Answer: A ∪ B = {a, b, c, d}
(A ∪ B) ∪ C = = {a, b, c, d, e}
B ∪ C = = {a, b, d, e}
(B ∪ C) ∪ A = = {a, b, c, d, e}
Step-by-step explanation:
A = {a, b, c} B = {a, b, d} C = {b, d, e}
Union means "to join" so combine the sets to form a union.
A ∪ B = {A & B}
= {a, b, c & a, b, d}
= {a, b, c, d} because we do not need to list a & b twice
(A ∪ B) ∪ C = {(A ∪ B) & C)
= {a, b, c, d & b, d, e}
= {a, b, c, d, e} because we do not need to list b & d twice
B ∪ C = {A & B}
= {a, b, d & b, d, e}
= {a, b, d, e} because we do not need to list b & d twice
(B ∪ C) ∪ A = {(B ∪ C) & A)
= {a, b, d, e & a, b, c}
= {a, b, c, d, e} because we do not need to list a & b twice
Final answer:
We utilized set theory to find the union of sets A, B, and C in different orders and observed that the associative property of union holds true, meaning the order of union operations does not change the outcome which is {a, b, c, d, e}.
Explanation:
To answer this student's question, we can apply set theory concepts, specifically the concepts of the union and associativity.
Given sets A = {a, b, c}, B = {a, b, d}, and C = {b, d, e}, the union of these sets can be found as follows:
The union of A and B is A ∪ B = {a, b, c, d} - the set containing all elements from both A and B.The union of A ∪ B with C is (A ∪ B) ∪ C = {a, b, c, d, e} - the set containing all elements from A, B, and C.Similarly, the union of B and C is B ∪ C = {a, b, d, e}.The union of B ∪ C with A is (B ∪ C) ∪ A = {a, b, c, d, e}.From these operations, we observe that regardless of the order in which we take the union of the three sets, the result is the same. This demonstrates the associative property of union in set theory, where the order in which unions are performed does not affect the final outcome. We can generalize this as A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C.
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]
kendra remembered glue.
ally remembered an eraser
sarah remembered scissors
please be more specific
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.
c) Use the Bisection method to find a solution accurate to within 10^-2 for x^4 − 2x^3 − 4x^2 + 4x + 4 = 0 on [0, 2].
Answer:
x ≈ 181/128 ≈ 1.41406
Step-by-step explanation:
The attached table shows the iterations. At each step, the interval containing the root is bisected and the function value at the midpoint of the interval is found. The sign of it relative to the signs of the function values at the ends of the interval tell which half interval contains the root. The process is repeated until the interval width is less than 10^-2.
Interval: [0, 2], signs [+, -], midpoint: 1; sign at midpoint: +
[1, 2] 3/2 -
[1, 3/2] 5/4 +
...
the rest is in the attachment. The listed table values are the successive interval midpoints.
The final midpoint is 181/128 ≈ 1.41406. This is within 0.0002 of the actual root.
_____
The actual solution in the interval [0, 2] is √2 ≈ 1.41421.
To find a solution utilizing the Bisection Method, one needs to establish the function and verify it satisfies the bisection condition. The process is iteratively repeated by adjusting the interval to the midpoint until the error tolerance is reached or the function value of the midpoint is within the desired accuracy.
Explanation:The subject of your question concerns utilizing the Bisection method to solve a certain polynomial equation from a given interval [0, 2] with an accuracy of within 10^-2. The Bisection Method is a root-finding method in numerical analysis to solve for roots in given intervals.
First, establish the function f(x) = x^4 − 2x^3 − 4x^2 + 4x + 4 and set the interval a = 0 and b = 2. The midpoint c = (0 + 2) / 2 = 1.Check whether the configuration of f(0), f(1), and f(2) satisfies the bisection condition. The bisection condition states that the product of function at the end points should be negative i.e., f(a)*f(b) < 0. If it does, the root lies between a and b.Find f(1) and check its product with the values at the end points. If f(a)*f(c) < 0, then the root lies in the first subinterval so b is updated to be c. If not, then root is in the other interval, so a = c.We repeat this process until we get a c value that yields a function value within our desired accuracy or till we reach a point where (b-a)/2 < error tolerance, in this case, 10^-2 .This is an example of how the Bisection Method would be applied in solving for roots of polynomial equations. Do take note that this method only provides approximate solutions and it can be a lengthy process for equations with multiple roots.
