Answers
The graph of cubic equations, a to d and the associated cubic equations are as follows;
Equation (B) ⇒ Graph a
Equation (C) ⇒ Graph b
Equation (D) ⇒ Graph c
Equation (A) ⇒ Graph d
What is a cubic equation?A cubic equation that has only one variable is an equation of the form a·x³ + b·x² + c·x + d, where a ≠ 0.
The evaluation of the graphs are as follows;
Graph a
The y-intercept of the graph a, is 1, and the x-intercept, which is a solution of the equation is x = -1,
Therefore, when x = 0, y = 1, and when y = 0, x = -1
The function in graph a represents the translation of the parent function, y = x³, one unit, upwards, which corresponds with the function;
y = x³ + 1
Therefore, graph a corresponds to cubic equation (B) y = x³ + 1
Graph b
The y-intercept is one (1), and the turning point is at the point x = -1, which represents a shift of one unit to the left, and which can be obtained by the equation; y = (x + 1)³
Therefore, graph b corresponds with the cubic equation (C) y = (x + 1)³
Graph c
The turning point of the graph and the y-intercept is shifted one unit down, which corresponds with the equation; (D) y = x³ - 1
Graph d
The turning point is shifted 1 unit to the right and is located on the x-axis, The y-intercept is located at the point y = 1, therefore, the graph corresponds with the equation y = (x - 1)³
Graph d corresponds to equation (A), y = (x - 1)³
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Related Questions
help solve this there is reward
you have ten mangoes and twelve carrots that cost 23$ in total, five mangoes and four carrots cost ten dollars what does one mango and one carrot cost
Answers
Answer:1 carrot cost .75 cents and a mango cost 1.40
Step-by-step explanation:
12 carrots + 10 mangoes = 23$
4 carrots + 5 mangoes = 10$
I know to get from 5 mangoes to 10 mangoes you have to multiply twice. so it wil come up to 20 dollars
8 carrots + 10 mangoes = 20$
divide by 2
that will leave 4 carrots and 3 dollars.
4/3 = .75
17 dollars is left
so 10 carrots is equal to 14 dollars
14/10 = 1.4 dollars
create an equation to describe the relationship between the rise in temperature (y) and he change in time ( x) for the period from 6 to 24 minutes math connections
Answers
y = mx + b where m is the rate of change and b is the y-intercept, describes the linear relationship between rise in temperature (y) and change in time (x) from 6 to 24 minutes.
To calculate the linear equation, we can use the Point-Slope Form of a Line formula, y = mx + b. To find the slope (m), we must calculate the rise (change in the y-value) over the run (change in the x-value). We must calculate the temperature difference between 6 minutes (T1) and 24 minutes (T2). Then, divide the temperature difference by the time difference of 18 minutes. The y-intercept (b) can be found by plugging in the known x and y values of 6 minutes and T1 respectively into the equation y = mx + b. Once we have the slope and y-intercept, we can plug them into the equation y = mx + b to calculate the equation that describes the linear relationship between the rise in temperature (y) and change in time (x) from 6 to 24 minutes.
y = mx + b, where
m = (T2 - T1)/(24 - 6) = (T2 - T1)/18
b = T1 - (m*6)
Therefore,
y = (T2 - T1)/18 * x + (T1 - (m*6))
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Jane wanted a single board cut into 8 equal pieces. The lumber company charges 60 cents for cutting a board into 4 equal pieces. How much will it charge for cutting Names board ?
Answers
The charge for cutting a board into 8 equal pieces is $1.2
How much will it charge for cutting Names board?Charges for cutting a board into 4 equal pieces = $0.60Charges for cutting a board into 8 equal pieces = xEquate charges for cutting board : Number of equal pieces
0.60 : 4 = x : 8
0.60/4 = x / 8
cross product
0.60 × 8 = 4 × x
4.8 = 4x
divide both sides by 4
x = 4.8 / 4
x = 1.2
Therefore, it will cost $1.2 for cutting a board into 8 equal pieces.
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Fifty part time college students were asked how many hours they work per week. The graphs below summarize their responses. Frequency 12+ 10 81 6 0 10 30 40 0 10 30 40 20 Hours 20 Hours What is the best measure of center for this data? Estimate the value of that measure of center.
