Questions: A research group using hospital records developed an approximate mathematical model relating systolic blood pressure and age. P(x)=50+25 ln (x+2) 0 ≤ x ≤ 65 P(x) is pressure measured in millimeters of mercury and x is age in years. The rate of change of pressure at the end of 10 years is 2.08 mm / year. (Type an integer or decimal rounded to the nearest hundredth as needed.) The rate of change of pressure at the end of 30 years is 0.78 mm / year. (Type an integer or decimal rounded to the nearest hundredth as needed.) The rate of change of pressure at the end of 60 years is 0.40 mm / year. (Type an integer or decimal rounded to the nearest hundredth as needed.)

A research group using hospital records developed an approximate mathematical model relating systolic blood pressure and age.

P(x)=50+25 ln (x+2) 0 ≤ x ≤ 65

P(x) is pressure measured in millimeters of mercury and x is age in years.

The rate of change of pressure at the end of 10 years is 2.08 mm / year.
(Type an integer or decimal rounded to the nearest hundredth as needed.)
The rate of change of pressure at the end of 30 years is 0.78 mm / year.
(Type an integer or decimal rounded to the nearest hundredth as needed.)
The rate of change of pressure at the end of 60 years is 0.40 mm / year.
(Type an integer or decimal rounded to the nearest hundredth as needed.)

Transcript text: A research group using hospital records developed an approximate mathematical model relating systolic blood pressure and age. \[ P(x)=50+25 \ln (x+2) \quad 0 \leq x \leq 65 \] $\mathrm{P}(\mathrm{x})$ is pressure measured in millimeters of mercury and x is age in years. The rate of change of pressure at the end of 10 years is $2.08 \mathrm{~mm} / \mathrm{year}$. (Type an integer or decimal rounded to the nearest hundredth as needed.) The rate of change of pressure at the end of 30 years is $0.78 \mathrm{~mm} / \mathrm{year}$. (Type an integer or decimal rounded to the nearest hundredth as needed.) The rate of change of pressure at the end of 60 years is $0.40 \mathrm{~mm} /$ year. (Type an integer or decimal rounded to the nearest hundredth as needed.)

Solution

Solution Steps

Step 1: Identify the Function

The given function for systolic blood pressure is $P(x) = A + B \ln(x + C)$, where:

$A$ is the base pressure in mmHg.
$B$ is the coefficient that scales the logarithmic term.
$C$ is the horizontal shift of the logarithmic function.
$x$ is the age in years.

Given parameters are $A = 50$, $B = 25$, and $C = 2$.

Step 2: Derive the Rate of Change

The rate of change of systolic blood pressure with respect to age is given by the derivative of $P(x)$ with respect to $x$, which is: $$P'(x) = \frac{d}{dx}(A + B \ln(x + C)) = \frac{B}{x + C}$$

Step 3: Calculate the Rate of Change at Age $x = 10$

Substituting $x = 10$ into the derivative, we get: $$P'(x) = \frac{B}{x + C} = \frac{25}{10 + 2} = 2.08$$

Final Answer:

The rate of change of systolic blood pressure at age $x = 10$ is approximately $P'(x) = 2.08$ mmHg per year.

Step 1: Identify the Function

The given function for systolic blood pressure is $P(x) = A + B \ln(x + C)$, where:

$A$ is the base pressure in mmHg.
$B$ is the coefficient that scales the logarithmic term.
$C$ is the horizontal shift of the logarithmic function.
$x$ is the age in years.

Given parameters are $A = 50$, $B = 25$, and $C = 2$.

Step 2: Derive the Rate of Change

The rate of change of systolic blood pressure with respect to age is given by the derivative of $P(x)$ with respect to $x$, which is: $$P'(x) = \frac{d}{dx}(A + B \ln(x + C)) = \frac{B}{x + C}$$

Step 3: Calculate the Rate of Change at Age $x = 30$

Substituting $x = 30$ into the derivative, we get: $$P'(x) = \frac{B}{x + C} = \frac{25}{30 + 2} = 0.78$$

Final Answer:

The rate of change of systolic blood pressure at age $x = 30$ is approximately $P'(x) = 0.78$ mmHg per year.

Step 1: Identify the Function

The given function for systolic blood pressure is $P(x) = A + B \ln(x + C)$, where:

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