SOLUTION: Certain chemotherapy dosages depend on a patient's surface area. According to the Mosteller model S = √(h*M) / 6 where h is the patient's height in m, M is the patient's

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Question 989339: Certain chemotherapy dosages depend on a patient's surface area. According to the Mosteller model
S = √(h*M) / 6
where h is the patient's height in m, M is the patient's mass in kg and S is the patient's approximate surface area in m^2. Assume that the patient's height is a constant 187 cm but he is on a diet. If a patient loses 3.1 kg per month, how fast is his surface area changing (in m^2/month) at the instant his mass is 71 kg?
Give your answer correct to 4 decimal places.
Note. You may be wondering why we have referred to the patient's mass here, rather than his weight. Technically, a person's weight is a force (recall: F = Ma, where F is the force, M is the mass, and a is the acceleration, which in this case is g, the acceleration due to gravity). Force (and therefore weight) is measured in Newtons, not kilograms. Scientists, and you are one, should know that. (In orbit, an astronaut experiences weightlessness, not masslessness.) - I am not sure why this matters?
I am so confused!
Initial = S = √(187*71) / 6 = 19.2043
After One month = S = √(187*69.7) / 6 = 19.0277
I am not sure how to put these together?
THANK YOU
Found 2 solutions by MathLover1, jim_thompson5910:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

S+=+sqrt%28h+M%29+%2F+6
S+=+%281%2F6%29sqrt%28h+M%29+
Since height is constant, differentiate with respect to mass:
dS+=+%281%2F6%29%281%2F2%29dM%2Fsqrt%28hM%29+=+dM%2F%2812sqrt%28hM%29%29
dS%2Fdt+=+%28dM%2Fdt%29%2F%2812sqrt%28hM%29%29
M+=+71kg
h+=+187cm=>h+=+1.87m
dM%2Fdt+=+-3.1%28kg%2Fmonth%29

dS%2Fdt+=+-3.1%2F%2812sqrt%281.87%2A71%29%29
dS%2Fdt+=+-3.1%2F%2812sqrt%28132.77%29%29
dS%2Fdt+=+-3.1%2F%2812%2A11.52258651518833%29
dS%2Fdt+=+-3.1%2F%28138.27103818226%29
dS%2Fdt+=+-3.1%2F%28138.27103818226%29
dS%2Fdt+=-0.0224%28m%5E2%2Fmonth%29


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let

x = mass in kilograms
S(x) = body surface area in square meters. This is a function of x.

The height is constant at 187 cm = 1.87 meters.

So

S%28x%29+=+%28sqrt%28h%2Am%29%29%2F6

S%28x%29+=+%28sqrt%281.87x%29%29%2F6

is our surface area function (in square meters).

-------------------------------------------------------
Take the derivative of S(x) with respect to t (time). Doing so gives you

S%28x%29+=+%28sqrt%281.87x%29%29%2F6

%28dS%29%2F%28dt%29+=+1.87%2F%2812%2Asqrt%281.87x%29%29%2A%28%28dx%29%2F%28dt%29%29

Plug in x = 71 (given mass) and dx%2Fdt=-3.1 (the amount of mass being lost per month)

%28dS%29%2F%28dt%29+=+1.87%2F%2812%2Asqrt%281.87x%29%29%2A%28%28dx%29%2F%28dt%29%29

%28dS%29%2F%28dt%29+=+1.87%2F%2812%2Asqrt%281.87%2A71%29%29%2A%28-3.1%29

%28dS%29%2F%28dt%29+=+-0.04192490398719

So the surface area is changing at an instantaneous rate of approximately -0.04192490398719 square meters per month when the mass is 71 kg.