Answers
The best measure of center for the frequency of the data is Mean. The value of the measure of the center is 30.
The Mean value refers to an intermediate value between a discrete set of numbers. It is a measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation, divided by the total number of observations. Fifty part time college students were asked how many hours they work per week.
Frequency 12 10 81 6 0 10 30 40 0 10 30 40 20 Hours 20 Hours.
we can calculate mean by adding all the frequencies and then divide it by the number of observations.
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) A line is described by the equation y=5/3 x-6. What is the y-intercept?
Answers
Answer:
y- intercept = - 6
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = [tex]\frac{5}{3}[/tex] x - 6 ← is in slope- intercept form
with y- intercept c = - 6
An LG Dishwasher, which costs $800, has a 14% chance of needing to be replaced in the first 2 years of purchase. A two-year extended warrantee costs $112.10 on a dishwasher. What is the expected value of the extended warranty assuming it is replaced in the first 2 years?
Answers
The expected value of the extended warranty is $687.90 assuming it is replaced in the first 2 years.
Here, we are assuming that the dishwasher is replaced in the first 2 years in order to calculate the expected value of the extended warranty.
Because there is 14% chance of replacement:
Value of warranty = 14%*$800 - $112.10
Value of warranty = $112.00 - $112.10
Value of warranty = -$0.10
Now, because it is assumed that replacement will be done in first 2 years, then,
Value of warranty will increase to $687.90 ($800 - $112.10).
Therefore, the expected value of the extended warranty is $687.90 assuming it is replaced in the first 2 years.
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↓
edpuzzle
YOUR TURN: FIND THE VALUE OF X AND Y:
12
!!!
MULTIPLE CHOICE QUESTION
Find the value of x:
3√4
8
4√3
4√8
8√4
please help me within today loll it’s due tonight 11:59pm!!
Answers
Answer:
Step-by-step explanation:
in similar triangles
sides are proportional.
g a popluation of squirrels lives in a firest with a carrying capacity of 1800. assume logisitic growth with growth constant k
Answers
The doubling time is approximately 83 years. This means that it will take approximately 83 years for the squirrel population to double from its initial size of 450 squirrels.
What is an exponential function?
An exponential function is a function in which the variable appears in the exponent. Exponential functions have the form y = ab^x, where a and b are constants and x is the variable. The constant b is called the base of the exponential function.
The given information states that the squirrel population P(t) follows a logistic growth model with a growth constant k = 0.8 yr-1 and a carrying capacity of 1800 squirrels. The formula for the squirrel population P(t) can be found using the logistic growth model:
P(t) = K / (1 + (K/P0 - 1) * e^(-kt))
where P0 is the initial population of squirrels, K is the carrying capacity, and t is time in years.
Substituting the given values into the formula, we get:
P(t) = 1800 / (1 + (1800/450 - 1) * e^(-0.8t))
Simplifying the equation, we get:
P(t) = 1800 / (1 + 3e^(-0.8t))
This is the formula for the squirrel population P(t).
To find the doubling time (the time it takes for the squirrel population to double), we need to find the time t at which the squirrel population reaches twice the initial population of 450 squirrels. This can be represented by the equation:
2 * 450 = 1800 / (1 + 3e^(-0.8t))
Solving for t, we get:
0.8t = ln(3)
t = ln(3) / 0.8
Hence, the doubling time is approximately 83 years. This means that it will take approximately 83 years for the squirrel population to double from its initial size of 450 squirrels.
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Two independent simple random samples are taken to test the difference between the means of two populations whose standard deviations are not known, but are assumed to be equal. The sample sizes are n1 - 25 and n2-35. The correct distribution to use is the 1) t distribution with 59 degrees of freedom 2) t distribution with 58 degrees of freedom. 3) t distribution with 61 degrees of freedom. 4) t distribution with 60 degrees of freedom e in tne abintres of students enroled n staustucs today if there is any diffe statistics teacher wants to see and those enrolled five years ago.
Answers
2- T-distribution with 58 degrees of freedom
When the estimated standard deviation rather than the actual standard deviation is used as the denominator, the t-distribution is a continuous probability distribution of the z-score.
The largest number of logically independent values—that is, values with the freedom to change—in the data sample is referred to as the degree of freedom. If there is a remaining requirement for the data sample, particular data sample items must be picked after the degree of freedom quantity has been decided.
