Calculate Laplace transform of the below: 0,5 < 0 The Impulse Response: u(t) = 300,t = 0 0,t> 0 - The unit step function: u(t) = 1,t > 0 - The unit ramp function (slope=1): r(t) = t, t > 0 The exponential function: f(t) = e-atu(t),t 20 # Cosine function: f(t) = cos(wt)u(t),t>=0.
Answers
1) The Laplace transform of a function f(t) is ∫[0 to ∞] e^(-st) * f(t) dt
2) Impulse Response = 1/s
3) Unit Step Function = 1/s
4) Unit Ramp Function = 1/s^2
5) The exponential function= 1/(s + a)
6) Cosine function = -s / (s^2 + w^2),
1) The Laplace transform of a function f(t) is defined as:
F(s) = L{f(t)} = ∫[0 to ∞] e^(-st) * f(t) dt,
where s is the complex frequency parameter.
2) Impulse Response:
The impulse response u(t) can be represented as a unit step function. Therefore, the Laplace transform of the impulse response is:
L{u(t)} = ∫[0 to ∞] e^(-st) * u(t) dt
= ∫[0 to ∞] e^(-st) * 1 dt
= ∫[0 to ∞] e^(-st) dt
= [-1/s * e^(-st)] [0 to ∞]
= -1/s * (e^(-s * ∞) - e^(-s * 0))
= -1/s * (0 - 1)
= 1/s,
where s > 0.
3) Unit Step Function:
The unit step function u(t) can be directly transformed using the definition of the Laplace transform:
L{u(t)} = ∫[0 to ∞] e^(-st) * u(t) dt
= ∫[0 to ∞] e^(-st) * 1 dt
= ∫[0 to ∞] e^(-st) dt
= [-1/s * e^(-st)] [0 to ∞]
= -1/s * (e^(-s * ∞) - e^(-s * 0))
= -1/s * (0 - 1)
= 1/s,
where s > 0.
4) Unit Ramp Function:
The unit ramp function r(t) = t can be transformed as follows:
L{r(t)} = ∫[0 to ∞] e^(-st) * r(t) dt
= ∫[0 to ∞] e^(-st) * t dt
= ∫[0 to ∞] t * e^(-st) dt.
To calculate this integral, we can use integration by parts. Let's assume u = t and dv = e^(-st) dt. Then, du = dt and v = (-1/s) * e^(-st). Applying integration by parts, we have:
∫[0 to ∞] t * e^(-st) dt = [-t * (1/s) * e^(-st)] [0 to ∞] - ∫[0 to ∞] (-1/s) * e^(-st) dt
= [(-t/s) * e^(-st)] [0 to ∞] + (1/s) * ∫[0 to ∞] e^(-st) dt
= [(-t/s) * e^(-st)] [0 to ∞] + (1/s) * (1/s),
where s > 0.
Since the term (-t/s) * e^(-st) approaches zero as t approaches infinity, the first part of the integral becomes zero. Therefore, we are left with:
L{r(t)} = (1/s) * (1/s)
= 1/s^2,
where s > 0.
5) Exponential Function:
The exponential function f(t) = e^(-at) * u(t) can be transformed as follows:
L{e^(-at) * u(t)} = ∫[0 to ∞] e^(-st) * e^(-at) * u(t) dt
= ∫[0 to ∞] e^(-st - at) dt
= ∫[0 to ∞] e^(-(s + a)t) dt
= [-1/(s + a) * e^(-(s + a)t)] [0 to ∞]
= -1/(s + a) * (e^(-(s + a) * ∞) - e^(-(s + a) * 0))
= -1/(s + a) * (0 - 1)
= 1/(s + a),
where s + a > 0.
6) Cosine Function:
The cosine function f(t) = cos(wt) * u(t) can be transformed as follows:
L{cos(wt) * u(t)} = ∫[0 to ∞] e^(-st) * cos(wt) * u(t) dt
= ∫[0 to ∞] e^(-st) * cos(wt) dt.
To evaluate this integral, we can use the Laplace transform of the cosine function, which is given by:
L{cos(wt)} = s / (s^2 + w^2), where s > 0.
