The total time taken by Amy to complete her assignments is 19/20 hours.
What is addition?Addition is one of the four basic operations of arithmetic, Addition in maths is a process of combining two or more numbers.
Given that, Amy finished her chemistry assignment in 1/5 hours. Then she completed her history assignment in 3/4 hours.
Total time taken by Amy = Time taken for chemistry assignment + Time taken for history assignment
= 1/5+3/4 = (15+4)/20
= 19/20
Hence, The total time taken by Amy to complete her assignments is 19/20 hours.
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What do I do now cause I’m stuck in the middle of the equation and not sure what else I may need to do besides make the x alone but I’m not sure how.
2x + 8 = 10
Answer:
x = 1
Step-by-step explanation:
2x + 8 = 10 Subtract 8 from both sides
2x + 8 - 8 = 10 - 8
2x = 2 Divide both sides by 2
[tex]\frac{2x}{2}[/tex] = [tex]\frac{2}{2}[/tex]
x = 1
Check:
2x + 8 = 10
2(1) + 8 = 10
2 + 8 = 10
10 = 10 checks.
I need help with this Question.
Six items purchased at a grocery store weigh 15 kilograms. One of the items is detergent weighing 872 grams. What is the total weight, in kilograms, of the other five items?
The total weight of the other 5 items is. kg.
Answer:
14.128
Step-by-step explanation:
you need to extract the 872g which is 0.872kg
from the 15 and that is 14.128kg
Given f(x) =2x2 + 1 and g g(x) = 3x-5 , find the following. f(g)(2))
The value of f(g(2)) from the given functions f(x) and g(x) is 3.
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
The given functions are f(x)=2x²+1 and g(x)=3x-5.
We need to find f(g(2)).
Put x=2 in the function g(x)=3x-5, we get
g(2)=3×2-5
= 6-5
= 1
So, g(2)=1
Put x=1 in the function f(x)=2x²+1, we get
f(g(2))=2(1)²+1
= 2+1
=3
Therefore, the value of f(g(2)) is 3.
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Solve the equation.
3^2x= 6,561
x=3
x=4
x=8
x= 16
Answer: 4
Step-by-step explanation
3^2×=6561
then think 3 raise to power what is 6561
the answer is 8
3^2×=3^8
three cancel three
2×=8
divide both sides by 2
then ×=4
PLEASE ANSWER FAST THIS IS TIMED
The graph of the linear inequation [tex]y \leq \frac{1}{2}x -5[/tex] has been described below
What is linear inequation?
Inequation shows the comparision between two algebraic expressions by connecting the two algebraic expressions by
[tex]> , < , \geq, \leq[/tex]
A one degree inequation is known as linear inequation.
The given linear inequation is
[tex]y \leq \frac{1}{2}x -5[/tex]
Here equality is maintained. So the line is a solid line
Now, (0, 0) does not satisfy the inequality.
So the solution is below the line
When graphing the inequality above, we will draw a solid line and the solution lies below the line
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A random sample of 16 pharmacy customers showed the waiting times below (in minutes).
20 27 24 23 19 21 18 17
22 15 18 32 16 20 19 19
Find a 90 percent confidence interval for μ, assuming that the sample is from a normal population.
The 90% confidence interval for the population mean is given as follows:
(18.75, 22.51).
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the following rule:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
In which the variables of the equation are presented as follows:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 90% confidence interval, with 16 - 1 = 15 df, is t = 1.7531.
From the sample given in this problem, the parameters are as follows:
[tex]\overline{x} = 20.63, s = 4.3, n = 16[/tex]
Then the lower bound of the interval is given as follows:
20.63 - 1.7531 x 4.3/4 = 18.75.
The upper bound of the interval is of:
20.63 + 1.7531 x 4.3/4 = 22.51.
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Suppose that the population of a certain town is made of 45% men and 55% women. Of the men, 40% wear glasses, and of the women 20% wear glasses. Given that a person chosen at random from this town wears glasses, what is the probability that the person is a woman?
