Activity 11.6.4

Suppose that the elevation of a plot of land is given by the function $h$, where we additionally know that $h(3,1) = 4.35$, $h_x(3,1) = 0.27$, and $h_y(3,1) = -0.19$. Assume that $x$ and $y$ are measured in miles in the east and north directions, respectively, from $(0,0)$.

Your GPS device says that you are currently at the point $(3,1)$. However, you know that the coordinates are only accurate to within $0.2$ miles; that is, $dx = \Delta x = 0.2$ and $dy= \Delta y = 0.2$. Estimate the uncertainty in your elevation using the linearization of $h$ for the input (3,1).

The pressure, volume, and temperature of an ideal gas are related by the equation $$\begin{gathered} P= P(T,V) = 8.31 \frac{T}{V}, \end{gathered}$$ 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 the linearization of $P$ at the point $(310,12)$ to estimate the change in the pressure when the volume increases to 12.3 liters and the temperature decreases to 305 K.

Use the table of values for the wind chill $w(v,T)$, in degrees Fahrenheit, as a function of temperature, also in degrees Fahrenheit, and wind speed, in miles per hour provided in this part 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.