Therefore, we have:
L{cos(wt) * u(t)} = ∫[0 to ∞] e^(-st) * (s / (s^2 + w^2)) dt
= (s / (s^2 + w^2)) * ∫[0 to ∞] e^(-st) dt
= (s / (s^2 + w^2)) * (-1/s * e^(-st)) [0 to ∞]
= (s / (s^2 + w^2)) * (0 - 1)
= -s / (s^2 + w^2),
where s > 0.
These are the Laplace transforms of the given functions.
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Related Questions
Hexadecimal numbers use the 16 "digits": 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. a) What is the base 10 value of the 3-digit hexadecimal number 2E5? Show your work. b) Find the probability that a 3-digit hexadecimal number with repeated digits allowed contains only letters, like ACC. (Note: Part (b) has nothing to do with part (a) of this problem.) Write your answer as a simplified fraction, not a decimal or percent. Explain briefly how you got it.
Answers
The base 10 value of the 3-digit hexadecimal number 2E5 is 741. The probability that a 3-digit hexadecimal number with repeated digits allowed contains only letters is 27/512.
a) To convert a hexadecimal number to its decimal equivalent, you can use the following formula:
(decimal value) =\((last digit) * (16^0) + (second-to-last digit) * (16^1) + (third-to-last digit) * (16^2) + ...\)
Let's apply this formula to the hexadecimal number 2E5:
(decimal value) = \((5) * (16^0) + (14) * (16^1) + (2) * (16^2)\)
= 5 + 224 + 512
= 741
Therefore, the base 10 value of the 3-digit hexadecimal number 2E5 is 741.
b) To find the probability that a 3-digit hexadecimal number with repeated digits allowed contains only letters, we need to determine the number of valid options and divide it by the total number of possible 3-digit hexadecimal numbers.
The number of valid options with only letters can be calculated by considering the following:
The first digit can be any letter from A to F, giving us 6 choices.The second digit can also be any letter from A to F, including the possibility of repetition, so we have 6 choices again.The third digit can also be any letter from A to F, allowing repetition, resulting in 6 choices once more.Therefore, the total number of valid options is 6 * 6 * 6 = 216.
The total number of possible 3-digit hexadecimal numbers can be calculated by considering that each digit can be any of the 16 possible characters (0-9, A-F), allowing repetition. So, we have 16 choices for each digit.
Therefore, the total number of possible 3-digit hexadecimal numbers is 16 * 16 * 16 = 4096.
The probability is then calculated as:
probability = (number of valid options) / (total number of possible options)
= 216 / 4096
To simplify the fraction, we can divide both numerator and denominator by their greatest common divisor, which in this case is 8:
probability = (216/8) / (4096/8)
= 27 / 512
Therefore, the probability that a 3-digit hexadecimal number with repeated digits allowed contains only letters is 27/512.
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You are starting a savings account for college. You put $1,000 in as your starting balance. You earn simple interest at 10% every year. You also must pay 30% income tax on the interest earned annually. Calculate the balance for the account after 5 years
Answers
Answer:
$1,350
Step-by-step explanation:
Simple interest formula
I = Prt
where:
I = interestP = principalr = interest rate (in decimal form)t = time (in years)Given:
P = $1,000r = 10% = 0.1t = 5 years⇒ Total interest earned = 1000 × 0.1 × 5 = $500
If you have to pay 30% income tax on the interest earned annually, then you will keep 70% of your earned interest.
⇒ 70% of $500 = 0.7 × $500 = $350
Account balance = principal + interest earned after tax
= $1,000 + $350
= $1,350
Find the unit rate for the cost of each energy drink if 5 energy drinks cost $15
$3 per energy drink
$5 per energy drink
$5.50 per energy drink
$75 per energy drink
Answers
Explanation: 15/3 is 5 so it would be 5 dollars have a nice day (:
8 cos(2x)+7
What is the period
Answers
the period is π in the given cos form 8 cos(2x)+7.
What is the period in trigonometry?The period of the function is the interval between repetitions of any function. A trigonometric function's period is the length of one whole cycle.
Given a trigonometric function 8 cos(2x)+7
Use the form a cos(bx−c)+d
a = 8
b = 2
c = 0
d = 7
to find the amplitude, period, phase shift, and vertical shift.
Amplitude: 8
The period of the function can be calculated using 2π / |b|.
The absolute value is the distance between a number and zero. The distance between 0 and 1 is 2.
Period: π
Phase Shift: None
Vertical Shift: 7
Therefore, In the provided cosine form, the period is equal to 8 cos(2x)+7.