The probability that the person is a woman is 40%.
what is the probability that the person is a woman?P(woman | glasses) = (20% * 55%) / (40% * 45% + 20% * 55%) = 0.55There is a 50% probability that a sperm will fertilize a boy and a 50% chance that a sperm will fertilize a female ovum each time.The probability that a randomly chosen person in this town wears glasses is equal to the sum of the probability of a man wearing glasses and the probability of a woman wearing glasses.P(Glasses) = P(M ∩ G) + P(W ∩ G)
P(Glasses) = 0.45 × 0.40 + 0.55 × 0.20
P(Glasses) = 0.27
The probability that the randomly chosen person is a woman is equal to the probability of a woman wearing glasses divided by the probability of a person wearing glasses.P(Woman) = P(W ∩ G) / P(Glasses)
P(Woman) = 0.55 × 0.20 / 0.27
P(Woman) = 0.44
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Find the length and width of a rectangle that has the given area and a minimum perimeter.
Area: 5A square cm
Find length and width of a rectangle with minimum perimeter that has an area of 5A square cm
length=
width=
By taking minimum perimeter the value of length and width will both be [tex]\sqrt{5A}[/tex].
What is perimeter?Perimeter is basically the sum of all sides of a figure.
How to calculate perimeter of rectangle?The perimeter of a rectangle is 2l + 2w where l = length and w = width of rectangle.
Since the perimeter needed in this question is minimum, the figure must be a square and not a rectangle. For a square, area = a^2
5A = a^2
a = l = w = [tex]\sqrt{5A}[/tex]
Hence, both the length and width are equal to [tex]\sqrt{5A}[/tex].
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URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
The solution is Option D.
The value of x from the triangle is x = 31
What are similar triangles?
If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.
Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides
Given data ,
Let the first triangle be ABC
The measure of side AB = 14.5
The measure of side BC = 15.5
Let the second triangle be XYZ
The measure of side XY = 29
The measure of side YZ = x
Now , if the triangle ABC is similar to triangle XYZ , then the corresponding sides of the triangles will be equal in ratio
So , the equation will be
XY / AB = YZ / BC
Substituting the values in the equation , we get
29 / 14.5 = x / 15.5
On simplifying the equation , we get
x/15.5 = 2
Multiply by 15.5 on both sides of the equation , we get
x = 2 x 15.5
x = 31
Therefore , the value of x is 31
Hence , the value of x of side YZ is 31
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the equation y = 1x/3 + 3 is shown graphed below. If the equation y = 2x - 4 was added to this one to form a system, what would be its solution?
The solution of the system of the equations, y = 1/3x + 3 and y = 2x - 4, as shown in the graph below is: (4.2, 4.4).
What is the Solution to a System of Equations?A system involving two equations has a solution, which are the coordinates of the point on a graph where the two equations intersect.
Given the equation, y = 1/3x + 3, which is represented by the red line in the graph below, when y = 2x - 4 (represented as the blue line) is added, a system is formed which will make both lines intersect at the point that has the coordinates (4.2, 4.4).
Therefore, the solution of the system y = 1/3x + 3 and y = 2x - 4 is (4.2, 4.4).
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root of 63x to the power of 11 yz to the power of 10
The value of the equation is [tex]A = (\sqrt{63x}^{(11yz)^1^0}[/tex]
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is given by
Root of 63x to the power of 11 yz to the power of 10
The value of root of 63x = [tex]\sqrt{63x}[/tex]
Now , the value of root of 63x to the power of 11 yz is
The value of root of 63x to the power of 11 yz = [tex](\sqrt{63x}^{(11yz)[/tex]
And ,
The value of root of 63x to the power of 11 yz to the power of 10 is
Value of root of 63x to the power of 11 yz to the power of 10 is [tex](\sqrt{63x}^{(11yz)^1^0}[/tex]
Therefore , the value of A is [tex](\sqrt{63x}^{(11yz)^1^0}[/tex]
Hence , The value of the equation is [tex]A = (\sqrt{63x}^{(11yz)^1^0}[/tex]
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To test if an observed frequency distribution with five classes is normally distributed, we need to find _____.a. The t-statisticb. The expected, normally distributed class frequenciesc. The class marksd. The class relative frequencies
Finding the t statistic is necessary to check the normality of a five-class observed frequency distribution.
what the T statistic means?The t-value gauges the magnitude of the difference in relation to the range of your sample's data. Or to put it another way, T is just the calculated difference shown in units of standard error. The evidence is stronger against the null hypothesis the larger the value of T.
