Problem 78 - Contemporary Precalculus

Question

According to the Center of Science in the Public Interest, the maximum healthy weight for a person who is 5 feet 5 inches tall is 150 pounds, and the maximum healthy weight for someone 6 feet 3 inches tall is 200 pounds. The relationship between weight and height here is linear. (a) Find a linear equation that gives the maximum healthy weight \(y\) for a person whose height is \(x\) inches over4 feet 10 inches. (Thus \(x=0\) corresponds to 4 feet \(10 \text { inches, } x=2 \text { to } 5 \text { feet, etc. }) \quad U_{S \in}\). (b) What is the maximum healthy weight for a person whose cise height is 5 feet? 6 feet? (c) How tall is a person who is at a maximum healthy.

Step-by-Step Solution

Answer

Answer: The maximum healthy weight for a person 5 feet tall is 125 pounds, and the maximum healthy weight for a person 6 feet tall is 185 pounds.

Step 1: Convert heights to inches and adjust for x-value

The given heights (5 feet 5 inches and 6 feet 3 inches) should be converted to inches and then adjusted for the x-value, where x=0 corresponds to 4 feet 10 inches. 5 feet 5 inches = 65 inches; 6 feet 3 inches = 75 inches. 4 feet 10 inches = 58 inches. Subtract 58 from each height to match the x-value requirement: 65 inches - 58 inches = 7; 75 inches - 58 inches = 17. So, our points become (7,150) and (17,200).

Step 2: Find the slope (m) of the line

To find the slope of the line, we will use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Using our points (7,150) and (17,200), we get: \(m = \frac{200-150}{17-7} = \frac{50}{10} = 5\)

Step 3: Find the y-intercept (b) of the line

Since we have the slope (m) and one point (7,150), we can use the point-slope form of a linear equation to find the y-intercept (b). \(y = mx + b\) Plug in the values for m, x, and y: \(150 = 5(7) + b\) Solve for b: \(150 = 35 + b \Rightarrow b = 115\)

Step 4: Write the linear equation

Now we have the slope (m) and y-intercept (b), so we can write the linear equation in slope-intercept form: \(y = 5x + 115\)

Step 5: Calculate the maximum healthy weight for 5 feet and 6 feet tall

For 5 feet: Convert 5 feet to inches (60 inches) and adjust the x value. 60 inches - 58 inches = 2 Now plug the x value into the linear equation: \(y = 5(2) + 115 \Rightarrow y = 10 + 115 = 125\) The maximum healthy weight for a person 5 feet tall is 125 pounds. For 6 feet: Convert 6 feet to inches (72 inches) and adjust the x value. 72 inches - 58 inches = 14 Now plug the x value into the linear equation: \(y = 5(14) + 115 \Rightarrow y = 70 + 115 = 185\) The maximum healthy weight for a person 6 feet tall is 185 pounds.

Step 6: Calculate the height for a person with maximum healthy weight 200 pounds

With a maximum healthy weight of 200 pounds, we want to find the corresponding x value: \(200 = 5x + 115\) Solve for x: \(85 = 5x \Rightarrow x = 17\) Now we convert the x value back to height: 17 + 58 inches = 75 inches 75 inches is 6 feet 3 inches. So, a person with maximum healthy weight of 200 pounds is 6 feet 3 inches tall.

Key Concepts

Height and Weight Relationship

Understanding the relationship between height and weight is crucial when determining a healthy weight for a person based on their height. In many health guidelines, this relationship is considered to be linear. This means that as a person's height increases, their healthy weight tends to increase proportionally. This kind of relationship can be easily represented using a linear equation, which helps in predicting the maximum healthy weight for any given height. In this particular exercise, the height is measured in inches, and it is assumed that starting from 4 feet 10 inches, the healthy weight increases linearly as the height increases. The idea is to find an equation that accurately models this relationship based on the data points provided: one for a height of 65 inches (5 feet 5 inches) with a maximum weight of 150 pounds, and another for a height of 75 inches (6 feet 3 inches) with a maximum weight of 200 pounds. By establishing such a model, we can easily calculate the healthy weight for any other height within this range.

Slope and Intercept Calculation

To create a linear equation that represents the relationship between height and weight, it is vital to understand two main components: the slope and the y-intercept.

Calculating the Slope

The slope of a line in a linear equation indicates the rate at which the weight changes with respect to changes in height. It can be calculated with the following formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,

• \( y_1 \) and \( y_2 \) are the weights.
• \( x_1 \) and \( x_2 \) are the heights, adjusted by subtracting the baseline height of 4 feet 10 inches (58 inches).

Applying the values from our points: (7,150) and (17,200), we find the slope to be 5.

Finding the Y-Intercept

The y-intercept is the point on the line where it crosses the y-axis, and it represents the weight when height \( x \) is zero. Using the point-slope form of the equation \( y = mx + b \), we substitute our known values to solve for \( b \): \[ 150 = 5 \times 7 + b \]Thus, \( b = 115 \).Consequently, the linear equation describing the relationship is \( y = 5x + 115 \), where \( y \) is the weight and \( x \) is the adjusted height measure.

Converting Units (inches and feet)

In this exercise, accurate unit conversion is essential for correctly determining the height in inches when dealing with feet and inches. Here's how you can convert heights and adjust them based on a reference point:

• Firstly, to convert feet into inches, remember that 1 foot equals 12 inches.
• Add the additional inches to your total after performing this conversion.

For example, a height of 5 feet 5 inches becomes:

• 5 feet = 5 × 12 = 60 inches
• Adding the extra 5 inches gives a total of 65 inches.

Similarly, 6 feet 3 inches translates to 75 inches in total.The exercise further involves adjusting these measurements by subtracting the baseline height of 4 feet 10 inches, which is 58 inches. This sets our reference height where the x-value is zero, allowing us to more straightforwardly use and apply the linear equation \( y = 5x + 115 \) to calculate various heights and their corresponding weights. By managing these conversions, the units become consistent, simplifying calculations significantly.