Circle P is shown. Line segments P T, P U, P R, and P Q are radii. Line segments T S and S U are secants. Angle T S U is 120 degrees, angle U P R is 59 degrees, and angle Q P T is degrees. Use the drop-down menus to complete the statements. A central angle of circle P is angle . The measure of is degrees.
Answers
A central angle of circle P is angle TPQ.
The measure of RU is 59 degrees.
What is an arc?In Geometry, an arc can be defined as a trajectory that is generally formed when the distance from a given point has a fixed numerical value. Generally speaking, the degree measure of an arc in a circle is always equal to the central angle that is present in the included arc.
Based on the information provided about Circle P, we can logically deduce that m∠TPU and m∠UPR are supplementary angles:
m∠TPU + m∠UPR = 180°
m∠TPU = 180° - 59°
m∠TPU = 121°.
m∠TPQ + m∠QPR = 180°
m∠QPR = 180° - 107°
m∠QPR = 73°.
Therefore, the central angle of circle P is m∠TPQ and the measure of arc RU is equals to 59 degrees.
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Answer:
1. TPQ
2. 58
Step-by-step explanation:
Related Questions
Please help me answer this math problem!
Answers
Answer:
P(t) = 30t + 4820
In 2009, the moose population will be 5,330 according to the model.
Step-by-step explanation:
4970 - 4820 = 150
150 / 5 = 30
P(t) = 30t + 4820
30 x 17 = 510
510 + 4820 = 5330
Referring to the figure, use the graph to estimate the
roots of the equation: y = x2 - 2x - 8
Answers
Answer:
The roots are -2 and 4
Step-by-step explanation:
Using the graph, we can estimate the roots of the equation by seeing where it crosses the x axis.
It appears to cross the x axis at -2 and 4.
The roots are -2 and 4
If two medians of a triangle are congruent, then what is the triangle?
Answers
The equation:
(x³+2x²) (3x²+2x+1)=x³ (3x² + 2x+1) + 2x² (3x² + 2x+1) is an example of which method/property?
A. Distributive Property
B. FOIL
C. Vertical multiplication
D. Collect like terms
Answers
The equation (x³+2x²) (3x²+2x+1)=x³ (3x² + 2x+1) + 2x² (3x² + 2x+1) is an example of the Distributive Property. The Distributive Property states that when multiplying two terms, you can multiply each term in one expression by each term in the other expression. In this equation, the terms in the first expression (x³+2x²) are multiplied by each term in the second expression (3x²+2x+1). This is why the equation is an example of the Distributive Property.
classify the following set of measurements as accurate, precise, both, or neither. checking for consistency in the weight of chocolate chip cookies: 17.27 g, 13.05 g, 19.46 g, 16.92 g
Answers
The dimensions listed above are thus neither exact nor precise.
In the given question the chocolate chip cookies weigh 17.27 g, 13.05 g, 19.46 g, and 16.92 g, respectively and we have to classify the following set of measurements as accurate, precise, both, or neither.
It is the "value" that a buyer is prepared to pay for an object, particularly a used or second-hand car; typically, the "real worth" of any used car is a predetermined price tag after determining its value based on its condition and usage.
The chocolate chip cookies weigh 17.27 g, 13.05 g, 19.46 g, and 16.92 g, respectively. This demonstrates that the weight of chocolate chips is not comparable to their actual worth.
The dimensions listed above are thus neither exact nor precise.
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I've been working on this problem for the last hour and can't quite seem to get it right.
A tank is full of water. Find the work required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft3. (Assume a = 4 ft, b = 5 ft, and c = 6 ft.)
Answers
The work required to pump the water which weighs 62.5 lb/ft3 out of the spout is 15000 foot-pounds.
To find the work required to pump the water out of the spout, we need to calculate the weight of the water in the tank and then multiply by the height of the spout to find the potential energy of the water.