Which T statistic value is optimal?In general, t-values that are more than +2 or less than -2 are acceptable. We are more confident in the coefficient's ability to predict outcomes the higher the t-value. Low t-values are a sign of the predictability of that coefficient's coefficient being unreliable.
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would you expect the proportion of stay-at-home residents to be higher in virginia than in arkansas? support your conclusion with the results obtained in parts (b) and (c). from the results obtained in parts (b) and (c), we would expect the number of stay-at-home residents to be higher correct: your answer is correct. in arkansas than virginia. the sample results show that, rounded to two decimal places, the estimated percentage of arkansas residents who were born there is %. the sample results also show that, rounded to two decimal places, the estimated percentage of virginia residents who were born there is %.
As per the estimated percentage, the test statistic is 0.6167 or 61.67%
The term percentage is defined as the number or ratio that can be expressed as a fraction of 100.
Here we have given that in Arkansas than Virginia. the sample results show that, rounded to two decimal places, the estimated percentage of Arkansas residents who were born there is %. the sample results also show that, rounded to two decimal places, the estimated percentage of Virginia residents who were born there is %.
And we need to find the test statistics.
While we looking into the given situation we have identified the following values,
=> H0 : p = 0.577
=> Ha : p ≠ 0.577
Now, we have estimate the proportion of stay-at-home residents in Arkansas is calculated as,
Here the Proportion of stay-at-home residents in Arkansas is written as,
=> p = 74/120 = 0.6167
Therefore, the test statistics is 0.6167.
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5. Simplify -2 + 6.45z - 6 + (-3.25z).
[tex]-2 + 6.45z - 6 + (-3.25z)[/tex]
Combine Like Terms:
[tex](6.45z+-3.25z)+(-2+-6)[/tex]
[tex]\fbox{3.2z - 8}[/tex]
which of the following is a function that is passed as an argument to another function? question 40 options: anonymous value callback restricted
callback restricted is a function that is passed as an argument to another function
It tests whether the condition returned by the callback function holds for at least one item in array.
What is an array?
It should be noted that an array means a data structure that can store a fixed size collection of elements.
In this case, the true statement of the array is that it tests whether the condition returned by the callback function holds for at least one item in array.
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How do I get n(E)/n(S)= 7,650/341,055 simplified to 170/7579
Answer:
Divide the numerator and denominator by 45.
Step-by-step explanation:
7650/341055
Divide the numerator and denominator by 5.
1530/68211
Divide the numerator and denominator by 3.
510/22737
Divide the numerator and denominator by 3.
170/7579
Find the vector v with an initial point at (5, 7) and a terminal point at (2, 10). write your answer in component form. then write the vector v as a linear combination of the unit vectors i and j. finally, find the length of the vector.
The length of the vector v is approximately 4.24. A vector is a mathematical object that has both magnitude and direction
Vectors are commonly used in physics, engineering, and other fields to represent quantities that have both magnitude and direction, such as velocity, acceleration, and force. They are also important in linear algebra, where they can be used to represent lines and planes in space, and to solve systems of linear equations.
The vector v can be expressed in component form as v = <2-5, 10-7> = <-3, 3>.
To express the vector v as a linear combination of the unit vectors i and j, we can write it as v = (-3)i + 3j.
To find the length of the vector, we can use the Pythagorean theorem:
length of v = sqrt((-3)^2 + 3^2) = sqrt(9 + 9) = sqrt(18) = 3*sqrt(2) = 4.24
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help i suck at math!!!!
Answer:
146
Step-by-step explanation:
the sides of a triangle always equal 180, so just subtract those numbers from 180
Gabe is practicing for his upcoming track meet. One lap around the track is 1/4 of a mile. Gabe sets a goal to
run more than 2 miles today. How many laps should Gabe run to meet his goal?
Answer:
20=-7
Step-by-step explanation:
j
Answer:
8 Laps
Step-by-step explanation:
This is because one lap is 1/4 so it would take 4 laps to make it a whole, which would be a mile.
We could now times this by two, since 1 mile is half of 2 miles.