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Sarah surveyed 100 people. She asked each person two
Sarah made the following two-way table to display information about the responces
of the people Sarah surveyed who listen to jazz, what is the probability that a randomly
selected person also plays a musical instrument?
Answers
Solve the proportion.
s+1/4 = 4/8
s=
Answers
Answer:
8 = 16
Step-by-step explanation:
1/4 = 4 /8
Cross multiply:
1 * 8 = 4 * 4
Simplifying
1 * 8 = 4 * 4
Multiply 1 * 8
8 = 4 * 4
Multiply 4 * 4
8 = 16
Solving
8 = 16
Factorise fully the following:
a) x² + 3x
b) 2x? - 8x
c) 6x + 9x
d) 12x - 4x?
Answers
Answer:
a) \(x(x+3)\)
b) \(2x(x-8)\)
c) \(3x(2+3x^{2})\)
d) \(4x^{2} (3x-1)\)
Step-by-step explanation:
a) \(x^{2} +3x=x(x+3)\)
b) \(2x^{2} -8x=2x(x-8)\)
c) \(6x+9x^{3} =3x(2+3x^{2})\)
d) \(12x^{3} -4x^{2} =4x^{2} (3x-1)\)
Factors of the following algebraic expression are
a. Factors of x² + 3x are x and x+3
b. Factors of 2x² -8x are 2, x and ( x-4)
c. Factors of 6x +9x³ are 3, x and (2 + 3x²)
d. Factors of 12x³ - 4x² are 4, x, x and (3x -41)
What is factor of an algebraic expression?The factor of an given algebraic expressions are one or more numbers or linear or other expressions that multiply to form the given algebraic expression and cannot be broken down further into simpler expressions.
For finding factors of an algebraic expression we can simply take out common expressions from all the terms of the expression and try to simplify it.
a. x² + 3x = x( x+3), (taking x common from both terms)
b. 2x² -8x = 2x ( x-4) (taking 2x common from both terms)
c. 6x +9x³ = 3x (2 + 3x²) ( taking 3x common from both terms)
d.12x³ - 4x² = 4x× x × (3x -41) ( taking 4x² common from both terms)
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A counterexample for the expression sin(y)*tan(y)= cos(y) is 0 degrees
Answers
Actually, 0 degrees is not a counterexample for the expression sin(y)*tan(y) = cos(y).
To see why, let's substitute y = 0 degrees into the expression:
sin(y)*tan(y) = cos(y)
sin(0)*tan(0) = cos(0)
0*tan(0) = 1
0 = 1
As we can see, the equation does not hold for y = 0 degrees. However, this does not make 0 degrees a counterexample, because 0 degrees is not in the domain of the tangent function.
The tangent function is undefined at odd multiples of 90 degrees (e.g. 90, 270, etc.), because at those angles the denominator of the tangent function becomes zero. Therefore, we cannot substitute y = 0 degrees into the expression sin(y)*tan(y) = cos(y), because it would result in division by zero.
In summary, 0 degrees is not a counterexample for the expression sin(y)*tan(y) = cos(y), because it is not in the domain of the tangent function.
Please help giving up 20 points for answers
Answers
Answer:
Question 2: 6x-21
Question 1: -9x+3
Step-by-step explanation:
Determine the area of the given region. y = x + sin x
Answers
The area of the given region, bounded by the curve y = x + sin(x) and the x-axis over the interval [0, π], is (1/2)(π)^2 + 2.
To find the area of the given region, we need to evaluate the definite integral of the function y = x + sin(x) with respect to x over the interval [0, π].
∫[0, π] (x + sin(x)) dx
To evaluate the integral, we can split it into two separate integrals:
∫[0, π] x dx + ∫[0, π] sin(x) dx
The first integral is the integral of x with respect to x, which is (1/2)x^2.
For the second integral, the antiderivative of sin(x) is -cos(x). Therefore, we have:
(1/2)x^2 - cos(x)
Now, we need to evaluate this expression at the upper and lower limits of integration, which are π and 0, respectively.
Plugging in the values, we have:
Area = [(1/2)(π)^2 - cos(π)] - [(1/2)(0)^2 - cos(0)]
Area = [(1/2)(π)^2 + 1] - [0 - 1]
Area = (1/2)(π)^2 + 2
Thus, the area of the given region, bounded by the curve y = x + sin(x) and the x-axis over the interval [0, π], is (1/2)(π)^2 + 2.