First, we need to find the volume of the tank, which is given by:
V = a * b * c = 4 ft * 5 ft * 6 ft = 120 ft^3
Next, we can find the weight of the water in the tank, using the fact that water weighs 62.5 lb/ft^3:
W = V * ρ = 120 ft^3 * 62.5 lb/ft^3 = 7500 lb
So the weight of the water in the tank is 7500 pounds.
Now, we can find the potential energy of the water by multiplying the weight by the height of the spout. Let's assume the spout is 2 feet above the top of the tank:
PE = W * h = 7500 lb * 2 ft = 15000 ft-lb
So the work required to pump the water out of the spout is 15000 foot-pounds (ft-lb). This means that if we used a pump that was 100% efficient, it would require 15000 ft-lb of work to pump all the water out of the tank. However, in reality, pumps are not 100% efficient, so the actual work required would be greater than 15000 ft-lb.
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22\5= how many /10?
44
22
11
Or
10
Answers
Answer:
the correct answer is 44
Step-by-step explanation:
22/5 = 4.4
44/ 10 = 4.4
An train station has determined that the relationship between the number of passengers on a train and the total weight of luggage stored in the
baggage compartment can be estimated by the least squares regression equation y = 200+ 23x. Predict the weight of luggage for a flight with
57 passengers.
Answer: pounds
Answers
The residual sum of squared values is reduced using the least-squares method. The predicted weight of luggage for a flight with 57 passengers is 1511 Pounds.
What is meant by least-squares method?
The least-squares method is a statistical technique used to find the best fit line for a set of data points. It works by minimizing the sum of the squared differences between the observed values of a dependent variable and the values predicted by a linear function of one or more independent variables.
In other words, the least-squares method finds the line that best represents the relationship between two or more variables, such that the sum of the squared errors (the difference between the observed values and the predicted values) is minimized. The resulting line is known as the regression line or best-fit line, and it can be used to make predictions about future values of the dependent variable based on the values of the independent variable.
Based on the given least squares regression equation, we have:
y: the total weight of luggage stored in the baggage compartmentx: the number of passengers on the train200: the intercept term of the regression equation23: the slope of the regression equationTo predict the weight of luggage for a flight with 57 passengers, we simply substitute x = 57 into the regression equation:
y = 200 + 23x
= 200 + 23(57)
= 200 + 1311
= 1511
Therefore, the predicted weight of luggage for a flight with 57 passengers is 1511 pound.
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Your credit card has a balance of $4500 and an interest rate of 22%. The credit card requires a minimum payment of 3%.
What is your minimum payment?
What is your interest for 1 month?
What is your new balance?
Answers
Answer: Your welcome!
Step-by-step explanation:
Your minimum payment would be $135 ($4500 x 0.03 = $135).
Your interest for 1 month would be $99 ($4500 x 0.22 = $990 x 0.1 = $99).
Your new balance would be $4634 ($4500 + $99 - $135 = $4664).
IV. Long Answer and Analysis (20 points) Do a simple Price Elasticity of Demand Calculation: ; A. If an original price was $5.00 and the amount bought (purchased) was an amount of 50 units (say of good like McDonalds Big Macs). What is the Revenue the producer receives at this point: $________ B. Now the price is lowered to $4.00 and now 200 Big Macs are purchased. What is the new revenue at this point: has it gone up or down?
C. using the formula for Price elasticity of Demand, calculate the actual fraction value of the change In price
Answers
If the quantity of big macs demanded falls from 2.0 million to 1.6 million as the price of whoppers falls from $1.40 to $.80, the cross-price elasticity of demand for big macs is 0.4.
What is cross-price elasticity?Cross-price elasticity quantifies how responsive a product's demand is to changes in a related product price. Some products on the market frequently have connections to one another.
This could imply that the demand for a product can be favourably or negatively impacted by a product's price change. The cross elasticity of demand, also known as the cross-price elasticity of demand, in economics quantifies the relationship between the percentage change in the quantity of a commodity sought and the percentage change in the price of another good, ceteris paribus. The cost of milk is one such example. People might switch to 2% milk if full milk prices rise. Similar to how whole milk gets more popular as 2% milk prices go up.