[tex]t[/tex] = Student's t-distribution
[tex]X{i}[/tex] = sample mean
μ = population mean
[tex]s[/tex] = sample standard deviation
[tex]n[/tex] = sample size
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The largest number of logically independent values—that is, values with the freedom to change—in the data sample is referred to as the degree of freedom. If there is a remaining requirement for the data sample, particular data sample items must be picked after the degree of freedom quantity has been decided.
When the estimated standard deviation rather than the actual standard deviation is used as the denominator, the t-distribution is a continuous probability distribution of the z-score.
A bag contains 3 gold marbles, 6 silver marbles, and 29 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1.
What is your expected value if you play this game?
Answers
The expected value of the game that you play will be negative 0.21.
What is the expected value?In parameter estimation, the expected value is an application of the weighted sum. Informally, the expected value is the simple average of a considerable number of individually determined outcomes of a randomly picked variable.
The expected value is given below.
E(x) = np
Where n is the number of samples and p is the probability.
A pack contains 3 gold marbles, 6 silver marbles, and 29 dark marbles. Somebody offers to play this game: You haphazardly select one marble from the pack. Assuming it is gold, you win $3. Assuming it is silver, you win $2. Assuming it is dark, you lose $1.
The expected value is given as,
E(x) = 3(3/38) + 2(6/38) - 1(29/38)
E(x) = 0.2368 + 0.3158 - 0.7632
E(x) = -0.21
The expected value of the game that you play will be negative 0.21.
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The function f(x) = Negative Startroot x EndRoot is shown on the graph
Which statement is correct?
The domain of the function is all real numbers less than or equal to −1.
The range of the function is all real numbers greater than or equal to 0.
The range of the function is all real numbers less than or equal to 0.
The domain of the function is all real numbers less than or equal to 0.
Answers
Answer:
C) The range of the function is all real numbers less than or equal to 0.
Step-by-step explanation:
Given function:
[tex]f(x)=-\sqrt{x}[/tex]
Domain
The domain of a function is the set of all possible input values (x-values).
As the square root of a negative number cannot be taken, x ≥ 0.
Therefore, the domain of the function is all real numbers greater than or equal to 0.
Range
The range of a function is the set of all possible output values (y-values).
As the domain is x ≥ 0, the range of the function is all real numbers less than or equal to 0.
Answer:
C) The range of the function is all real numbers less than or equal to 0.
If f(x)=3x-1/2,find f^-1(x)
Answers
Answer: x=3y-1/2 or y=(x+1/2)/3
Step-by-step explanation:
f^-1(x) is just the inverse so all you need to do is switch x and y.
right now you have y=3x-1/2
so switch the x and y to get your answer of x=3y-1/2
if you're asked to solve for y, follow these steps:
1. x +1/2 = 3y-1/2 +1/2 -> x+1/2=3y
2. divide both sides by 3 -> (x+1/2)/3=y
Use the given conditions to find the values of all six trigonometric functions. (If an answer is undefined, enter UNDEFINED.) sec(x) = − 9/5, tan(x) < 0
sin(x)=
cos(x)=
tan(x)=
csc(x)=
sec(x)=
cot(x)=
Answers
Answer:
Step-by-step explanation
sec x = -9/5.
cos x = 1/ secx
So, here
cos x = 1 / -9/5
= -5/9
sin x = √(1 - cos^2 x)
= √(1 - 25/81)
= √(56/81)
= √56/9
tan x = sinx / cos x
= √56/9 / -5/9
= -√56/5
cosec x = 1 / sin x
= 1 / √56/9
= 9/√56.
cot x = 1/tanx
= 1/-√56/5
= -5/√56
TRUE/FALSE. a negative covariation means that there is a negative change in the investment associated with it. group of answer choices
Answers
The answer is false. The covariance statistic gauges how closely two variables are related to one another.
What is covariayion?a connection between two quantitative variables in which the corresponding values of the second variable tend to rise when the value of the first variable tends to rise (or fall) (or decrease).
The direction of the link between two variables is measured by covariance. A positive covariance indicates that both variables frequently exhibit high or low values together. While two variables have a negative covariance, they are often low when one is high.
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Find the length of the third side. If necessary, write in simplest radical form. 3. 4 Need Answer ASAP
Answers
The answer is √7.