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Incomplete Question
Determine the area of the given region.
y = x + sin x
y
4
3
2
1
元
Note that for the given region, x varies from 0 to π
Hence, to find the area of the
π
given region, integrate
y = x + sin x between x = 0 and x = π
π
The angle represents the phase shift, determined by the initial conditions of the experiment or the position of the weight at t
Answers
At the initial condition, this weight approached the equilibrium position from a positive direction of 5 units. Also, the weight was descending (down) to the ground, with the spring stretching.
How to describe the initial condition of this experiment?First of all, we would derive an expression for the position with respect to the angle (phase shift) under the appropriate conditions:
Asin(ωt + ø) = c₂sinωt + c₁cosωt.
At t = 0, we have:
y = c₂sin(0) + c₁cos(0).
y = 0 + c₁
y = 0 + 5
y = 5 units.
Therefore, we can logically deduce that at the initial condition, this weight approached the equilibrium position from a positive direction of 5 units. Also, the weight was descending (down) to the ground, with the spring stretching.
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Complete Question:
The angle Φ represents the phase shift, determined by the initial conditions of the experiment or the position of the weight at t = 0. If the weight is at its maximum positive position (weight is above equilibrium) at t = 0, then Φ = 0. If the weight is at its maximum negative position (spring is stretched and weight is below equilibrium) at t = 0, then Φ = π. If the weight is traveling in the negative direction and passing through equilibrium at t = 0, then Φ = π/2. In the response box, describe the initial condition of our experiment; specifically, describe the position of the weight and the direction in which it was traveling.
Please help!!!!I’ll mark you as brainliest!!!!!
Answers
Answer: A. $970
Step-by-step explanation:
An individual earning $60,000 would pay 10% on the first $9,700 they earned then pay for remaining amount earned depending on which tax brackets the amount falls within.
4) Express the results of the following calculations with the correct number of significant figures: (a)
5.233
3.41−0.23
×0.205 (b)
4.223−0.08
5.556×2.3
5) Tungsten, the element used to make filaments in light bulbs, has a melting point of 6192∘F. Convert this temperature to degrees Celcius and to kelvin. 6) Aspirin has a density of 1.40 g/cm
3
. What is the volume in cubic centimeters of an aspirin tablet weighing 250mg ? Of a tablet weighing
Answers
(a) 5.2333.41−0.23 × 0.205
= (5.23) * (3.18 - 0.23) * (0.205)
= 8.48013
Rounded to the correct number of significant figures, the result is: 8.48
(b) 4.223-0.085.556×2.3
= (4.14) / (5.556) * (2.3)
= 1.759619378
= 1.76
Rounded to the correct number of significant figures, the result is: 1.76
5) To convert the melting point of tungsten from Fahrenheit to Celsius and Kelvin:
Melting point in Fahrenheit: 6192°F
To convert to Celsius:
°C = (°F - 32) * 5/9
°C = (6192 - 32) * 5/9
°C ≈ 3434.44°C
Rounded to the correct number of significant figures, the result is: 3434°C
To convert to Kelvin:
K = °C + 273.15
K = 3434.44 + 273.15
K ≈ 3707.59K
Rounded to the correct number of significant figures, the result is: 3708K
6) For the volume calculation of the aspirin tablet
Tablet weight: 250 mgTo find the volume, we use the formula:
Volume = Mass / Density
Volume = 250 mg / 1.40 g/cm³
Volume = 250 mg / 1.40 g/cm³ * (1 g / 1000 mg) * (1 cm³ / 1 mL)
Volume ≈ 178.571 cm³
Rounded to the correct number of significant figures, the result is: 179
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what are the missing numbers?
Answers
Which inequality represents all values of × for which the quotient below is lefined?