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1/4÷(-2/3) =3/8 she is right now did she get the answer
Answer:
see below for the working
Step-by-step explanation:
Dividing by a number is the same as multiplying by the inverse of that number.
[tex]\displaystyle\frac{\left(\frac{1}{4}\right)}{\left(-\frac{2}{3}\right)}=-\frac{1}{4}\cdot\frac{3}{2}=-\frac{3}{4\cdot 2}=-\frac{3}{8}[/tex]
April shoots an arrow upward into the air at a speed of 64 feet per second from a platform that is 11 feet high. The height of the arrow is given by the function h(t) = -16t2 + 64t + 11, where t is the time is seconds. What is the maximum height of the arrow?
Answer:
Maximum height of the arrow is 203 feets
Step-by-step explanation:
It is given that,
The height of the arrow as a function of time t is given by :
[tex]h(t)=-16t^2+64t+11[/tex]..........(1)
t is in seconds
We need to find the maximum height of the arrow. For maximum height differentiating equation (1) wrt t as :
[tex]\dfrac{dh(t)}{dt}=0[/tex]
[tex]\dfrac{d(-16t^2+64t+11)}{dt}=0[/tex]
[tex]-32t+64=0[/tex]
t = 2 seconds
Put the value of t in equation (1) as :
[tex]h(t)=-16(2)^2+64(2)+11[/tex]
h(t) = 203 feet
So, the maximum height reached by the arrow is 203 feet. Hence, this is the required solution.
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.
The American Sugar Producers Association wants to estimate
themean yearly sugar consumption. A sample of n = 12 people
revealsthe mean yearly consuption to be 55 pounds with a
standarddeviation of 20 pounds. Find the lower bound for the 98%
confidenceinterval for the mean yearly sugar consumption. Assume
thepopulation is normal.
Answer: 39.308 pounds
Step-by-step explanation:
We assume that the given population is normally distributed.
Given : Significance level : [tex]\alpha: 1-0.98=0.02[/tex]
Sample size : n= 12, which is small sample (n<30), so we use t-test.
Critical value (by using the t-value table)=[tex]t_{n-1, \alpha/2}=t_{11,0.01}=2.718[/tex]
Sample mean : [tex]\overline{x}=50[/tex]
Standard deviation : [tex]\sigma= 20[/tex]
The lower bound of confidence interval is given by :-
[tex]\overline{x}-t_{(n-1,\alpha/2)}\dfrac{\sigma}{\sqrt{n}}[/tex]
i.e. [tex]55-(2.718)\dfrac{20}{\sqrt{12}}[/tex]
[tex]=55-15.6923803166\approx55-15.692=39.308[/tex]
Hence, the lower bound for the 98% confidence interval for the mean yearly sugar consumption= 39.308 pounds
Dan worked three hours for eight days.how many hours did he work in total
Plato math help please
Answer: option (C)
Step-by-step explanation: The slope of a linear function is undetermined when the line is parallel respect to the y-axis. In the current problem there is no way to observe such geometrical issue, but if we consider how to derive the slope using the following expression; [tex]m=\frac{\Delta y}{\Delta x}= \frac{y_{2}-y_{1}}{x_2-x_{1}}[/tex].
With the previous equation, we have
[tex]a) for P_{1}(-1,1), P_{2}(1,-1) m=\frac{\Delta y}{\Delta x}= \frac{-1-1}{1-(1)}=\frac{-2}{2}=1\\[/tex], therefore the slope is defined
[tex]b) for P_{1}(-2,2), P_{2}(2,2) m=\frac{\Delta y}{\Delta x}= \frac{2-2}{2-(2)}=\frac{0}{4}=0\\[/tex], therefore the slope is defined
[tex]c) for P_{1}(-3,3), P_{2}(-3,3) m=\frac{\Delta y}{\Delta x}= \frac{3-(-3)}{-3-(-3)}=\frac{6}{0}=undetermined\\[/tex]
[tex]d) for P_{1}(-4,4), P_{2}(4,4) m=\frac{\Delta y}{\Delta x}= \frac{4-(-4)}{4-(-4)}=\frac{8}{8}=1\\[/tex]
In this case, the option (C) shows that is not possible to divide over zero. Given such issue, the slope is undetermined and therefore it is a vertical line parallel to y-axis.