What is Pythagorean theorem?
The Pythagorean theorem, sometimes known as Pythagoras' theorem, is a basic relationship between a right triangle's three sides in Euclidean geometry. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
Step-by-step explanation:
Using the pythagorean theorem, we know that
a^2 + b^2 = c^2
In this case,
c = 4, which means that a^2 + b^2 = 16
Since we know that a = 3, this means that a^2 is also 9. So, we know that b^2 = 16-9
Therefore, b^2 = 7
So, the answer is √7.
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A line passes through the point 6,2 and has a slope of -4 over 3.
Write an equation in slope-intercept form for this line.
Answers
Answer:
y = -4/3x + 10
Step-by-step explanation:
A line with a slope of -4 over 3 that passes through the point (6,2) can be written in slope-intercept form as:
y = (-4/3)x + b
To find the value of b, we can plug in the coordinates of the given point (6,2) into the equation and solve for b:
2 = (-4/3)(6) + b
2 = -8 + b
b = 10
Therefore, the equation of the line in slope-intercept form is:
y = (-4/3)x + 10
For linear function f, f(2)=0 and f(4)=-6.
Write function f in slope-intercept form.
Hint
Examples:
f(5)=0 means (5,0) is a point on the line.
f(3)=-2 means (3,-2) is a point on the line.
How do you find the slope of the line when you have 2 points?
Answers
The function f in slope - intercept form is y = -3x + 6 .
Given :
For linear function f, f(2)=0 and f(4)=-6.
Examples :
f(5)=0 means (5,0) is a point on the line.
f(3)=-2 means (3,-2) is a point on the line.
so f ( 2 ) = 0 means ( 2 , 0 )
f ( 4 ) = -6 means ( 4 , -6 )
slope m = -6 - 0 / 4 - 2
= -6 / 2
= -3
substitute m and ( 2 , 0 ) in y = mx + b
0 = -3 * 2 + b
b = 6
substitute m and b value
y = -3x + 6
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Perform the indicated operations.
c +6b/(-2bc-6ab- 3a²) + 2b/(a² + 2ab) - b/(ac - 3a²)
Answers
The simplified expression is c²a³ - 3[tex]a^4[/tex]c + a²bc² - a³bc - 3[tex]a^4[/tex]b+ 4ab²c² - 22a²b²c + 36a³b + 12a³b² / (-2bc-6ab- 3a²)(a² + 2ab)(ac - 3a²).
What is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
Given:
c +6b/(-2bc-6ab- 3a²) + 2b/(a² + 2ab) - b/(ac - 3a²)
Now, simplifying
(c+ 6b) (a² + 2ab)(ac - 3a²) + 2b(-2bc-6ab- 3a²)(ac - 3a²)- b (-2bc-6ab- 3a²)(a² + 2ab) / (-2bc-6ab- 3a²)(a² + 2ab)(ac - 3a²)
= ca² + abc + 6a²b + ab² (ac - 3a²) + (4b²c - 12ab² - 6a²b)(ac - 3a²) - ( -2b²c - 6ab² - 3a²b)(a² + 2ab) / (-2bc-6ab- 3a²)(a² + 2ab)(ac - 3a²)
= c²a³ - 3[tex]a^4[/tex]c + a²bc² - 3a³bc + 6a³bc - 18[tex]a^4[/tex]b + 4ab²c² - 12a²b²c - 12 a²b²c+ 36a³b - 6a³bc + 18[tex]a^4[/tex]b + 2a²b²c + 4a³bc + 6a³b² + 12a²b³ - 3[tex]a^4[/tex]b + 6a³b² / (-2bc-6ab- 3a²)(a² + 2ab)(ac - 3a²)
= c²a³ - 3[tex]a^4[/tex]c + a²bc² - a³bc - 3[tex]a^4[/tex]b+ 4ab²c² - 22a²b²c + 36a³b + 12a³b² / (-2bc-6ab- 3a²)(a² + 2ab)(ac - 3a²)
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Please kindly answer the question with a quick answer.
Answers
Answer:
Step-by-step explanation:
what is your question?
Write a polynomial function of least degree with zeroes
–
1, 5,9
Write your answer using the variable x and in standard form with a leading coefficient of 1
Answers
By multiplying together (x-1) and (x-5) we get x^2 - 5x - 1x + 5, which simplifies to x^2 - 6x + 5.