4 x 2 = 8
Which would make it 8
Find the distance traveled by a particle with position (x, y) as t varies in the given time interval.
x = 3sin^2(t), y = 3cos^2(t), 0 = t = 5p
Find the length of the curve.
By The length of the curve can be found using the formula for the arc length of a parametric curve, the distance traveled by the particle is 30π.
The length of the curve can be found using the formula for the arc length of a parametric curve:
L = ∫0→5p √(dx/dt)2 + (dy/dt)2 dt
We can calculate dx/dt and dy/dt as follows:
dx/dt = 6sin(t)cos(t)
dy/dt = -6sin(t)cos(t)
Substituting these values into the formula for the arc length gives us:
L = ∫0→5p √[(6sin(t)cos(t))2 + (-6sin(t)cos(t))2] dt
= ∫0→5p 6√(sin2(t)cos2(t)) dt
= 6∫0→5p |sin(t)cos(t)| dt
= 6∫0→5p sin(t)cos(t) dt
= 6[(1/2)sin2(t) + (1/2)cos2(t)]5p
= 6[(1/2)(1) + (1/2)(1)]5p
= 6(1)5p
= 30π
Therefore, the distance traveled by the particle is 30π.
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The accompanying data file shows the midterm and final scores for 32 students in a statistics course Click here for the Excel Data File
a. Estimate a students final score as a function of hiss/her midterm score. (Round your answers t decimal places.) Final27.5818+ 0.6774 Midterm b. Find the standard error of the estimate. (Round your answer to 2 decimal places.) 14.947 c. Interpret the coefficient of determination. (Round your answer to 2 decimal places.) R means that % of the sample Variation â–¼ | in Final grades is explained by the sample regression equation
In this case, the coefficient of determination is 0.5297, which means that 53% of the sample variation in Final grades is explained by the sample regression equation.
The data file contains the midterm and final scores of 32 students in a statistics course. To estimate a student's final score as a function of their midterm score, we used a regression analysis. The regression equation was Final = 27.5818 + 0.6774 Midterm. This equation can be used to predict the final score of a student based on their midterm score. The standard error of the estimate was 14.947. This indicates the accuracy of the estimated final score. The coefficient of determination was 0.5297, which means that 53% of the sample variation in final grades is explained by the regression equation. This tells us that the regression equation is a relatively good predictor of a student's final score.
Given:
Midterm Score = x
Final Score = y
Regression Equation: y = 27.5818 + 0.6774x
Standard Error of the Estimate = 14.947
Coefficient of Determination = 0.5297
Calculating Final Score:
y = 27.5818 + 0.6774x
y = 27.5818 + 0.6774(x)
y = 27.5818 + 0.6774(x)
y = 27.5818 + 0.6774(x)
y = 27.5818 + 0.6774x
Final Score = 27.5818 + 0.6774(Midterm Score)
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11) Solve for n in s=(n-2) 180
A) n=180 s+2
B) n=(s-2) . 180
C) n=s/180-2
D) n=s/180+2
For the given equation the value of n is equal to option D.
n = (s/180) + 2.
As given in the question,
Given equation is equal to :
s = (n-2)180
Now take all the terms on one side and term n on other side of the equation we have,
Divide both the side by 180 we get,
s/ 180 = [(n-2)180]/180
s/180 = n -2
Add 2 both the side of the equation we get,
(s/180 ) + 2 = n -2 + 2
(s/180 ) + 2 = n
so here the value is equal to
n = (s/180) +2.
Therefore, the value of n in the equation is equal to option D.
n = (s/180) + 2.
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Given: AB AC and DA bisects /BAC.
Prove: ZDBC ZDCB.
Step
Statement
AB AC
DA bisects BAC
AD AD
ZBAD ZDAC
A
Reason.
Given
Reflexive Property
Select a Resison
B
Solve for proof
Use photo to understand better PLEASE HELP
The correct option is C ie 3.
D is the midpoint of BC.
What is Congruence of triangles?
Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure.
What are the congruency properties of a triangle?
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
RHS (Right angle-Hypotenuse-Side)
In triangles BAD and CAD,
(i) AD=AD (common)
(ii)∠DAB=∠CAD (Given, AD bisects ∠BAC)
(iii)AB = AC (given)
Thus, by SAS property both the triangles BAD and CAD are congruent .
