee is of only 69 businesses that have replied to the committee. So the study is based on this population of 69 businesses only.
Hence, option c is the correct answer.
Classify the following data. Indicate whether the data is qualitative or quantitative, indicate whether the data is discrete, continuous or neither. You order a pizza. The kind of pizza you order is recorded by entering the appropriate number on an order form. The numbers used are given below. 1) Pepperoni 2) Mushroom 3) Black Olive 4) Sausage
Answer:
Qualitative and Neither
Step-by-step explanation:
Quantitative data is such a data which can be measured or calculated i.e. which is defined only in terms of numbers. In simple words we can say that numeric data is the Quantitative data.
On the other hand, qualitative data describes the characteristics, attribute or quality of the objects. This type of data is not measured or calculated.
The data which we are dealing with is "Kind of pizza". The kind/flavor of pizza is an attribute or characteristic of pizza. So from here it is clear that the data is Qualitative. Though numbers are assigned to different flavors, these numbers are just for identification of the flavor on the order form.
The terms discrete and continuous can only be used when the data is Quantitative. Qualitative data cannot be referred to as discrete or continuous data, even if some numbers are assigned to data.
Therefore, the answers are: Qualitative and Neither
kendra remembered glue.
ally remembered an eraser
sarah remembered scissors
please be more specific
Multi step equation
-3(4-x)+3x=3(10-5x)
Answer:
x=2
Step-by-step explanation:
−3(4−x)+3x=3(10−5x)
(−3)(4)+(−3)(−x)+3x=(3)(10)+(3)(−5x)
−12+3x+3x=30+−15x
(3x+3x)+(−12)=−15x+30
6x+−12=−15x+30
6x−12=−15x+30
6x−12+15x=−15x+30+15x
21x−12=30
Step 3: Add 12 to both sides.
21x−12+12=30+12
21x=42
6. answer the question below
Answer:
-½
Step-by-step explanation:
Simplify \frac{25}{100}
100
25
to \frac{1}{4}
4
1
.
-\sqrt{\frac{1}{4}}−√
4
1
2 Simplify \sqrt{\frac{1}{4}}√
4
1
to \frac{\sqrt{1}}{\sqrt{4}}
√
4
√
1
.
-\frac{\sqrt{1}}{\sqrt{4}}−
√
4
√
1
3 Simplify \sqrt{1}√
1
to 11.
-\frac{1}{\sqrt{4}}−
√
4
1
4 Since 2\times 2=42×2=4, the square root of 44 is 22.
-\frac{1}{2}−
2
1
Done
Decimal Form: -0.5
Answer: D.)
Step-by-step explanation:
The first step would be to reduce the fraction in the square root, so divide both the numerator and denominator by 25 to get 1/4.
Then calculate the root, any root of one equals one so that stays as is. The exponential form of 4 would be 2^2.
Then reduce the index of the radical and exponent with 2 and you get answer D
A thin tube stretched across a street counts the number of pairs of wheels that pass over it. A vehicle classified as type A with two axles registers two counts. A vehicle classified as type B with nine axles registers nine counts. During a 2-hour period, a traffic counter registered 101 counts. How many type A vehicles and type B vehicles passed over the traffic counter? List all possible solutions.
The problem is a step-by-step calculation with several possible combinations of type A and type B vehicles when a total of 101 counts are registered. A systematic approach is required to find all possible whole number solutions.
Explanation:This problem is an example of a diophantine problem or a linear equation in two variables. If we denote the number of type A vehicles by 'a', and the number of type B vehicles by 'b', the problem can be represented by the equation 2a + 9b = 101.
As you are looking for all possible solutions, you have to do a systematic search. You will find that:
If there were 0 type B vehicles, there would have to be 50.5 type A vehicles, which isn't possible as we can't have half a vehicle.If there was 1 type B vehicle, there would have to be 46 type A vehicles.If there were 2 type B vehicles, there would be 41.5 type A vehicles, which again isn't possible.If there were 3 type B vehicles, there would be 37 type A vehicles.If there were 4 type B vehicles, there would be 32.5 type A vehicles which isn't possible.If there were 5 type B vehicles, there would be 28 type A vehicles.Continuing in this manner, you can find all possible whole number of vehicle combinations.Learn more about Linear Equation here:https://brainly.com/question/32634451
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An election for a new treasurer was held at a recent trade conference. In the election, Jamal received 40% of the votes, and his only opponent, Tran, received 4,320 votes. How many people voted for Jamal in the election?