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An experiment observing the growth of two germ strand populations finds these patterns.
• The population of the first germ is represented by the function A(x) = (1.3)^x+9
• The population of the second germ is represented by the function B (x) = ^4x+1
Which function best represents function R, the ratio of the population of the second germ to the population of the first germ?
A. R(x) = (1.3)^3x-8
B
An experiment observing the growth of two germ strand populations finds these patterns.
• The population of the first germ is represented by the function A(x) = (1.3)^x+9
• The population of the second germ is represented by the function B (x) = ^4x+1
Which function best represents function R, the ratio of the population of the second germ to the population of the first germ?
A. R(x) = (1.3)^3x-8
B. R(x) = (1.3)^5x+10
C.R(x) = (2.6) ^3x-8
D. R(X) =(2.6) ^4x^2+9
Answers
The right answer, based on the provided assertion, is (C) R(x) = 2.6³ˣ⁻⁸.
What does an arithmetic the expression mean?Mathematical expressions consist of at least two integers or factors, a minimum of one math procedure, and a sentence. It's possible to multiply, divide, add, or remove with this mathematical procedure. An expression's form is as follows: Expression: (Math Operator, Number/Variable, Math Operator)
To find the ratio of the population of the second germ to the population of the first germ, we can divide the function for the population of the second germ by the function for the population of the first germ. That is:
R(x) = B(x) / A(x)
Plugging in the given functions, we get:
R(x) = (x+1) / (1.3)ˣ⁺⁹
To simplify this expression, we can use the properties of exponents and rewrite it as:
R(x) = 2.6ˣ⁻⁸
Therefore, the function that best represents the ratio of the population of the second germ to the population of the first germ is R(x) = 2.6ˣ⁻⁸
So, the correct option is (C) R(x) = (2.6)³ˣ⁻⁸.
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1. Cuando Raúl y Esperanza llegaron a una fiesta quedaban 3/10 del pastel, así que se dividieron esa porción en partes iguales. ¿Qué parte del pastel completo le tocó a cada uno?
Answers
Answer:
13
Step-by-step explanation:
first multiply then divide and subtract
Forty percent of what number is 18?
Answers
Answer:
45
Step-by-step explanation:
Y = forty percent of blank to make 18
18 = 40/100 x Y
Y = 18 x 100/40
Y = 18 x 100 divided by 40
Y = 45
10. Usando el metodo de la iteracion del punto fijo
¿Para que raiz el porcentaje de error es del
16.1%?
*
(4 puntos)
f (x) = x − (√5x – e^x)comenzando en xo = 0.5
O 1.7278
O 1.7351
O 1.4482
O 0.9226
Answers
Answer: Your welcome!
Step-by-step explanation:
O 1.7278
El porcentaje de error del 16.1% se refiere a la raíz de la ecuación f(x) = x - (√5x - e^x), comenzando en xo = 0.5. Utilizando el método de iteración del punto fijo, se obtiene que la raíz de esta ecuación es 1.7278.
Answer:
Step-by-step explanation:
Para encontrar la raíz de la ecuación f(x) = x - (√(5x) - e^x) con un porcentaje de error del 16.1%, podemos utilizar el método de iteración de punto fijo con la siguiente fórmula:
x1 = f(x0) + x0
Donde x0 es el valor inicial, f(x) es la función y x1 es el siguiente valor de la raíz aproximada.
Primero, evaluamos la función en x0 = 0.5:
f(0.5) = 0.5 - (√(5*0.5) - e^0.5) = -0.2749
Luego, usamos la fórmula para encontrar el siguiente valor:
x1 = f(0.5) + 0.5 = 0.2251
Continuamos iterando hasta que el porcentaje de error sea menor o igual al 16.1%. Podemos calcular el porcentaje de error relativo por iteración con la siguiente fórmula:
Error relativo (%) = |(x_nuevo - x_viejo) / x_nuevo| * 100%
donde x_nuevo es el valor actual de la raíz aproximada y x_viejo es el valor anterior de la raíz aproximada.
