A study of the blood pressure of women has shown that for women aged 45–54 with a BMI greater than 21, the average systolic blood pressure was given by SBP = 0.68BMI+110 where SBP is measured in mmHg and BMI in kg/m^2. Recall that BMI =m/h^2, where m is the mass of the person and h is the person’s height. Calculate the expected rate of change in systolic blood pressure for a 1.6m woman in her early 50s with a mass of 80kg as a result of losing weight at a rate of 0.5kg per week. What are the units of your answer?

Question

A study of the blood pressure of women has shown that for women aged 45–54 with a BMI greater than 21, the average systolic blood pressure was given by SBP = 0.68BMI+110 where SBP is measured in mmHg and BMI in kg/m^2. Recall that BMI =m/h^2, where m is the mass of the person and h is the person’s height. Calculate the expected rate of change in systolic blood pressure for a 1.6m woman in her early 50s with a mass of 80kg as a result of losing weight at a rate of 0.5kg per week. What are the units of your answer?

Answer 1

We start by finding the current BMI of the woman:

BMI = m/h^2 = 80 kg / (1.6 m)^2 = 31.25 kg/m^2

We can then use the given equation to find her current systolic blood pressure:

SBP = 0.68BMI + 110 = 0.68(31.25) + 110 = 132.5 mmHg

Now we can use calculus to find the expected rate of change in SBP as a result of weight loss:

SBP' = d(SBP)/dt = d(SBP)/dm * dm/dt

We can find d(SBP)/dm by taking the derivative of the given equation with respect to BMI:

d(SBP)/d(BMI) = 0.68

We can also find dm/dt since the woman is losing weight at a rate of 0.5 kg/week:

dm/dt = -0.5 kg/week

Putting it all together, we get:

SBP' = d(SBP)/dt = d(SBP)/dm * dm/dt = 0.68 * (-0.5) = -0.34 mmHg/week

So the expected rate of change in SBP for this woman is -0.34 mmHg/week. The units are mmHg per week, which represents the change in systolic blood pressure over one week of weight loss.