Answer:
Jamal received 2,880 votes
Step-by-step explanation:
In the election,
Jamal received 40% of all votes
Tran received 100% - 40% = 60% that is 4,320 votes
Let x be the number of votes Jamal received. Then
x - 40%
4,320 - 60%
Write a proportion:
[tex]\dfrac{x}{4,320}=\dfrac{40}{60}\\ \\\text{Cross multiply}\\ \\60\cdot x=40\cdot 4,320\\ \\x=\dfrac{40\cdot 4,320}{60}=\dfrac{2\cdot 4,320}{3}=\dfrac{2\cdot 1,440}{1}=2,880[/tex]
Estimate the number of steps you would have to take to walk a distance equal to the circumference of the Earth. (We estimate that the length of a step for an average person is about 18 inches, or roughly 0.5 m. The radius of the Earth is 6.38 ✕ 106 m.)
Answer:
8771408311 steps ( approx )
Step-by-step explanation:
Given,
The radius of the earth,
[tex]r=6.38\times 10^6[/tex]
So, the circumference of the earth,
[tex]S=2\pi (r)[/tex]
[tex]=2\times \frac{22}{7}\times (6.38\times 10^8)[/tex]
[tex]=4.0102857143\times 10^9\text{ meters}[/tex]
∵ 1 meter = 39.3701 inches
[tex]\implies S =1.578853496\times 10^{11}\text{ inches }[/tex]
Also, the length of one step = 18 inches
Hence, the total number of steps = [tex]\frac{1.578853496\times 10^{11}}{18}[/tex]
[tex]= 8.7714083111\times 10^9[/tex]
[tex]\approx 8771408311[/tex]
What is the probability of selecting a red queen from a deck of cards?
2/26
3/52
1/52
1/26
Write the equation in the slope-intercept form and then find the slope m and y-intercept b of the corresponding line.
4x + 7y + 4 = 7
Answer:
[tex]m=-\frac{4}{7}[/tex]
[tex]b=\frac{3}{7}[/tex]
Step-by-step explanation:
Slope intercept form of a line is
[tex]y=mx+b[/tex] ... (1)
where, m is slope and b is y-intercept.
The given equation is
[tex]4x+7y+4=7[/tex]
We need to find the slope intercept form of given line.
Subtract 4 from both sides.
[tex]4x+7y=7-4[/tex]
[tex]4x+7y=3[/tex]
Subtract 4x from both sides.
[tex]7y=3-4x[/tex]
Divide both sides by 7.
[tex]y=\frac{3-4x}{7}[/tex]
[tex]y=\frac{3}{7}-\frac{4}{7}x[/tex]
Arrange the terms.
[tex]y=-\frac{4}{7}x+\frac{3}{7}[/tex] .... (2)
Form (1) and (2) we get
[tex]m=-\frac{4}{7}[/tex]
[tex]b=\frac{3}{7}[/tex]
Therefore, the slope of the line is -4/7 and y-intercept is 3/7.
The line contains the point (9,-9) and has the same y-intercept as y + 1 = 4 (x - 2). Write the equation of this line in slope-intercept form.
Answer:
The equation for this line, in slope-intercept form, is given by:
[tex]y = - 9[/tex]
Step-by-step explanation:
The equation of a line in the slope-intercept form has the following format:
[tex]y = ax + b[/tex]
In which a is the slope of the line and b is the y intercept.
Solution:
The line has the same y-intercept as [tex]y + 1 = 4 (x - 2)[/tex].
So, we have to find the y-intercept of this equation
[tex]y + 1 = 4 (x - 2)[/tex]
[tex]y = 4x - 8 - 1[/tex]
[tex]y = 4x - 9[/tex]
This equation, has the y-intercept = -9. Since this line has the same intercept, we have that [tex]b=-9[/tex].
Fow now, the equation of this line is
[tex]y = ax - 9[/tex]
The line contains the point [tex](9,-9)[/tex]
This means that when [tex]x = 9, y = -9[/tex]. We replace this in the equation and find a
[tex]y = ax - 9[/tex]
[tex]-9 = 9a - 9[/tex]
[tex]9a = 0[/tex]
[tex]a = \frac{0}{9}[/tex]
[tex]a = 0[/tex]
The equation for this line, in slope-intercept form, is given by:
[tex]y = - 9[/tex]
Solve the system of linear equations using the Gauss-Jordan elimination method. 2x + 3y 6212 3x + (x, y. z)
Answer:
The solution of the system of linear equations is [tex]x=3, y=4, z=1[/tex]
Step-by-step explanation:
We have the system of linear equations:
[tex]2x+3y-6z=12\\x-2y+3z=-2\\3x+y=13[/tex]
Gauss-Jordan elimination method is the process of performing row operations to transform any matrix into reduced row-echelon form.