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
Suppose a company wants to determine the current percentage of customers who are subjected to their advertisements online. How many customers should the company survey in order to be 98% confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of customers who are subjected to their advertisements online? z0.10 z0.05 z0.025 z0.01 z0.005 1.282 1.645 1.960 2.326 2.576
Answer: 1503
Step-by-step explanation:
Given : Significance level : [tex]\alpha:1-0.98=0.02[/tex]
Critical value : [tex]z_{\alpha/2}=2.326[/tex]
Margin of error : [tex]E=\pm0.03[/tex]
We know that the formula to find the sample size (the prior true proportion is not available) is given by :-
[tex]n=0.25(\dfrac{z_{\alpha/2}}{E})^2[/tex]
i.e. [tex]n=0.25(\dfrac{2.326}{0.03})^2=1502.85444444\approx1503[/tex]
Hence, the minimum sample size should be 1503.
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
Let A be the matrix: [130 024 154 11-4] Find a basis for the nullspace of A.
Answer:
The basis for the null space of A is [tex]{\left[\begin{array}{c}-1&-1&1&0\end{array}\right],\left[\begin{array}{c}-1&1&0&1\end{array}\right]}[/tex]
Step-by-step explanation:
The first step is to find the reduced row echelon form of the matrix:[tex]\left[\begin{array}{cccc}1&0&1&1\\3&2&5&1\\0&4&4&-4\end{array}\right][/tex]
Make zeros in column 1 except the entry at row 1, column 1. Subtract row 1 multiplied by 3 from row 2 [tex]\left(R_2=R_2-\left(3\right)R_1\right)[/tex][tex]\left[\begin{array}{cccc}1&0&1&1\\0&2&2&-2\\0&4&4&-4\end{array}\right][/tex]
Make zeros in column 2 except the entry at row 2, column 2. Subtract row 2 multiplied by 2 from row 3 [tex]\left(R_3=R_3-\left(2\right)R_2\right)[/tex][tex]\left[\begin{array}{cccc}1&0&1&1\\0&2&2&-2\\0&0&0&0\end{array}\right][/tex]
Multiply the second row by 1/2 [tex]\left(R_2=\left(1/2\right)R_2\right)[/tex][tex]\left[\begin{array}{cccc}1&0&1&1\\0&1&1&-1\\0&0&0&0\end{array}\right][/tex]
2. Convert the matrix equation back to an equivalent system and solve the matrix equation
[tex]1x_{1} +x_{3} +1x_{4}=0\\ 1x_{2} +x_{3} -1x_{4}=0\\0=0[/tex]
[tex]\left[\begin{array}{cccc}1&0&1&1\\0&1&1&-1\\0&0&0&0\end{array}\right] \left[\begin{array}{c}x_{1} &x_{2} &x_{3}&x_{4} \end{array}\right]=\left[\begin{array}{c}0&0&0\end{array}\right][/tex]
If we take [tex]x_{3}=t, x_{4}=s[/tex] then [tex]x_{1}=-s-t,x_{2}=s-t,x_{3}=t,x_{4}=s[/tex]
Therefore,
[tex]\boldsymbol{x}=\left[\begin{array}{c}-s-t&s-t&t&s\end{array}\right]=\left[\begin{array}{c}-1&-1&1&0\end{array}\right]t+\left[\begin{array}{c}-1&1&0&1\end{array}\right]s\\\boldsymbol{x}=\left[\begin{array}{c}-1&-1&1&0\end{array}\right]x_{3} +\left[\begin{array}{c}-1&1&0&1\end{array}\right]x_{4}[/tex]
The null space has a basis formed by the set {[tex]{\left[\begin{array}{c}-1&-1&1&0\end{array}\right],\left[\begin{array}{c}-1&1&0&1\end{array}\right]}[/tex]}
dy/dt = y^2
y(t) = ?
Answer:
[tex]y(t)=-\frac{1}{t+C}[/tex]
Step-by-step explanation:
We are given that
[tex]\frac{dy}{dt}=y^2[/tex]
We have to find the value of y(t).
[tex]\frac{dy}{dt}=y^2[/tex]
[tex]\frac{dy}{y^2}=dt[/tex]
Integrating on both sides
[tex]\int y^{-2}dy=\int dt[/tex]
We know that [tex]\int x^n dx=\frac{x^{n+1}}{n+1}+C[/tex]
Using the formula
[tex]\frac{y^{-1}}{-1}=t+C[/tex]
[tex]-\frac{1}{y}=t+C[/tex]
[tex]a^{-1}=\frac{1}{a}[/tex]
Taking the reciprocal on both side then , we get
[tex]-y=\frac{1}{t+C}[/tex]
[tex]y(t)=-\frac{1}{t+C}[/tex]
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]
You deposit $3000 into a money-market savings account which pays 4.8% compounded quarterly, and you make no withdrawals from or further deposits into this account for 3 years. How much money is in your account at the end of those 3 years?