Después de algunas iteraciones, encontramos que la respuesta que tiene un porcentaje de error del 16.1% es 1.7351.
Evaluate h - 3 for h = 1
Answers
The result of the function is h = 1 is equal to -2
Evaluating linear equationsLinear equations are equations that has a leading degree of 1.
Given the linear equation f(h) = h - 3
We need to determine the resulting value when h is equal to 1
f(1) = 1 - 3
f(1) = -2
Hence the value of the function given if the value of h is 1 is -2
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g consider the following sets of vectors. show that each set (i) contains the zero vector, (ii) is closed under addition and scalar multiplication. then find a basis for each set and give the dimension.
Answers
(i) A basis for the set is {(2, 3, 4), (3, -2, 1)} and the dimension is 2.
(ii) A basis for the set is {(1, 1, 1, 1), (2, 0, 0, -1)} and the dimension is 2.
(i) {(2, 3, 4), (3, -2, 1)}
(i) The set contains the zero vector (0, 0, 0). It is closed under addition because for any two vectors in the set, the sum of the two vectors is also in the set. It is closed under scalar multiplication because for any vector in the set and any scalar, the scalar multiplied by the vector is also in the set.
A basis for the set is {(2, 3, 4), (3, -2, 1)} and the dimension is 2.
(ii) {(1, 1, 1, 1), (2, 0, 0, -1)}
(ii) The set contains the zero vector (0, 0, 0, 0). It is closed under addition because for any two vectors in the set, the sum of the two vectors is also in the set. It is closed under scalar multiplication because for any vector in the set and any scalar, the scalar multiplied by the vector is also in the set.
A basis for the set is {(1, 1, 1, 1), (2, 0, 0, -1)} and the dimension is 2.
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The complete question is:
Consider the following sets of vectors. show that each set (i) contains the zero vector, (ii) is closed under addition and scalar multiplication. then find a basis for each set and give the dimension.
(i) {(2, 3, 4), (3, -2, 1)}
(ii) {(1, 1, 1, 1), (2, 0, 0, -1)}
What are the coordinates of R' for the Dilation D(0,5 P) (PQRS)?
Answers
The coordinate of R' after dilation of (0.5, P) is at R(- 1.5, - 1.5).
What are transformations?Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.
Dilation: The preimage is scaled up or down to create the image.
Reflection: The picture is a preimage that has been reversed.
Rotation: Around a given point, the preimage is rotated to create the final image.
Translation: The image is translated and moved a fixed amount from the preimage.
Given, The coordinate point of R is (1, 1).
Now, After a dilation of 0.5 it would be (1×0.5, 1×05) ⇒ (0.5, 0.5).
Again a dilatation of P(- 3, - 3) would be, - 3[(0.5, 0.5)] ⇒ (- 1.5, - 1.5)
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The number 345,600 can be expressed as 2^a*3^b*5^b for integers a, b and c. What is the value of the product abc?
Answers
For a number 345,600 which is expressed as 2ᵃ × 3ᵇ × 5ᶜ, ae integers a, b and c. Then the value of the product abc is equals to the 54.
We have, a number 345,600, which can be expressed as 2ᵃ × 3ᵇ × 5ᶜ. That is it can be written as 345,600 = 2ᵃ × 3ᵇ × 5ᶜ, which is prime factorazation of number. To determine the value of a , b , c, we have to do prime factorization of number 345,600. Prime factorzation of a number contains all prime factors of a number. So,
345600 = 3456×100 = 576× 6 × 100
= 24² × 2×3 × 10² = ( 2×2×2×3)² × 2× 3×(2× 5)²
= 2⁶ × 3² × 2× 3×2²× 5²
345,600 = 2⁹ × 3³ × 5²
but 345,600 = 2ᵃ × 3ᵇ × 5ᶜ,
=> 2ᵃ × 3ᵇ × 5ᶜ = 2⁹ × 3³ × 5²
On comparing the powers of 2, 3, and 5 in above equation, we get a = 9 , b = 3 , c = 2.
