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 some base point \((0,0)\text{.}\)
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 \(0.2\) units; 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)
Table 11.5.12 is a 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. Suppose your anemometer (an instrument for measuring wind speed) says the wind is blowing at \(25\) miles per hour and your thermometer (an instrument for measuring the temperature)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.