By cpct , ∠ADB=∠ADC = x (let)
Since ∠ADB and ∠ADC lies in a straight line
So by the property of straight lines the sum is 180°.
Now ∠ADB+∠ADC= 180°
⇒x+x= 180°
⇒2x= 180°
⇒x=90°
∴AD ⊥ BC
By cpct ,BD=DC
So D is the midpoint of BC.
Hence option C that is 3 is the right answer.
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1. A taxi service charges a $ 1.95 flat rate plus $ 0.55 per mile. Jason only has $ 12 to spend on a ride. Which graph best represents the number of miles Jason can afford to travel?
Answer:
B
Step-by-step explanation:
Subtract $1.95 from $12 to account for the flat rate that Jason must pay, then divide that by 0.55, or the per mile charge. The answer, or the highest amount of miles he can go with $12, is 18.27272727. The answer that best represents the number of miles he can go is B because he can go up to 18 miles, including 18 miles, hence the closed circle.
write an indirect proof in paragraph form. given: triangle rvt and svt are equilateral. triangle rvs is not equilateral
The indirect proof that defines triangle RVS is not equilateral is explained below.
How triangle look like?
In math, A triangle is a closed, 2-dimensional shape with 3 sides, 3 angles, and 3 vertices and also a polygon.
Here we need to find the indirect proof for the triangle RVS which is not equilateral.
Here we know that, We need to write an indirect proof in paragraph proof.
For, the given statement, △RVT and SVT are equilateral;
And △RVS is not equilateral.
Then the explanation for this one is written as,
Let us assume temporarily that △RST is equilateral.
And we have given that △RVT and SVT are equilateral, then the sides of △RVT, SVT, and RST are congruent.
Therefore, RV≅VSRS which follows that △RVS is equilateral.
In contradicts the given that △RVS is not equilateral, then the temporary assumption must be false. It follows that △RST is not equilateral.
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a norman window has the shape of a rectangle surmounted by a semicircle. (thus the diameter of the semicircle is equal to the width of the rectangle, labeled x.) a window with the shape of a rectangle surmounted by a semicircle. the diameter of the semicircle is equal to the width of the rectangle. the rectangle has width x. if the perimeter of the window is 32 feet, find the exact value of x (in ft) so that the greatest possible amount of light is admitted.
Answer:
The exact value of x so that the greatest possible amount of light is emitted is 8.96ft.
What is the perimeter of a semicircle?
The perimeter of a semicircle is half that of a circle, that is [tex]\pi r[/tex], where r is the radius of the circle.
Step-by-step explanation:
Given the width of the rectangle is x, let the height of the rectangle be y.
The radius of the semicircle is x/2.
The perimeter of the window is equal to [tex]2y+x+\pi \frac{x}{2} =32[/tex].
The area of the window is [tex]xy+\frac{1}{2}(\pi )(\frac{x}{2})^2=xy+\frac{\pi x^2}{8}[/tex].
We can insert into the formula for the area to obtain the area in terms of only x by solving the equation for the perimeter of y in terms of x.
[tex]2y=32-(1+\frac{\pi }{2} )x=32-(\frac{2+\pi }{2} )x\\y=16-(\frac{2+\pi }{4} )x[/tex].
Now the area of the window in terms of only x is:
Area=[tex]16x-(\frac{2+\pi }{4} )x^2+\frac{\pi x^2}{8}[/tex].
Since it is given in the question that maximum light should be admiitted, we will differentiate the Area and set it to 0.
[tex]16-2(\frac{2+\pi }{4} )x+\frac{\pi x}{4}=0\\ 16=x(1+\frac{\pi }{4})\\ x=\frac{16}{1+\frac{\pi }{4}} \\x=8.96ft[/tex].
Therefore the exact value of x is 8.96ft.
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Your
gross pay is $2,759.00. Your involuntary deductions are FICA (7.65%), federal
withholding (12%), and state withholding (7%). How much are you allowed for housing and
fixed expenses
The amount allowed for housing and expenses is $2023.73.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
Example:
50% = 50/100 = 1/2
25% = 25/100 = 1/4
20% = 20/100 = 1/5
10% = 10/100 = 1/10
We have,
Gross pay = $2759
Deduction:
7.65%
= 7.65/100 x 2759
= $211.06
12%
= 12/100 x 2759
= $331.08
7%
= 7/100 x 2759
= $193.13
The remaining amount.