The first step is to transform the system of linear equations into the matrix form. A system of linear equations can be represented in matrix form (Ax=b) using a coefficient matrix (A), a variable matrix (x), and a constant matrix(b).
From the system of linear equations that we have, the coefficient matrix is
[tex]\left[\begin{array}{ccc}2&3&-6\\1&-2&3\\3&1&0\end{array}\right][/tex]
the variable matrix is
[tex]\left[\begin{array}{c}x&y&z\end{array}\right][/tex]
and the constant matrix is
[tex]\left[\begin{array}{c}12&-2&13\end{array}\right][/tex]
We also need the augmented matrix, this matrix is the result of joining the columns of the coefficient matrix and the constant matrix divided by a vertical bar, so
[tex]\left[\begin{array}{ccc|c}2&3&-6&12\\1&-2&3&-2\\3&1&0&13\end{array}\right][/tex]
To transform the augmented matrix to reduced row-echelon form we need to follow these row operations:
multiply the 1st row by 1/2[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\1&-2&3&-2\\3&1&0&13\end{array}\right][/tex]
add -1 times the 1st row to the 2nd row[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\0&-7/2&6&-8\\3&1&0&13\end{array}\right][/tex]
add -3 times the 1st row to the 3rd row[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\0&-7/2&6&-8\\0&-7/2&9&-5\end{array}\right][/tex]
multiply the 2nd row by -2/7[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\0&1&-12/7&16/7\\0&-7/2&9&-5\end{array}\right][/tex]
add 7/2 times the 2nd row to the 3rd row[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\0&1&-12/7&16/7\\0&0&3&3\end{array}\right][/tex]
multiply the 3rd row by 1/3[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\0&1&-12/7&16/7\\0&0&1&1\end{array}\right][/tex]
add 12/7 times the 3rd row to the 2nd row[tex]\left[\begin{array}{ccc|c}1&3/2&-3&6\\0&1&0&4\\0&0&1&1\end{array}\right][/tex]
add 3 times the 3rd row to the 1st row[tex]\left[\begin{array}{ccc|c}1&3/2&0&9\\0&1&0&4\\0&0&1&1\end{array}\right][/tex]
add -3/2 times the 2nd row to the 1st row[tex]\left[\begin{array}{ccc|c}1&0&0&3\\0&1&0&4\\0&0&1&1\end{array}\right][/tex]
From the reduced row echelon form we have that
[tex]x=3\\y=4\\z=1[/tex]
Since every column in the coefficient part of the matrix has a leading entry that means our system has a unique solution.
3. You have decided to wallpaper your rectangular bedroom. The dimensions are 12 feet 6 inches by 10 feet 6 inches by 8 feet 0 inches high. The room has two windows, each 4 feet by 3 feet and a door 7 feet by 3 feet. Determine how many rolls of wallpaper are needed to cover the walls, allowing 10% for waste and matching. Each roll of wallpaper is 30 inches wide and 30 feet long. How many rolls of wallpaper should be purchased? * OA. 4 rolls OB. 5 rolls OC. 6 rolls OD. 7 rolls OE. 8 rolls
Answer:
B. 5 rolls
Step-by-step explanation:
The areas of the room, not including the ceiling, are discriminated as follows:
Longer walls: [tex](12.5\times8)\times2=200ft^2[/tex] (6 inches equals one foot)
Shorter walls: [tex](10.5\times8)\times2 = 168ft ^ 2[/tex] (6 inches equals one foot)
Window area: [tex](4\times3)\times2 = 24ft ^ 2[/tex]
Door area: [tex](7\times3) = 21ft ^ 2[/tex]
Area that will be effectively covered:
Total area to wallpaper: [tex]200 + 168 -24 -21 = 323ft ^ 2[/tex]
Amount of paper needed: [tex]323\times1.1 = 355.3ft ^ 2[/tex]
[tex]30in = 24in + 6in = 2ft + 0.5ft = 2.5ft.[/tex] That is, the area of a roll of paper is [tex]2.5\times30 = 75ft ^ 2[/tex]
Number of rolls needed:
[tex]\frac{355.3}{75} = 4.73[/tex] rolls
Answer:
Correct answer is B. 5Rolls
Step-by-step explanation:
First you have to put all values in same unit of measurement, knowing that 1 foot=12 inches, we apply 3 rule:12 inches is 1 feet
6 inches is 6/12 feet= 0,5 feet.
