Question

The wind-chill index is modeled by the function \(W = 13.12 + 0.6215T - 11.37{V^{0.16}} + 0.3965T{V^{0.16}}\) where \(T\) is the temperature (in \(^oC\)) and \(v\) is the wind speed ( in \(km/h)\). The wind speed is measured as \(26\;km/h\), with a possible error of \( \pm 2\;km/h\), and the temperature is measured as \( - 1{1^^\circ }C\), with a possible error of \( \pm {1^^\circ }C\). Use differentials to estimate the maximum error in the calculated value of \(w\) due to the measurement errors in \(T\) and \(V\).\(\)

Short answer

The maximum error is \(1.61538\).

Step-by-step solution

Given and To Find.

\(w = 13.12 + 0.62157 - 11.37{v^{0.16}} + 0.39657{v^{0.16}}\)

Wind speed\( = 26\;km/h\)

Temperature\( = - 1{1^o}C\)

Error\( = \pm {1^o}C\)

The maximum error (To find)

Find Values.

\(dw = {f_T}\left( {{T_1}v} \right)dT + {f_v}\left( {{T_2}} \right)dv\)

Where,

\(f(T,v) = 13.12 + 0.6215T - 11.37{v^{0.16}} + 0.3965T{v^{0.16}}\)

\({f_T}(T,V) = 0.{6^{2/5}} + 0.3965{V^{0.16}}\)

\({f_v}(T,v) = - 11.37(0.16){v^{ - 0.84}} + 0.3465(0.16)7{v^{ - 0.84}}\)

\( = - 1.8192{v^{ - 0.84}} + 0.063447{v^{ - 0.84}}\)

Find Maximum Error.

\(dw = \left( {0.6215 + 0.3965{v^{0.16}}} \right)dT + \left( { - 1.8192{v^{ - 0.84}} + 0.063447{v^{ - 0.84}}} \right)dv\)

Where,

\(v = 26\)

\(T = - 11\)

\(\left| {\Delta T} \right| \le 1\)

\(|\Delta v| \le 2\)

\(dw \le \left| {0.6215 + 0.3965{{(26)}^{0.16}}} \right|\left| 1 \right| + \left| { - 1.8192{{(26)}^{ - 0.84}} + 0.6344( - 11){{(26)}^{ - 0.84}}} \right|\left| 2 \right|\)

\( \le 1.61538\)

Therefore, The maximum error is \(1.61538\).