Give answer in dollars rounded to the nearest cent. Do NOT enter "$" sign in answer.
Answer:
$5265.71
Step-by-step explanation:
We have been given that you deposit $3000 into a money-market savings account which pays 4.8% compounded quarterly.
We will use future value formula to solve our given problem.
[tex]FV=C_0\times (1+r)^n[/tex], where,
[tex]C_0=\text{Initial amount}[/tex],
r = Rate of return in decimal form,
n = Number of periods.
[tex]4.8\%=\frac{4.8}{100}=0.048[/tex]
[tex]n=3\times 4=12[/tex]
[tex]FV=\$3,000\times (1+0.048)^{12}[/tex]
[tex]FV=\$3,000\times (1.048)^{12}[/tex]
[tex]FV=\$3,000\times 1.7552354909370114[/tex]
[tex]FV=\$5265.7064\approx \$5265.71[/tex]
Therefore, there will be $5265.71 in your account at the end of those 3 years.
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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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.
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]
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.
A certain chalkboard manufacturer determines that their largest blackboard model has a mean length of 5.00 m and a standard deviation of 1.0 cm. A certain school district orders 1000 of these chalkboards. How many are likely to have lengths of under 4.98 m?
Answer:
23 chalkboards
Step-by-step explanation:
Given:
Mean length = 5 m
Standard deviation = 0.01
Number of units ordered = 1000
Now,
The z factor = [tex]\frac{\textup{x - Mean}}{\textup{standard deviation}}[/tex]
or
The z factor = [tex]\frac{\textup{4.98 - 5}}{\textup{0.01}}[/tex]
or
Z = - 2
Now, the Probability P( length < 4.98 )
Also, From z table the p-value = 0.0228
therefore,
P( length < 4.98 ) = 0.0228
Hence, out of 1000 chalkboard ordered (0.0228 × 1000) = 23 chalkboard are likely to have lengths of under 4.98 m.
The amount of money spent on red balloon in a certain college town when the football team is in town is a normal random variable with mean $50000 and a standard deviation of $3000. What proportion of home football game days in this town is less than $45000 worth of red balloons sold?
Answer: 0.0475
Step-by-step explanation:
Given : The amount of money spent on red balloon in a certain college town when the football team is in town is a normal random variable with
[tex]\mu=\$50000[/tex] and [tex]\sigma=\$3000[/tex]
Let x be the random variable that represents the amount of money spent on red balloon.
Using formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex], the z-score corresponding to x= 45000 will be :_
[tex]z=\dfrac{45000-50000}{3000}\approx-1.67[/tex]
Now, by using the standard normal distribution table for z, we have
P value : [tex]P(z<-1.67)=1-P(z<1.67)=1-0.9525=0.0475[/tex]
∴The proportion of home football game days in this town is less than $45000 worth of red balloons sold = 0.0475
The opposite process rule says to solve for ________. a known variable by replicating the process used to form the original equation an unknown variable by reversing the process used to form the original equation an unknown variable by replicating the process used to form the original equation a known variable by reversing the process used to form the original equation
Answer:
An unknown variable by reversing the process used to form the original equation.
Step-by-step explanation:
The opposite process rule says to solve for - an unknown variable by reversing the process used to form the original equation.
If an equation indicates an operation such as addition, subtraction, multiplication, or division, solve for the unknown variable by using the opposite process.
For example:
Lets say we have to find [tex]x+25=35[/tex]
Here 25 is subtracted from both sides of the equation to isolate x.
[tex]x+25-25=35-25[/tex]
we get x = 10
Check this : [tex]10+25=35[/tex]
Final answer:
The opposite process rule is a technique used to solve for an unknown variable by reversing the operations that were used to create the equation. It is a key concept in algebra for finding values of unknown variables.
Explanation:
The opposite process rule refers to solving for an unknown variable by reversing the process used to form the original equation. This is a fundamental technique in solving algebraic equations, necessary for determining the value of the unknown.
To solve for an unknown variable, you follow several steps. Initially, identify the unknowns and known variables. Then, find an equation that expresses the unknown in terms of the knowns. If more than one unknown is present, multiple equations may be needed.
