Question 1104045: Hello, please help me with this question, thank you...!
Human's diastolic blood pressure (mm Hg) varies directly as the concentration of red blood cells (RBC/mL) and inversely as the square of the diameter (mm) of the arterial blood vessels. In a middle-aged man, the diastolic blood pressure is 70 mm Hg when the diameter of an artery feeding the hand is 3.1 mm and the concentration of red blood cells is 4.5×109 RBC/mL.What will his blood pressure be when same artery has narrowed to 3.0 mm in diameter, and the RBC concentration is 4.6×109 RBC/mL?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! NOTE: In the USA,  , so the concentrations of red blood cells in the problem are 4.5 billion/mL and 4.6 billion/mL, and since I have been living in the USA for almost 40 years, I am allowed to read the numbers that way. (Before living in the USA, I would have read those values as 4.5 and 4.6 million cells per microliter).
THE COMMON SENSE CALCULATION:
        .
Explanation:
"Human's diastolic blood pressure
varies directly as the concentration of red blood cells" means that
if the number of red blood cells per milliliter increases by a factor of
  ,
the diastolic blood pressure will increase by the same factor.
If that was the only change, multiplying times the ratio
would give us the new higher blood pressure.
"Human's diastolic blood pressure
varies inversely as the square of the diameter" of the artery means that
if the diameter squared gets  smaller,
the diastolic blood pressure will become higher.
If the narrowing of the arteries was the only change,
multiplying times the  ratio
would give us the new, higher blood pressure.
With both changes for the worse happening together,
we multiply times both ratios (both change factors).
It makes sense that both ratios/factors are greater than 1,
as we know that both contribute to increasing the diastolic blood pressure.
The calculator's answer is a never ending decimal starting with 76.405,
but blood pressure is read as whole numbers, so we round that to 76.
AS A MATH PROBLEM:
We have three variables to name.
I assume we are allowed to name them any way we want,
and use the values in whatever units we choose.
Naturally, we try to do that so as to make it easy to remember what means what,
and easier to calculate.
Let
 be the diastolic blood pressure, measured in mm Hg,
 be the concentration of red blood cells,
measured in billions of red blood cells per milliliter, and
 be the diameter of an artery feeding the hand, in mm.
JUST THE NUMBERS:
 -->  -->  , rounded up to 
 -->  , rounded down to 
Explanation:
"Human's diastolic blood pressure (mm Hg) varies directly as the concentration of red blood cells (RBC/mL) and inversely as the square of the diameter (mm) of the arterial blood vessels"
translates as
 where  is a constant that we will find from the clue given.
For the middle-aged man in the question,
 when  and  .
Substituting those values into ,
we get an equation that we solve to find the constant  for that man.
Then we use the equation with the value found,
 and  ,
to calculate the new diastolic blood pressure.
NOTE:
In case the teacher insisted on expressing red blood cell counts as number of cells per milliliter,
we would get
and the new diastolic blood pressure would still be the same.
 ,
or if the teacher insisted
 .
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