Activity 11.5.4.
(a)
Suppose that the elevation of a plot of land is given by the function \(h\text{,}\) where we additionally know that \(h(3,1) = 4.35\text{,}\) \(h_x(3,1) = 0.27\text{,}\) and \(h_y(3,1) = -0.19\text{.}\) Assume that \(x\) and \(y\) are measured in miles in the east and north directions, respectively, from \((0,0)\text{.}\)
Your GPS device says that you are currently at the point \((3,1)\text{.}\) However, you know that the coordinates are only accurate to within \(0.2\) miles; that is, \(dx = \Delta x = 0.2\) and \(dy= \Delta y = 0.2\text{.}\) Estimate the uncertainty in your elevation using differentials.
(b)
The pressure, volume, and temperature of an ideal gas are related by the equation
\begin{equation*}
P= P(T,V) = 8.31 \frac{T}{V},
\end{equation*}
where \(P\) is measured in kilopascals, \(V\) in liters, and \(T\) in kelvin. Find the pressure when the volume is 12 liters and the temperature is 310 K. Use differentials to estimate the change in the pressure when the volume increases to 12.3 liters and the temperature decreases to 305 K.
(c)
Use the table of values for the wind chill \(w(v,T)\text{,}\) in degrees Fahrenheit, as a function of temperature, also in degrees Fahrenheit, and wind speed, in miles per hour provided in part 11.5.3.b for this part. Suppose your anemometer says the wind is blowing at \(25\) miles per hour and your thermometer shows a reading of \(-15^\circ\) degrees. However, you know your thermometer is only accurate to within \(2^\circ\) degrees and your anemometer is only accurate to within \(3\) miles per hour. What is the wind chill based on your measurements? Estimate the uncertainty in your measurement of the wind chill.
2
An instrument for measuring wind speed.
3
An instrument for measuring the temperature.