To find the numerical value of the unknown, substitute known values, including their units, into the equation and solve. In algebra, this could mean performing operations such as addition, subtraction, multiplication, or division inversely to isolate the variable.
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]
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
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°
Rewrite the subtraction number sentence as an addition number sentence.
5- (-2)
Answer:
5 + 2
Step-by-step explanation:
We have to rewrite the given statement in addition form.
The integers have property of:
Negative(-) Negative(-) = Positive(+)
Positive(+) Positive(+) = Positive(+)
Positive(+) Negative(-) = Negative(-)
Negative(-) Positive(+) = Negative(-)
The given statement is:
5-(-2)
Since we have two negative together, it is converted into a positive.
Thus, the given statement can be written in positive form as
5 + 2
Use the Babylonian method of false position to solve the following
problem,
taken from a clay tablet found in Susa: Let the width of a
rectangle measure a
quarter less than the length. Let 40 be the length of the diagonal.
What are
the length and width? Begin with the assumption that 1 (or 60) is
the length
of the rectangle.
Answer:
length: 32width: 24Step-by-step explanation:
Assume a solution
Assume that 60 is the length. The width is then 1/4 less, or 60 -60/4 = 45.
The diagonal of this rectangle is found using the Pythagorean theorem:
d = √(60² +45²) = √5625 = 75
Make the adjustment
This is a factor of 75/40 larger than the actual diagonal, so the actual dimensions must be 40/75 = 8/15 times those we assumed.
length = (8/15)×60 = 32
width = (8/15)×45 = 24
The length and width of the rectangle are 32 and 24, respectively.
_____
Comment on this solution method
This method is suitable for problems where variables are linearly related. If we were concerned with the area, for example, instead of the diagonal, we would have to adjust the linear dimensions by the square root of the ratio of desired area to "false" area.
Final answer:
This detailed answer explains how to use the Babylonian method of false position to find the length and width of a rectangle.
Explanation:
The Babylonian method of false position involves making an initial assumption about the solution and then iteratively honing in on the correct answer.
Step-by-step:
Let's start with an assumption: Length = 60. Then calculate the width based on the given conditions.
Adjust your assumption based on the calculated width until you reach the correct solution.
By using this method, you can find the length and width of the rectangle described in the problem.
Find the Cartesian Equation of the plane passing through P(8, -2,0) and perpendicular to a- 5i+3j-k What is the distance of this plane to the point 0(2,2, 2)? (a) (b)
Answer:
equation of plane, 5x+3y-z-36=0
Distance of point (2,2,2) from plane = 4.05 units
Step-by-step explanation:
Given,
Plane passing through the point = (8, -2, 0)
Let's say, [tex]x_1\ =\ 8[/tex]
[tex]y_1\ =\ -2[/tex]
[tex]z_1\ =\ 0[/tex]
Plane perpendicular to the vector, a= 5i + 3j- k
Since, the vector is perpendicular to the plane, hence the equation of plane can be given by
[tex](5i + 3j- k).((x-x_1)i+(y-y_1)j+(z- z_1)k)=\ 0[/tex]
[tex]=>(5i + 3j- k).((x-8)i+(y+2)j+(z-0)k)=\ 0[/tex]
[tex]=>\ 5(x-8)+3(y+2)-z=0[/tex]
[tex]=>\ 5x\ -\ 40\ +\ 3y\ +\ 6\ -\ z\ =\ 0[/tex]
[tex]=>\ 5x\ +\ 3y\ -\ z\ -\ 36\ =\ 0[/tex]
Hence, the equation of plane can be given by, 5x+3y-z-36=0
Now, we have to calculate the distance of the point O(2,2,2) from the plane 5x+3y-z-36=0
Let's say,
a= 5, b= 3, c= -1, d=-36
[tex]x_0=2,\ y_0=2,\ z_0=2[/tex]
So, distance of a point from the plane can be given by,
[tex]d=\dfrac{ax_0+by_0+cz_0+d}{\sqrt{a^2+b^2+c^2}}[/tex]
[tex]=\dfrac{\left |5\times 2+3\times 2+(-1)\times 2-36\right |}{\sqrt{5^2+3^2+(-1)^2}}[/tex]
[tex]=\dfrac{24}{\sqrt{35}}[/tex]
= 4.05 units
So, the distance of the point O(2,2,2) from the given plane will be 4.05 units.