Now we have to add (x-9). When we multiply that with the polynomial we just got, we get x^3 - 6x^2 + 5x - 9x^2 + 54x - 45, which simplifies to x^3 - 15x^2 + 59x - 45.
So the function is y = x^3 - 15x^2 + 59x - 45
a teacher figures that final grades in the statistics department are distributed as: a, 25%; b, 25%; c, 40%; d, 5%; f, 5%. at the end of a randomly selected semester, the following number of grades were recorded. find only the test statistics to test the claim that the grade distribution for the department is different than expected. be sure to set up the equation. do not conduct the hypothesis testing. grade a b c d f number 42 36 60 8 14
Answers
The critical value is 13.28 when the distribution of final marks in the statistics department.
Given that,
According to a teacher, the distribution of final marks in the statistics department is: A, 25%; B, 25%; C, 40%; D, 5%; and F, 5%. The following number of grades were recorded at the conclusion of a randomly chosen semester.
We have to recognize the important value.
We know that,
Given a=0.01,
The critical value of chi-square with 0.99,
df=n-1=4 is
13.28 (From chi-square table)
Therefore, The critical value is 13.28 when the distribution of final marks in the statistics department.
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A student is assessing the correlation between the number of workers in a factory and the number of units produced daily. The table shows the data:
Number of workers
0 10 20 30 40 50 60 70 80 90
Number of units 2 52 102 152 202 252 302 352 402 452
(
Part A: Is there any correlation between the number of workers in a factory and the number of units produced daily? Justify your answer. (4 points)
Part B: Write a function that best fits the data. (3 points)
Part C: What do the slope and y-intercept of the plot indicate? (3 points)
Answers
Answer: Part A: Yes there is. As the number of workers increases, so does the production.
Part B: y = 5x + 2
Part C: The slope indicates that for every person there is 5 units being produced, and the y-intercept indicates that 2 units are being made every day, regardless of if there are no workers ( this, however is unrealistic but it is how the data is )
Step-by-step explanation:
I am very confused can someone help with this problem?
Answers
The value of b must be equal to -14, so that the function has at least one inflection point. Hence, option B is correct.
What is an expression?Expressions are mathematical operations that involve at least two terms connected by an operator and can be either integers, variables, or both. The operations with reflection coefficients are addition, subtraction, multiplication, and division.
As per the instruction given in the question,
The given function,
f(x) = x⁴ + bx² + 1
Differentiate with respect to x,
f'(x) = 4x³ + 2xb
Again differentiate with respect to x,
f''(x) = 12x² + 2b
For inflection point, f''(x) = 0
12x² + 2b = 0
2b = -12x²
b = -6x²
x = √-b/6, x ∈ [1, 2]
Let's assume that b = -28
x = √-(-28)/6
x = √28/6
x = √14/3, here x ∉ [1, 2], hence it is incorrect.
If b = -14
x = √-(-14)/6
x = √14/6
x = √7/3 ≈ 1.82, here x ∈ [1, 2], hence it is correct.
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Kaylee wants to find an exact solution to the system shown. Which method would be the most straightforward for her to use?
Answers
The method that would be the most straight forward for her to use would be the substitution method because you can easily solve one equation for a variable. That is option A.
What is substitution method for solving equations?The substitution method for solving equations is the method that can be used in a quadratic equation whereby one equation is solved and substituted into the next equation.
The given equation such as;
4x - 5y = 7 ---> eq 1
4X -9y = -4---> eq 2
From eq1 make 4x the subject of formula
4x = 7 + 5y
Substitute for 4x in equation 2;
7 + 5y - 9y = -4
7 - 4y = -4
7+4 = 4y
11 = 4y
y = 11/4
y = 2.75
Then substitute y = 2.75 into equation 1;
4x - 5y = 7
4x - 5(2.75) = 7
4x - 13.75 = 7
4x = 7 + 13.75
4x = 20.75
X = 20.75/4
X = 5.2
Therefore with the substitution method the value of X and y can be determined.
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Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
Asymptote code below. An equilateral triangle, a regular octagon, and a regular -gon, all with the same side length, also completely surround a point. Find .
Answers
The regular polygon has 20 sides.
