Problem 61
Question
A normal child's systolic blood pressure may be approximated by the function $$ p(x)=m(\ln x)+b $$ where \(p(x)\) is measured in millimeters of mercury, \(x\) is measured in pounds, and \(m\) and \(b\) are constants. Given that \(m=19.4\) and \(b=18\), determine the systolic blood pressure of a child who weighs \(92 \underline{\text { lb. }}\)
Step-by-Step Solution
Verified Answer
The systolic blood pressure of a child who weighs 92 pounds is approximately \(105.62\) millimeters of mercury.
1Step 1: Identify the given values
We are given the following information:
- The function representing a child's systolic blood pressure, \(p(x)=m(\ln x) + b\)
- Constants: \(m=19.4\) and \(b=18\)
- Weight: \(x=92\)
2Step 2: Substitute the values into the function
Let's substitute the given values into the function:
\(p(92)=19.4(\ln 92) + 18\)
3Step 3: Simplify the expression
Next, we need to simplify the expression inside the parentheses:
\(\ln 92 \approx 4.5218\)
Now, we can substitute this value back into the expression:
\(p(92)=19.4(4.5218) + 18\)
4Step 4: Solve for p(92)
Now, let's solve for \(p(92)\) which represents the systolic blood pressure of a child who weighs 92 pounds:
\(p(92)=19.4\times 4.5218 + 18\)
\(p(92) \approx 87.62292 + 18\)
\(p(92) \approx 105.62292\)
5Step 5: Interpret the result
The systolic blood pressure of a child who weighs 92 pounds is approximately 105.62 millimeters of mercury.
Key Concepts
Logarithmic FunctionsMathematical ModelingStep-by-Step Solution
Logarithmic Functions
Logarithmic functions are mathematical expressions that help determine how one quantity relates multiplicatively to another. In simple terms, a logarithm answers the question: "Exponent to what base equals a given number?" The notation \( \ln \) is often used to denote the natural logarithm, which has the number \( e \) (approximately 2.718) as its base.
In this exercise, the logarithmic function \( p(x)=m(\ln x) + b \) estimates a child's systolic blood pressure, where \( \ln x \) is computed with respect to the weight \( x \) measured in pounds. The logarithm offers a straightforward way to model how variability in weight might translate to blood pressure. When employing logarithmic functions, it doesn't simply predict a linear relationship but instead captures more complex, potentially exponential growth that suits biological data.
Understanding the properties of logarithms is vital here. For instance, the natural logarithm \( \ln(x) \) grows slower than any power of \( x \), offering balance between parameters that can span several orders of magnitude, such as weight and health indicators like blood pressure.
In this exercise, the logarithmic function \( p(x)=m(\ln x) + b \) estimates a child's systolic blood pressure, where \( \ln x \) is computed with respect to the weight \( x \) measured in pounds. The logarithm offers a straightforward way to model how variability in weight might translate to blood pressure. When employing logarithmic functions, it doesn't simply predict a linear relationship but instead captures more complex, potentially exponential growth that suits biological data.
Understanding the properties of logarithms is vital here. For instance, the natural logarithm \( \ln(x) \) grows slower than any power of \( x \), offering balance between parameters that can span several orders of magnitude, such as weight and health indicators like blood pressure.
Mathematical Modeling
Mathematical modeling involves using mathematical expressions and constructs to represent, simulate, or study real-world phenomena. Here, the function \( p(x)=m(\ln x) + b \) models a child's systolic blood pressure using input weight \( x \) and constants \( m \) and \( b \). These constants help fine-tune the model to match observed data, turning it into a tool for estimation and prediction.
In this context, constants have specific roles:
Modeling helps in predicting outcomes, such as blood pressure from known variables like weight. By providing approximate predictions, it serves as a useful resource for clinicians and health researchers, allowing assessments without exhaustive clinical tests.
In this context, constants have specific roles:
- \( m = 19.4 \) adjusts the contribution of the natural logarithm of weight. It's like a scaling factor that translates logarithmic results into pressure units.
- \( b = 18 \) acts as a base level or intercept, shaping the function's output when the weight's logarithmic influence is null or minimal.
Modeling helps in predicting outcomes, such as blood pressure from known variables like weight. By providing approximate predictions, it serves as a useful resource for clinicians and health researchers, allowing assessments without exhaustive clinical tests.
Step-by-Step Solution
Understanding a step-by-step problem solution ensures a comprehensive grasp of mathematical procedures. Here's how you tackle the function \( p(x)=m(\ln x) + b \) to find the blood pressure for a child weighing 92 pounds:
Begin by identifying all given values, which include the constants \( m = 19.4 \), \( b = 18 \), and the weight \( x = 92 \) pounds. Next, substitute these values into the function, rewriting it as:
\( p(92)=19.4(\ln 92) + 18 \)
Now compute \( \ln 92 \), which is approximately \( 4.5218 \). This value represents the logarithmic impact of the child’s weight. Substituting, you get:
\( p(92)=19.4(4.5218) + 18 \)
Simplify and solve the expression: multiply 19.4 by 4.5218, resulting in \( 87.62292 \). Adding 18 yields the systolic blood pressure:\( p(92) \approx 105.62 \) mmHg.
This result demonstrates combining concepts seamlessly, from recognizing relevant mathematical principles to performing accurate arithmetic, ultimately providing insights into health metrics like blood pressure.
Begin by identifying all given values, which include the constants \( m = 19.4 \), \( b = 18 \), and the weight \( x = 92 \) pounds. Next, substitute these values into the function, rewriting it as:
\( p(92)=19.4(\ln 92) + 18 \)
Now compute \( \ln 92 \), which is approximately \( 4.5218 \). This value represents the logarithmic impact of the child’s weight. Substituting, you get:
\( p(92)=19.4(4.5218) + 18 \)
Simplify and solve the expression: multiply 19.4 by 4.5218, resulting in \( 87.62292 \). Adding 18 yields the systolic blood pressure:\( p(92) \approx 105.62 \) mmHg.
This result demonstrates combining concepts seamlessly, from recognizing relevant mathematical principles to performing accurate arithmetic, ultimately providing insights into health metrics like blood pressure.
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