Reference no: EM13719029
Q1. The goal of this question is to find the date and time of the earliest sunrise and the date and time of the latest sunset. You should achieve this by interpolating the above data points.
(a) Explain how you define the nodes xk that represent the dates above. (Note that there are several options for your answer here! Choose only one.)
(b) Explain how you define the times of sunrise and sunset represented by function values at the nodes xk. (Again, there are several options for your answer here, but you should choose only one.)
(c) Define a global interpolating polynomial Pr that passes through all data points given for the time of sunrise. Also define a global interpolating polynomial Ps that passes through all data points given for the time of sunset.
- Specify the coefficients of Pr in the form Pr(x) = aN xN +.............+ a1x + a0.
- Specify the coefficients of Ps in the form Ps(x) = bNxN + .......+ b1x + b0.
- Use Pr to find an approximate date for the earliest sunrise and state at what time the sun is estimated to rise on that date. Clearly explain how you go about this.
Use Ps to find an approximate date for the latest sunset and state at what time the sun is estimated to set on that date.
(d) Include a plot with your answer that shows the graphs of Pr and Ps in one figure, along with the data points as well as the locations on the graphs of the earliest sunrise and latest sunset. Here, you may use the units as chosen in parts (a) and (b).
(e) How accurate do you think your predictions are? Note that it is easy to find data about sunrise and sunset online; you are welcome to compare your findings with this data.
Q2. Now consider the resulting lengths of daylight for each data point. The goal is to find the date and length of the longest day, that is, the day with the largest length of daylight.
(a) Explain how you can use Pr and Ps from Q1 to define an interpolating polynomial PL that represents the length of daylight.
- Specify the coefficients of PL in the form PL(x) = cNxN +..........+ c1x + c0.
- Use PL to find an approximate date and length for the longest day.
(b) Compare your findings with the plot from Q1(d) and discuss any discrepancies.
Q3. Consider the integral
x0∫x3 PL (x) dx,
where x0 represents 13 November and x3 represents 24 January and PL is the interpolated polynomial from Q2a.
(a) Approximate the integral with the composite trapezoidal rule using PL only at the measurement data itself. Explain how many subdivisions you take and why.
(b) Use the composite trapezoidal rule with N = 100 subdivisions to approximate the integral.
(c) The error of the trapezoidal rule can be bounded using an upper bound for the second derivative of the integrand. Describe how you can use PL to find such a bound and give estimated bounds on the errors of your calculations in parts (a) and (b).
(d) Explain how to interpret the meaning of this integral in terms of daylight and which units you used for your answers.
|
Calculate the power developed by the turbin
: An ideal Rankine cycle uses water as its working fluid. The mass flow rate of water through the cycle is 150 kg/s. saturated vapour enters the turbine at 15 MPa and saturated liquid enters the pump at 10 kPa. Cooling water passes through the condense..
|
|
Quantitatively amount of work needed in these two cases
: You can transform a long bar of stainless steel from an initial 25 mm in diameter to 23 mm diameter by either machining or by drawing. Compare quantitatively the amount of work needed in these two cases.
|
|
Explain about transaction and transform mapping
: explain about transaction and transform mapping
|
|
Width from plate for values of the critical reynolds number
: Consider atmospheric air at 25 degrees Celsius and a velocity at 25 m/s flowing over both sides of a 1 eter long flat plate that is maintained at 125 degrees Celsius. Determine the rate of heat transfer per unit width from the plate for values of th..
|
|
Explain how you define the times of sunrise and sunset
: Explain how you define the nodes xk that represent the dates above - Define a global interpolating polynomial Pr that passes through all data points given for the time of sunrise.
|
|
Diameter impeller of a radial-flow water pump rotates
: A 400-mm-diameter impeller of a radial-flow water pump rotates at 120rad/s and has a discharge of 0.5m3/s. Determine the discharge for a similar pump that has an impeller diameter of 100mm and operates at 90rad/s.
|
|
Engine assuming gallon gasoline produces
: As a 75 slug car travels along a horizontal, 3200 ft section of road, it changes speed from 5 mph to 60 mph. During this acceleration, how efficient is the car's engine assuming 1 gallon gasoline produces 120 MJ of energy when completely combusted in..
|
|
Temperature of the refrigerant exiting the compressor
: Refrigerant R134a is compressed under steady conditions from a saturated vapour at 100 kPa to 500 kPa. The compressor’s isentropic efficiency is 85%. Heat losses from the compressor can be ignored, as can kinetic energy and potential energy effects.
|
|
Heat exchanger required to achieve the desired heating
: A counter-flow double pipe heat exchanger is heat water from 20 degrees Celsius to 80 degrees Celsius at the rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at 160 degrees Celsius at a mass flow rate of 2 kg/s.
|