Answers
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given expression
\(\sqrt{28(x-1)}\div\sqrt{8x^2}\)STEP 2: Simplify the expression
For the quotient to be defined, the numerator and the denominator must be greater than zero, this means that:
\(\begin{gathered} \sqrt{8x^2}>0 \\ \mathrm{Square\:both\:sides} \\ \left(\sqrt{8x^2}\right)^2>0^2 \\ 8x^2>0 \\ x<0\quad \mathrm{or}\quad \:x>0 \end{gathered}\)For the numerator,
\(\begin{gathered} \mathrm{Square\:both\:sides} \\ \left(\sqrt{28\left(x-1\right)}\right)^2>0^2 \\ \mathrm{Simplify} \\ 28x-28>0 \\ x>1 \\ \mathrm{Combine\:the\:intervals} \\ x>1\quad \mathrm{and}\quad \:x\ge \:1 \\ x>1 \end{gathered}\)Merging both interval, we have:
\(\begin{gathered} x<0,x>0,x>1 \\ x>1 \end{gathered}\)Hence, the answer is given as:
\(x>1\)Determine: ∫ydx if :
y=sin⁴4x
Answers
The integral of y = sin⁴4x is **2/5*sin8x + C**, where C is an arbitrary constant.
The integral can be found using the following steps:
1. First, we can use the identity sin²2x = 1 - cos²2x to rewrite y as sin⁴4x = (1 - cos²8x)².
2. Then, we can use the double angle formula cos2x = 2cos²x - 1 to rewrite the expression in terms of cosx.
3. Finally, we can integrate the expression using the reverse power rule and the sum rule for integrals.
The following is the integration process in detail:
```
∫ydx = ∫sin⁴4x dx
= ∫(1 - cos²8x)² dx
= ∫(1 - 2cos²8x + cos⁴8x) dx
= ∫1dx - 2∫cos²8x dx + ∫cos⁴8x dx
= x - 2∫(1 + cos²4x)/2 dx + ∫cos⁴8x dx
= x - ∫1/2 dx - ∫cos²4x dx + ∫cos⁴8x dx
= x - 1/2x + 1/8*sin8x + C
= 2/5*sin8x + C
```
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95 times 10 to the 4 power
Answers
There's a lack of info regarding the question, however if you are looking for the answer in scientific notation and expanded form, here you go.
Scientific Notation:
\(95\) × \(10^4\)
Expanded Form:
\(950,000\)
Not what you're looking for? Feel free to comment on the matter and we'll see what else we can do to assist you further.
I really need help with this
Answers
Answer:
x= 10
Step-by-step explanation:
The angles are corresponding angles, and if corresponding angles are equal, the lines are parallel
4x = 3x+10
Subtract 3x from each side
4x-3x = 3x+10-10
x = 10
This week, Ivania ran a total of 20 km, which is 6.5 km farther than she ran last week. If w represents the number of kilometers she ran last week, what is the equation that models this situation?
Answers
Answer:
Step-by-step explanation:
6.5 + w = 20
13.5 km
Answer:
C first then for second one is b
Step-by-step explanation:
I just did the assignment
In Circle H below if m
Answers
Answer: um what do u mean?
what is the value of result after the following statement executes? result = (3 * 5) 24 / (15 - (7 - 4));
Answers
The value of result after executing the statement is 30.
To evaluate the expression and find the value of result, let's break down the operations step by step, following the order of operations (PEMDAS/BODMAS):
Inside the parentheses, we have:
3 × 5 = 15
Next, the expression becomes:
15 × 24 / (15 - (7 - 4))
Inside the inner parentheses:
7 - 4 = 3
Now, the expression becomes:
15 × 24 / (15 - 3)
Inside the parentheses:
15 - 3 = 12
The expression simplifies to:
15 × 24 / 12
Multiplication:
15 × 24 = 360
Division:
360 / 12 = 30
Therefore, the value of result after executing the statement is 30.
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The mathematical expression 'result = (3 * 5) * 24 / (15 - (7 - 4))' simplifies to 'result = 15 * 24 / 12', leading to a final value of 30.
Explanation:The task here is to understand and solve the provided mathematical expression, which is result = (3 * 5) 24 / (15 - (7 - 4)). However, this expression seems to contain a typo - there's an operation missing between (3 * 5) and 24. Let's assume that the operation is multiplication, turning the expression into result = (3 * 5) * 24 / (15 - (7 - 4)).
First, solve the operations in the parenthesis. Hence, 3 * 5 equals 15 and 15 - (7 - 4) equals 12. The modified expression is now result = 15 * 24 / 12.
Then, performing the multiplication and division in order from left to right gives the result as 30.
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Which phrase refers to the legal act of lowering an individual’s tax liability?Which phrase refers to the legal act of lowering an individual’s tax liability?