= 2759 - 211.06 - 331.08 - 193.13
= $2023.73
Thus,
The amount allowed is $2023.73.
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The sketch shows a curve with equation y = ab* where a and b are constants and b>0 The curve passes through the points (0,5) and (2,45) Calculate the value of a and b.
NEED PROPER ANSWER AND WORKING OUT
WILL MARK BRAINLIEST
The number line above shows the solution graph of which inequality?
A) 2x+9 > 25
B) 2x+9 ≥ 25
C) 2x+9 < 25
D) 2x +9 ≤ 25
Answer:
2x+9<25 or u just can say c
Consider a particle in a rigid box of length L. For each of the states n = 1, n = 2 and n = 3:
a. Sketch graphs of ǀ (x) ǀ2. Label the points x = 0 and x = L.
b. Where, in terms of L, are the positions at which the particle is most likely to be found?
c. Where, in terms of L, are the positions at which the particle is least likely to be found?
d. Determine, by examining your ǀ (x) ǀ2 graphs, if the probability of finding the particle in the left one-third of the box is less than, equal to, or greater than 1 3. Explain your reasoning.
e. Calculate the probability that the particle will be found in the left one-third of the box.
For states, n = 1, 2, and 3 the probability of finding the particle at left 1/3rd of the box is P₁ = (3/4) * (2/L)² P₂ = (1/4) * (2/L)² and P₃ = (1/3) * (2/L)² respectively
a.
The wave function for the particle in a rigid box of length L for the state
n = 1, is given by:
|ψ₁(x)|² = (2/L)² * sin²(πx/L)
the wave function for the state n = 2, is given by:
|ψ₂(x)|² = (2/L)² * sin²(2πx/L)
And the wave function for the state n = 3, is given by:
|ψ₃(x)|² = (2/L)² * sin²(3πx/L)
The graphs of |ψ₁(x)|², |ψ₂(x)|², and |ψ₃(x)|² is attached.
b.
The points of maximum probability density, which occur at x = 0 and x = L for all three states are the positions at which the particle is most likely to be found.
c.
The points of minimum probability density, occur at x = L/2 for the state
n = 1, and at x = L/4 and x = 3L/4 for the state n = 2. For the state n = 3, the points of minimum probability density occur at x = L/3, x = 2L/3, and
x = L are the positions at which the particle is least likely to be found.
d.
Since the probability density is highest at x = 0, which is in the left one-third of the box for the state n = 1 the probability of finding the particle in the left one-third of the box is greater than 1/3.
Since the probability density is highest at x = 0 and x = L, which are both outside the left one-third of the box for the state n = 2, the probability of finding the particle in the left one-third of the box is less than 1/3.
Since the probability density is highest at x = 0, which is in the left one-third of the box for the state n = 3, the probability of finding the particle in the left one-third of the box is equal to 1/3.
e.
We need to integrate the probability density over the region of interest to calculate the probability that the particle will be found in the left one-third of the box for each of the states n = 1, n = 2, and n = 3.
the probability for the state n = 1, is given by:
P₁ = ∫ |ψ₁(x)|² dx = (2/L)² * ∫ sin²(πx/L) dx
The probability for the state n = 2, is given by:
P₂ = ∫ |ψ_2(x)|² dx = (2/L)² * ∫ sin²(2πx/L) dx
The probability for the state n = 3, is given by:
P₃ = ∫ |ψ₃(x)|² dx = (2/L)² * ∫sin²(3πx/L) dx
Here all integrals have limits 0 and L/3
Solving these integrals gives:
P₁ = (3/4) * (2/L)²
P₂ = (1/4) * (2/L)²
P₃ = (1/3) * (2/L)²
Thus, the probabilities that the particle will be found in the left one-third of the box are:
P₁ = (3/4) * (2/L)²
P₂ = (1/4) * (2/L)²
P₃ = (1/3) * (2/L)²
for the states n = 1, n = 2, and n = 3, respectively.
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