Therefore, the value of product of a, b, and c
= abc = 9×3×2 = 54
Hence, the value of product is 54.
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Complete question:
The number 345,600 can be expressed as 2ᵃ × 3ᵇ × 5ᶜ for integers a, b and c. What is the value of the product abc?
what is the kinetic energy of 1500 kg pick-up truck traveling at 31.0 m/sec.
Answers
The kinetic energy of a 1500 kg pick-up truck traveling at 31.0 m/sec will be 720 kJ.
What is kinetic energy?If the body of the mass (m) is moving with a velocity (v), then the body possesses the energy which is known as kinetic energy. The SI unit of the kinetic energy is Joule. Then the formula is given as,
KE = 1/2 x mv²
The kinetic energy of a 1500 kg pick-up truck traveling at 31.0 m/sec is given as,
KE = 1/2 x 1500 x 31²
KE = 720,750 Joule
KE = 720 kJ
The kinetic energy of a 1500 kg pick-up truck traveling at 31.0 m/sec will be 720 kJ.
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A car drove at 65 miles per
hour for 7 hours. How
many miles are left if the
entire trip is 540 miles?
Answers
Answer: 85 miles left
Step-by-step explanation:
65 miles x 7 hours = 455 miles
540 miles - 455 miles = 85 miles
In a study of television viewing habits, it is desired to estimate the average number of hours teenagers spend watching per week. If it is reasonable to assume σ = 3.2 hours, how large a sample is needed so that it will be possible to assert with 95 percent confidence that the sam mean is off by at most 20 minutes?
Answers
We need a sample size of at least 397 in order to be 95% confident that the sample mean is within 20 minutes of the true mean.
To calculate the sample size needed, we need to use the formula:
n = (zα/2σ/E)2
where zα/2 is the critical value associated with the desired confidence level (in this case 95%), σ is the standard deviation (3.2 hours in this case), and E is the margin of error (20 minutes in this case).
Thus, n = (1.96*3.2/20)2, which simplifies to 397.
E = 20
σ = 3.2
n = (1.96*3.2/20)2
Therefore, we need a sample size of at least 397 in order to be 95% confident that the sample mean is within 20 minutes of the true mean.
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X
5
Z
10
5√3
Y
Find sin cos tan
Answers
The trigonometry ratios have their values to be
sin(x) = √3/2cos(x) = 0.5tan(x) = √3How to determine the trigonometry ratiosThe complete question is added as an attachment
From the question, we have the following parameters that can be used in our computation:
Hypotenuse = 10
Adjacent = 5
Opposite = 5√3
Using the above as a guide, we have the following:
sin(x) = Opposite/Hypotenuse
cos(x) = Adjacent/Hypotenuse
tan(x) = Opposite/Adjacent
Using the above as a guide, we have the following:
sin(x) = 5√3/10 = √3/2
cos(x) = 5/10 = 0.5
tan(x) = 5√3/5 = √3
The above are the values of the trigonometry ratios based on the attached figure
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Kendallyn wants to know the height of a sailboat mast. if the sailboat mast casts a shadow 45 meters long and a nearby marble column that is 8 meters tall casts a shadow 15 meters long, then the sailboat mast is___ meters long. Write your answer as a whole number or a decimal.
Answers
The sailboat mast casts a shadow 45 meters long and a nearby marble column that is 8 meters tall casts a shadow 15 meters long, then the height of the sailboat mast is 24 meters.
We can use proportions to solve this problem. Let's let h be the height of the sailboat mast, then we have:
h/45 = 8/15
Cross-multiplying, we get:
15h = 8 * 45
Simplifying, we get:
15h = 360
Dividing by 15, we get:
h = 24
Therefore, the height of the sailboat mast is 24 meters. This means that the sailboat mast is 3 times as tall as the marble column, since its shadow is 3 times as long.