12 inches is 1 foot
30 inches is 30/12 feet=2,5feet.
2. Second,you have to calculate the total surface where you will wallpaper.
So you have to calculate the dimension of 2 different rectangles and substract the surfaces that you don't have to wallppaper (door and windows).
Let's calculate the surface of the rectancles. Let's put all units in feets.[tex]Area rectangle 1 =10.5feet*8feet=84ft^{2} \\Area rectangle2=12.5feet*8feet=100ft^{2}\\Total Area of walls=(Area rectangle 1 *2) + (Area rectangle2*2)\\Total Area of walls=(84ft^{2} *2)+(100ft^{2})\\\\Total Area of walls=368ft^{2}[/tex]
3. Now we have to calculate the area to substract from the total area, since you will not wallpaper the door and windows:
[tex]Windows=4feet*3feet*2=24ft^{2} \\Door=7feet*3feet=21ft^{2}[/tex]
4. Total area to wallpaper is Total surface of the room minus door and windows surface:
[tex]wallpaperArea=368ft^{2} -45ft^{2}\\wallpaperArea=323ft^{2}[/tex]
5. Now you have to add 10% waste to the calculated surface:
[tex]323ft^{2} +(323*0.10)=355.3ft^{2}[/tex]
6. So, you have the real area that will wallpaper considering 10% waste, it is 326.23 square feet. To calculate how many rolls you will need, you have to calculate the surface that each roll covers and then divide total surface by roll surface.
[tex]Roll surface=2,5feet*30feet=75feet^{2}[/tex]
[tex]Rolls needed=355.3ft^{2}/75ft^{2}=4.73[/tex]
7. As the number of rolls is not integer you have to round, then the answer is you will need 5 rolls of wallpaper.
find m angle A
A
10
B
20
C
16
D
14
Answer:
D. 14°
Step-by-step explanation:
The Law of Sines tells you sides are in proportion to the sine of the opposite angle:
sin(A)/BC = sin(B)/AC
sin(A) = BC/AC·sin(B)
A = arcsin(BC/AC·sin(B)) = arcsin(7/28·sin(75°)) ≈ 13.974° ≈ 14°
A batch contains 37 bacteria cells. Assume that 12 of the cells are not capable of cellular replication. Six cells are selected at random, without replacement, to be checked for replication. Round your answers to four decimal places (e.g. 98.7654). (a) What is the probability that all six cells of the selected cells are able to replicate? (b) What is the probability that at least one of the selected cells is not capable of replication?
The probability that all six cells are able to replicate is approximately 0.0051, while the probability that at least one cell is not capable of replication is approximately 0.9949.
Explanation:To solve this problem, we need to use the concept of probability and combinations.
(a) Probability that all six cells are able to replicate:There are 37 - 12 = 25 cells capable of replication. Out of these, we need to select 6 cells. The probability of selecting a cell capable of replication is 25/37 for the first selection, multiplied by 24/36 for the second selection, and so on, until 20/32 for the sixth selection. So, the probability is:
P(all 6 cells able to replicate) = (25/37) * (24/36) * (23/35) * (22/34) * (21/33) * (20/32) ≈ 0.0051
(b) Probability that at least one cell is not capable of replication:The probability that at least one cell is not capable of replication is equal to 1 minus the probability that all six cells are able to replicate. So, the probability is:
P(at least one cell not able to replicate) = 1 - P(all 6 cells able to replicate) ≈ 0.9949
Learn more about Probability here:https://brainly.com/question/32117953
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What is the probability of selecting a brown marble from a jar of marbles?
Seven of the marbles are brown, two of the marbles are white, and one of the marbles is green.
9/10
1/10
3/10
7/10
Answer:
Quality A is greater
Step-by-step explanation:
Answer:
7/10
Step-by-step explanation:
The total number of marbles is 7 + 2 + 1 = 10. So the probability of selecting a brown marble is 7/10