The regular pentagon is a regular polygon with five sides
The regular decagon is a regular decagon with ten sides
The angle sum around a point adds to 360 degrees.
Each interior angle of the square is 90 degrees
Each interior angle of the regular pentagon is 108 degrees.
The Interior angle of our unknown polygon is 360 - 108 - 90 = 162 degrees.
A quick calculation shows us that the exterior angle of this polygon is 18 degrees.
The sum of the external angles of any regular polygon is 360 degrees
Divide 360 by 18 and you get 20 times.
Thus our unknown polygon has 20 sides.
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A $810 washing machine in a laundromat is depreciated for tax purposes at a rate of $90 per year. Find a function
for the depreciated value of the washing machine as it varies with time. When does the depreciated value reach $0?
Answers
Step-by-step explanation:
This function can be modeled linearly because we know that the cost decreases by $90 every year. That means our cost is equal to the original value minus $90 for every year, which is t.
So, c=810-90t. And this function equals zero when t is equal to 9 years.
a survey was conducted with a large group of teenagers from 6 different states and it asked them for their favourite sport out of a large list of sports. the study aims to look at whether a teenager's state of residence is associated with their sporting preferences. assume that p is the proportion of teenagers who selected the sport most preferred in their state of residence, such that p1 is that for state 1, p2 is that for state 2 and so on. select from the following all statements that are true as well as the hypothesis that corresponds to this study:
Answers
H0: There is no link between state and preferred sport.
H1: There is a connection between state and preferred sport.
Given,
The question to be answered is whether the factor "state" and the factor "sporting choice" are related.
Therefore, the focus of the study is on "association" and whether or not state-to-state variations in athletic preference distributions exist. The most appropriate test in this instance is the test for association.
Unless there is evidence to the contrary, two things are assumed to not be related. Since the null hypothesis is assumed to be true, the following two hypotheses are the proper pair:
H0: There is no connection between a person's state and their preferred sport.
H1: The choice for sports and state are related.
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20. college majors there are three sections of english 101. in section i, there are 25 students, of whom 5 are mathmatics majors. in section ii, there are 20 students
Answers
The probability that the student is from Section I, given that he or she is a mathematics major, is 0.214 and the probability that the student is from Section 1, is 0.294.
In the give question;
There are three sections of English 101.
In Section 1, there are 25 students, of whom 3 are mathematics majors.
In Section II, there are 20 students, of whom 7 are mathematics majors.
In Section lIl, there are 40 students, of whom 4 are mathematics majors.
We have to find the probability that the student is from Section I, given that he or she is a mathematics major and the probability that the student is from Section 1.
Total number of students in all sections = 25+20+40 = 85
Total number of students major in mathematics = 3+7+4 = 14
The probability that the student is from Section I, given that he or she is a mathematics major is represented as
P(I | M) = P(I∩M)/P(M)
P(I∩M) = (Student in section II - Student in section III)/Total student in all sections
P(I∩M) = (7 - 4)/85 = 3/85
P(M) = Total student of mathematics/Total student in all sections
P(M) = 14/85
Now putting the values
P(I | M) = (3/85)/(14/85)
P(I | M) = 3/14
P(I | M) = 0.214
Now finding the probability that the student is from Section 1.
Probability that student is from section I = Total students in section I/ total students
Probability that student is from section I = 25/85
Probability that student is from section I = 0.294
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The right question is:
There are three sections of English 101. In Section 1, there are 25 students, of whom 3 are mathematics majors, In Section II, there are 20 students, of whom 7 are mathematics majors. In Section lil, there are 40 students, of whom 4 are mathematics majors. A student in English 101 is chosen at random. Find the probability that the student is from Section I, given that he or she is a mathematics major. The probability that the student is from Section 1 is (Simplify your answer. Round to three decimal places as needed.)
consider the initial value problem find the eigenvalue , an eigenvector , and a generalized eigenvector for the coefficient matrix of this linear system. -4 , 1 0 , c 0 find the most general real-valued solution to the linear system of differential equations. use as the independent variable in your answers.
Answers
The eigenvalue for this coefficient matrix is c, the eigenvector is (1, 0) and the generalized eigenvector is (1, -4).The most general real-valued solution to the linear system of differential equations is given by:y(x) = c1e^(-4x) + c2xe^(-4x),
To find the eigenvalue, we need to solve the characteristic equation of the coefficient matrix, which is given by det(A - cI) = 0. In this case, the characteristic equation is -c^2 + 4c = 0, which has a single solution of c = 4. Thus, the eigenvalue is c = 4.