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A television and DVD player cost total of $1192. The cost of the television is three times the cost of the DVD player. Find the cost of each, item.
Cost of television:
Cost of DVD player:
Answers
The cost of the DVD player and the cost of the television will be $298 and $894, respectively.
What is the solution to the equation?An answer to a formula is any variable value that fulfills the equal outcomes, that is, it tends to make the Left Hand Side (LHS) and Right Hand Side (RHS) of the formula equal. To solve an equation, you must locate the feasible solution) to that formula.
A television and DVD player cost a total of $1192. The cost of the television is three times the cost of the DVD player.
Let 'x' be the cost of the DVD player and 'y' be the cost of the television. Then the equations are given as,
x + y = 1,192 ...1
y = 3x ...2
From equations 1 and 2, then we have
x + 3x = 1192
4x = 1192
x = $298
Then the value of 'y' is given as,
y = 3 (298)
y = $894
The cost of the DVD player and the cost of the television will be $298 and $894, respectively.
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Part 2
Bc my autocorrect messed up my other question.
Make a word using all of these letters
RrCsVb
This doesn't have to be an actual word but one that is memorable. Please use all of the words
Answers
Answer:
CrabSVR
BRscrV
ScarbRV
VRbScCr
Step-by-step explanation:
I can suggest that some of these words might be memorable because they are short and have a combination of consonants and vowels that make them sound unique and catchy. For example, "ScrubVR" has a rhythmic and playful sound to it, while "BRscrV" has a more abbreviated and tech-like feel to it. Ultimately, what makes a word memorable is highly subjective and varies from person to person.
Answer: he right
Step-by-step explanation:
Evaluate the following expression by drawing the unit circle and the appropriate right triangle. The angle is in radians.
cos(5π/3)
Answers
The answer for the angle cos (5π/3) is 1/2.
We know that in the polar coordinate system,
x = r cosθ ; y = r sinθ
Since, we are approximating using a unit circle, therefore, r = 1
As we can see in the graph of unit circle, we take the angle in radians anti-clockwise as 5π/3.
As we want the value of cos(5π/3), we look for the x coordinate.
We get this from the end point of the angle that we have drawn on the unit circle in anti-clockwise motion.
The x co-ordinate is 1/2, hence we get the value of cos(5π/3) as 1/2.
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what is the equation of the secant line connecting the point (1,f(1)) to the point (0.9,f(0.9))? the equation of this secant line is y
Answers
(a) The equation of the secant line connecting (1, f(1)) and (0.9, f(0.9)) is y = -x + 2.1.
(b) The slope of the tangent line to f at x is f'(x) = -cos(x), and the equation of the tangent line at (x, f(x)) is y = -cos(x)*x + sin(x) + cos(x)*x.
In contrast to the tangent line, which meets a curve just once and has the same slope as the curve at that point, the secant line is a straight line that connects two locations on a curve. In this task, we are required to determine the equations of the secant and tangent lines at various places given a function f(x).
Using the two provided locations, we determine the slope of the secant line in section (a), which is equal to the difference between the y- and x-coordinates. The equation of the line is therefore written using the point-slope form, and it is y = -x + 2.1.
In part (b), we use the definition of the derivative to find the slope of the tangent line at a given point x. We then use the point-slope form of a line to write the equation of the tangent line, which has the same slope as the derivative and passes through the point (x, f(x)). In this case, the slope of the tangent line is -cos(x), and the equation of the tangent line is
y = -cos(x)*x + sin(x) + cos(x)*x.
The complete question is provided in the image below.
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4,400 is placed in an account with an annual interest rate of 8.25%. How much will be in the account after 22 years to the nearest cent
Answers
After 22 years the amount in the account will be 12,386.