Answers
Answer:
Tax avoidance - legal
Tax evasion - illegal
pls help I will give thanks and mark brainiest
Answers
Answer:
2
Step-by-step explanation:
because if -3 was matched up with two before then you have to make it consistent and make sure that the pattern is continous.
Answer: 2
step by step explanation: -3 = 2
how do i solve this equation
Answers
Answer:
\(\frac{9b}{a^{2} }\)
Step-by-step explanation:
\(\frac{a^{2} }{b}(\frac{3b}{a^{2} })^{2}\)
\(\frac{9a^{2}b^{2} }{a^{4}b }\)
\(\frac{9b}{a^{2} }\)
What is pump cavitation and how can you prevent it. Discuss the pump cavitation in terms of Net positive suction head NPSH
Answers
Pump cavitation refers to the formation and collapse of vapor bubbles in a pump due to low pressure conditions. It can cause damage to the pump and decrease its efficiency. Preventing pump cavitation is crucial for maintaining optimal pump performance and avoiding potential issues.
Pump cavitation occurs when the pressure at the pump inlet drops below the vapor pressure of the liquid being pumped, causing the formation of vapor bubbles.
When these bubbles move to regions of higher pressure within the pump, they collapse, creating tiny shockwaves that can erode the pump impeller and other components.
This erosion can lead to reduced pump efficiency, increased vibration, and even mechanical failure.
To prevent pump cavitation, it is important to ensure an adequate Net Positive Suction Head (NPSH). NPSH is a measure of the available pressure at the pump inlet above the vapor pressure of the fluid. It determines the margin of safety against cavitation.
Maintaining a sufficient NPSH value is crucial to prevent cavitation. Preventing pump cavitation involves several measures. Firstly, selecting a pump with appropriate specifications and operating it within its recommended range can help ensure sufficient NPSH.
Additionally, proper system design, including adequate pipe sizing, minimizing pressure losses, and avoiding sudden changes in flow velocity, can contribute to preventing cavitation.
Proper maintenance, such as regular inspection of impellers and suction piping, can also help identify and address any issues that may lead to cavitation.
Overall, preventing pump cavitation involves maintaining a sufficient NPSH through proper pump selection, system design, and regular maintenance to ensure smooth and efficient pump operation.
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A light beam strikes a piece of glass with an incident angle of 45.00 ∘
. The beam contains two colors: 450.0 nm and an unknown wavelength. The index of refraction for the 450.0 -nm light is 1.482. Assume the glass is surrounded by air, which has an index of refraction of 1.000 . Determine the index of refraction n u
for the unknown wavelength if its refraction angle is 0.8000 ∘
greater than that of the 450.0 nm light.
Answers
Answer: The index of refraction for the unknown wavelength is approximately 1.355.
Step-by-step explanation:
We can use Snell's law to relate the incident angle and refracted angle to the indices of refraction:
n1 sinθ1 = n2 sinθ2
where n1 and θ1 are the index of refraction and incident angle of the light in air, and n2 and θ2 are the index of refraction and refracted angle of the light in glass. Since the incident angle is 45.00 degrees, we have:
sinθ1 = sin(45.00) = √2/2
Since we know the index of refraction for the 450.0 nm light is 1.482, we can solve for the refracted angle θ2:
1.000 * √2/2 = 1.482 * sinθ2
sinθ2 = 1.000 * √2/2 / 1.482 = 0.4951
θ2 = sin^(-1)(0.4951) = 29.07 degrees
Now, we can use Snell's law again to relate the index of refraction to the refracted angle for the unknown wavelength:
n1 sinθ1 = n3 sinθ3
where n3 is the index of refraction for the unknown wavelength, and θ3 is the refracted angle for the unknown wavelength. We know that θ3 is 0.8000 degrees greater than θ2:
θ3 = θ2 + 0.8000 = 29.87 degrees
Substituting all the known values into Snell's law, we get:
1.000 * √2/2 = n3 * sin(29.87)
n3 = 1.000 * √2/2 / sin(29.87) = 1.355
Therefore, the index of refraction for the unknown wavelength is approximately 1.355.
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Mr. Nice's backyard has an area of 225 square feet. What is the length of one side of his backyard?