To find the eigenvector, we need to solve the linear system (A - cI)v = 0. For this coefficient matrix, the linear system is (A - 4I)v = 0, which has the solution v = (1, 0). Thus, the eigenvector is (1, 0).
To find the generalized eigenvector, we need to solve the linear system (A - cI)w = v, where v is the eigenvector. In this case, the linear system is (A - 4I)w = (1, 0), which has the solution w = (1, -4). Thus, the generalized eigenvector is (1, -4).
Finally, the most general real-valued solution to the linear system of differential equations is given by y(x) = c1e^(-4x) + c2xe^(-4x), where c1 and c2 are arbitrary constants.
Characteristic equation: det(A - cI) = 0
-c2 + 4c = 0
c = 4
Eigenvector: (A - 4I)v = 0
v = (1, 0)
Generalized eigenvector: (A - 4I)w = v
w = (1, -4)
Most general solution: y(x) = c1e^(-4x) + c2xe^(-4x), where c1 and c2 are arbitrary constants.
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Your iron works has contracted to design and build a 500-ft^3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is made by welding thin stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible. How will you take weight into account?
What dimensions do you tell the shop to use?
Answers
The dimensions we will tell the shop to use are ⇒ Volume = 10 x 10 x 5 foot
The surface area of the box = [tex]225[/tex] [tex]ft^{2}[/tex]
Weight of the metal = 1125 kg
According to the question,
Given value = [tex]500[/tex] [tex]ft^{3}[/tex] tanks is to be built which has to be a square-based rectangle steel holding tank.
For this, we have to know the total surface area of the box,
Surface area (A) = 4 x sides of the rectangle + square base -- equation 1
Let's take the width of the square base = p
And let height = h
Since we know that,
Rectangle sides = p x h
Square base ⇒ p x p = [tex]p^{2}[/tex]
Substituting these values in equation 1,
Surface area ( A ) = 4 x ( p x h ) + [tex]p^{2}[/tex] ---- equation 2
The formula for calculating the volume of the box,
V = p x p x h
V = [tex]p^{2}[/tex] x h --- equation 3
Referring to the question that the value of the volume is already given such that volume = [tex]500[/tex] [tex]ft^{3}[/tex]
So, V = [tex]500[/tex] [tex]ft^{3}[/tex]
Putting the value of V in equation 3,
500 = [tex]p^{2}[/tex] x h
h = [tex]\frac{500}{p^{2} }[/tex] ---- equation 4
Substituting the value of h from equation 4 to equation 2,
A = 4 x ( p x [tex]\frac{500}{p^{2} }[/tex] ) + [tex]p^{2}[/tex]
⇒ A = [tex]\frac{2000}{P}[/tex] + [tex]p^{2}[/tex]
We will be taking the derivative of the above equation,
⇒ A' = - [tex]\frac{2000}{p^{2} }[/tex] + 2p
Now we will be minimizing A,
⇒ 0 = - [tex]\frac{2000}{p^{2} }[/tex] + 2[tex]p[/tex]
⇒ [tex]\frac{2000}{p^{2} }[/tex] = 2[tex]p[/tex]
⇒ 2000 = 2[tex]p^{3}[/tex]
⇒ 1000 = [tex]p^{3}[/tex]
⇒ p = [tex]\sqrt[3]{1000}[/tex]
⇒ p = 10 foot
Substituting the value of p in equation 4,
⇒ h = [tex]\frac{500}{10^{2} }[/tex]
⇒ h = [tex]\frac{500}{100}[/tex]
⇒ h = 5 foot
Substituting values in the area of the square box,
A = 4 x (10 x 5) + [tex]5^{2}[/tex]
A = 200 + 25
A = 225 [tex]ft^{2}[/tex]
The weight of the metal,
⇒ weight = A x weight per square foot
⇒ weight = 225 x 5
⇒ weight = 1125 kg
Therefore, The dimensions we will tell the shop to use are ⇒ Volume = 10 x 10 x 5 feet
The surface area of the box = 225 [tex]ft^{2}[/tex]
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