(Chemistry: Wind Chill) At 0℃ the heat loss H (in kilocalories per square meter per hour) from a person’s body can be modeled by
H = 33(10√v - v + 10.45)
where v is the wind speed (in meters per second).
Find dH/dv and interpret its meaning.
I am pretty confident the derivative is 33(5/√v - 1), my problem is what is the interpretation? Doing the math is the easy part, what does it mean? Any help?
..by equating the heat loss in still air at the "wind chill" temperature to the actual loss at 0 deg C and the actual wind speed.
To find the derivative of H with respect to v, we will use the power rule for differentiation. The power rule states that for a function of the form f(x) = ax^n, the derivative with respect to x is given by f'(x) = anx^(n-1).
In this case, H = 33(10√v - v + 10.45). To find dH/dv, we will differentiate each term separately:
- Differentiating 33 with respect to v gives us 0 since it is a constant.
- Differentiating 10√v with respect to v gives us (10/2)√v = 5√v.
- Differentiating -v with respect to v gives us -1.
- Differentiating 10.45 with respect to v gives us 0 since it is a constant.
Combining these results, we get dH/dv = 33(5√v - 1).
Now, let's interpret the meaning of dH/dv. The derivative represents the rate of change of H with respect to v. In this case, dH/dv = 33(5√v - 1) tells us how the heat loss from a person's body changes as the wind speed (v) varies.
When the wind speed increases, the derivative dH/dv will tell us whether the heat loss is increasing or decreasing. If dH/dv is positive, that means the heat loss is increasing with the increasing wind speed. If dH/dv is negative, it means the heat loss is decreasing as the wind speed increases.
In this case, dH/dv = 33(5√v - 1), where 5√v - 1 > 0 for positive values of v. This means that as the wind speed increases, the heat loss from a person's body will increase. The larger the value of v, the faster the heat loss will increase.
Therefore, the interpretation of dH/dv = 33(5√v - 1) is that it represents the rate of increase of heat loss from a person's body with respect to the wind speed.