A.13 feet
B.14 feet
C.15 feet
D.16 feet
Answers
C. 15 feet
Step-by-step explanation:Area formula
A = l²
225 = l²
l = √225
l = √15²
l = 15 feet
Knowing that you need a sample pool of 1019 students to ultimately get about 500 students in your sample, you are in a position to draw a systematic sample from the student directory at your university. Further, 9,500 students are listed in the directory. What is the sampling interval? Interpret your results.Show all your work.
Answers
Answer:
Sampling interval = 9.323
Step-by-step explanation:
Given that:
Population size, N = 9500
Require sample, n = 1019
Sampling interval :
Population size / required sample size
Sampling interval = 9500 / 1019 = 9.3228
Interpretation :
In other to obtain a sample of 1019 drawn systematically from the population, a sample will be drawn at every 9.323 units from the population of 9500
La suma dels quadrats de tres nombres naturals consecutius i múltiples de 3 és igual a 450. Planteja una equació de segon grau i calcula els tres nombres.
Answers
Answer:
The 3 numbers are 9, 12, and 15.
Step-by-step explanation:
I will answer in English, below the answer you can see a translation in Catalan.
This can be translated to:
The sum of the squares of three consecutive and multiple natural numbers of 3 is equal to 450. Pose a quadratic equation and calculate the three numbers.
First, a multiple of 3 can be written as:
3*n
The consecutive multiple of 3 is:
3*n + 3
The consecutive multiple of 3 is:
3*n + 3 + 3
Then the sum of their squares is:
(3*n)^2 + (3*n + 3)^2 + (3*n + 6)^2
And we know that this is equal to 450, then we need to solve the equation:
(3*n)^2 + (3*n + 3)^2 + (3*n + 6)^2 = 450
let's solve this:
9*n^2 + (9*n^2 + 2*(3*n)*3 + 9) + (9*n^2 + 2*(3*n)*6 + 36) = 450
27*n^2 + 54*n + 45 = 450
we can write this as:
27*n^2 + 54*n + 45 - 450 = 0
27*n^2 + 54*n - 405 = 0
The solutions of this equation are given by the Bhaskara's formula, and the solutions are:
\(n = \frac{-54 \pm \sqrt{54^2 -4*27*(-415)} }{2*27} = \frac{-54 \pm 216 }{54}\)
we know that our numbers are naturals, then the numbers are positives, which means that we only care for the positive solution of n, which is:
n = (-54 + 216)/54 = 3
Then the 3 numbers are:
3*n = 3*3 = 9
(3*n + 3) = (3*3 + 3) = 12
(3*n + 6) = (3*3 + 6) = 15
In Catalan:
En primer lloc, es pot escriure un múltiple de 3 com:
3 * n
El múltiple consecutiu de 3 és:
3 * n + 3
El múltiple consecutiu de 3 és:
3 * n + 3 + 3
Llavors, la suma dels seus quadrats és:
(3 * n) ^ 2 + (3 * n + 3) ^ 2 + (3 * n + 6) ^ 2
I sabem que això és igual a 450, llavors hem de resoldre l’equació:
(3 * n) ^ 2 + (3 * n + 3) ^ 2 + (3 * n + 6) ^ 2 = 450
resolem això:
9 * n ^ 2 + (9 * n ^ 2 + 2 * (3 * n) * 3 + 9) + (9 * n ^ 2 + 2 * (3 * n) * 6 + 36) = 450
27 * n ^ 2 + 54 * n + 45 = 450
podem escriure això com:
27 * n ^ 2 + 54 * n + 45 - 450 = 0
27 * n ^ 2 + 54 * n - 405 = 0
Les solucions d’aquesta equació vénen donades per la fórmula de Bhaskara, i les solucions són:
\(n = \frac{-54 \pm \sqrt{54^2 -4*27*(-415)} }{2*27} = \frac{-54 \pm 216 }{54}\)
sabem que els nostres nombres són naturals, aleshores els nombres són positius, cosa que significa que només ens importa la solució positiva de n, que és:
n = (-54 + 216) / 54 = 3
Llavors els 3 nombres són:
3 * n = 3 * 3 = 9
(3 * n + 3) = (3 * 3 + 3) = 12
(3 * n + 6) = (3 * 3 + 6) = 15
Omar bought a pastry from the bakery near his apartment. The pastry was $2.80, and he paid 5% sales tax. How much did Omar pay in all?
Answers
Answer:2.94
Step-by-step explanation:2.80*1.05=2.94