The heat index I is a measure of how hot it feels when the relative humidity is H (as a percentage) and the actual air temperature is T (in degrees Fahrenheit). An approximate formula for the heat index that is valid for (T ,H) near (90, 40) is I (T,H) = 45.33 + 0.6845T + 5.758H − 0.00365T 2 − 0.1565HT + 0.001HT2 (a) Calculate I at (T ,H) = (95, 50). (b) Which partial derivative tells us the increase in I per degree increase in T when (T ,H) = (95, 50)? Calculate this partial derivative.

Answer:a) [tex]I(95,50) = 73.19[/tex] degreesb) [tex]I_{T}(95,50) = -7.73[/tex]Step-by-step explanation:An approximate formula for the heat index that is valid for (T ,H) near (90, 40) is:[tex]I(T,H) = 45.33 + 0.6845T + 5.758H - 0.00365T^{2} - 0.1565TH + 0.001HT^{2}[/tex]a) Calculate I at (T ,H) = (95, 50).[tex]I(95,50) = 45.33 + 0.6845*(95) + 5.758*(50) - 0.00365*(95)^{2} - 0.1565*95*50 + 0.001*50*95^{2} = 73.19[/tex] degrees(b) Which partial derivative tells us the increase in I per degree increase in T when (T ,H) = (95, 50)? Calculate this partial derivative.This is the partial derivative of I in function of T, that is [tex]I_{T}(T,H)[/tex]. So[tex]I(T,H) = 45.33 + 0.6845T + 5.758H - 0.00365T^{2} - 0.1565TH + 0.001HT^{2}[/tex][tex]I_{T}(T,H) = 0.6845 - 2*0.00365T - 0.1565H + 2*0.001H[/tex][tex]I_{T}(95,50) = 0.6845 - 2*0.00365*(95) - 0.1565*(50) + 2*0.001(50) = -7.73[/tex]