Problem 95 - Contemporary Precalculus

Question

According to data from the Center for Science in the Public Interest, the healthy weight range for a person depends on the person's height. For example, Height \(5 \mathrm{ft} 8 \mathrm{in}\) \(6 \mathrm{ft} 0 \mathrm{in}\) Healthy Weight Range ( \(\mathbf{l b}\) ) \(143 \pm 21\) \(163 \pm 26\)

Step-by-Step Solution

Answer

Answer: The healthy weight range for someone with a height of 5 ft 8 in is between 122 lb and 164 lb. For someone with a height of 6 ft 0 in, the healthy weight range is between 137 lb and 189 lb.

Step 1: Convert heights to inches

Start by converting the given heights into inches. There are 12 inches in a foot, so multiply the number of feet by 12 and then add the remaining inches. The first height is \(5\) ft \(8\) in, and the second height is \(6\) ft \(0\) in.

Step 2: Calculate the healthy weight range for height \(5\) ft \(8\) in

According to the data, the healthy weight range for a person with a height of \(5\) ft \(8\) in is \(143 \pm 21\) lb. To find the upper and lower bounds of the healthy weight range, simply add and subtract \(21\) from \(143\): Lower bound: \(143 - 21 = 122 \; \mathrm{lb}\) Upper bound: \(143 + 21 = 164 \; \mathrm{lb}\) Therefore, the healthy weight range for someone with a height of \(5\) ft \(8\) in is between \(122\) lb and \(164\) lb.

Step 3: Calculate the healthy weight range for height \(6\) ft \(0\) in

According to the data, the healthy weight range for a person with a height of \(6\) ft \(0\) in is \(163 \pm 26\) lb. To find the upper and lower bounds of the healthy weight range, simply add and subtract \(26\) from \(163\): Lower bound: \(163 - 26 = 137 \; \mathrm{lb}\) Upper bound: \(163 + 26 = 189 \; \mathrm{lb}\) Therefore, the healthy weight range for someone with a height of \(6\) ft \(0\) in is between \(137\) lb and \(189\) lb.

Key Concepts

Measurement Conversions

Measurement conversions are essential in many areas, including precalculus, where understanding various units and being able to convert between them is necessary for solving a range of problems. In our example, we need to convert a person's height from feet and inches to inches.
1 foot equals 12 inches. To convert feet and inches entirely to inches, you perform the following:

• Multiply the number of feet by 12 to convert them into inches.
• Add the remainder inches to this product to find the total height in inches.

Take the height 5 feet 8 inches as an example:

• First, convert the feet portion: 5 feet × 12 = 60 inches.
• Next, add the 8 inches: 60 + 8 = 68 inches total.

For a height of 6 feet 0 inches:

• Convert the feet: 6 feet × 12 = 72 inches.
• No additional inches are added here since it's 0 inches.

Thus, understanding these simple conversion steps allows you to work through measurement-based problems accurately.

Data Interpretation

Data interpretation is the process of analyzing information, often presented numerically, to glean insights, make decisions, or solve problems. In our example, we interpret a healthy weight range provided for a person's height.
The data gives ranges for two individuals based on height. These ranges are presented as a mean value with a possible variation:

• Height 5 ft 8 in: 143 ± 21 lb.
• Height 6 ft 0 in: 163 ± 26 lb.

To interpret these ranges, the plus-or-minus (±) symbol tells us how much above or below the given value the range can vary. This helps identify what is considered healthy for the specified height.
Being able to understand and calculate with these ranges is crucial for interpreting real-world data in fields like health science.

Range Calculations

Range calculations involve finding the set of possible values lying within a certain range, often represented by a center value and a range of deviation. This is common in statistics, physics, and other quantitative sciences.
In our exercise, we calculate a weight range using the formula:

• Lower bound: center value minus the deviation.
• Upper bound: center value plus the deviation.

Let's see this with the given data:

• For height 5 ft 8 in: 143 ± 21 lb, the weight range becomes 122 lb (lower) to 164 lb (upper).
• For height 6 ft 0 in: 163 ± 26 lb, the weight range becomes 137 lb (lower) to 189 lb (upper).

This method allows calculating precise boundaries of ranges, which is critical in assessments and decision-making processes.