What is simple interest?Simple interest is an interest amount calculated on the principal amount with certain interest rate. It is given by the formula,
[tex]S.I.=\frac{P*n*r}{100}[/tex]
Where P is the principal amount, n is the number of years, r is the rate of interest.
4,400 is placed in an account i.e., Principal amount, P=4400
With an annual interest rate, r=8.25%
Number of years, n=22 years.
Amount in the account after 12 years will be,
By using simple interest formula,
[tex]S.I.=\frac{P*n*r}{100}[/tex]
= [tex]\frac{4400*8.25*22}{100}[/tex]
=7,986
Interest amount = 7,896.
Total amount after 22 years= interest amount + principal amount
= 7896+4400
= 12,386.
Hence, after 22 years the amount in the account will be 12,386.
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Find the area of the isosceles triangle
Answers
The area of the isosceles triangle is 96 ft²
How to determine the areaThe formula for area of a triangle is expressed as
Area = 1/2bh
Where , b is the base and h is the height
We have that;
Base = 16ft
Using the Pythagorean theorem, we have;
20² - 16² =h²
Find the squares
h = √144
h= 12ft
Substitute the values
Area = 1/2 × 12 × 16
Multiply the values
Area = 96 ft²
Hence, the value is 96 ft²
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Let us now consider the Schrödinger Equation for an electron confined to a two dimensional box, 0
and 0
. That is to say, within this rectangle the electron wavefunction behaves as a free particle ( V(x,y)=0
), but the walls are impenetrable so the wavefunction ψ(x,y,t)=0
at the walls.
3.11.1.svg
Figure 1
: A particle in a a×b
2-dimensional box with infinite height can only be found within the box (highlighted). (CC BY-NC; Ümit Kaya)
Extending the (time-independent) Schrödinger equation for a one-dimensional system
−ℏ22md2ψ(x)d x2+V(x)ψ(x)=Eψ(x)(1)
to a two-dimensional system is not difficult
−ℏ22m(∂2ψ(x,y)∂x2+∂2ψ(x,y)∂y2)+V(x,y)ψ(x,y)=Eψ(x,y).(2)
Equation 2
can be simplified for the particle in a 2D box since we know that V(x,y)=0
within the box and V(x,y)=[infinity]
outside the box, ii.e.
V(x,y)=⎧⎩⎨0[infinity][infinity]0≤x≤aand0≤y≤bx<0andx>ay<0andy>b
So Equation 2
becomes
−ℏ22m(∂2ψ(x,y)∂x2+∂2ψ(x,y)∂y2)=Eψ(x,y).(3)
Since the Hamiltonian (i.e. left side of Equation 3
) is the sum of two terms with independent (separate) variables, we try a product wavefunction like in the Separation of Variables approach used to separate time-dependence from the spatial dependence previously. Within this approach we express the 2-D wavefunction as a product of two independent 1-D components
ψ(x,y)=X(x)Y(y).(4)
This ansatz separates Equation 3
into two independent one-dimensional Schrödinger equations
−ℏ22m(d2X(x)dx2)=εxX(x).(5)
−ℏ22m(d2Y(y)dy2)=εyY(y).(6)
where the total energy of the particle is the sum of the energies from each one-dimensional Schrödinger equation
E=εx+εy(7)
The differential equations in Equations 5
and 6
are familiar as they were found for the particle in a 1-D box previously. They have the general solution
X(x)=Axsin(kxx)+Bxcos(kxx)(8)
Y(y)=Aysin(kyy)+Bycos(kyy)(9)
Answers
For a particle in a 2D box with impenetrable walls, the time-independent Schrödinger equation can be simplified to equation 3, which is a separable differential equation.
Using the Separation of Variables approach, the 2D wavefunction can be expressed as the product of two independent 1D components, given by equation 4.
The resulting differential equations in equations 5 and 6 have the same form as those found for the particle in a 1D box and their solutions are given by equations 8 and 9. The total energy of the particle is given by equation 7.
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