Question

The height of a horse is usually measured in hands. One hand is exactly \(1 / 3 \mathrm{ft}\) (a) How tall (in feet) is a horse of \(19.2\) hands? (b) How tall (in meters) is a horse of \(17.8\) hands? (c) A horse of \(20.5\) hands is to be transported in a trailer. The roof of the trailer needs to provide \(3.0 \mathrm{ft}\) of vertical clearance. What is the minimum height of the trailer in feet?

Short answer

Question: Convert the following horse heights and find the minimum height of the trailer: (a) 19.2 hands to feet (b) 17.8 hands to meters (c) Minimum height of the trailer considering a 20.5 hands horse height and a 3.0 feet required clearance Answer: (a) 6.4 feet (b) 1.808 meters (c) 9.8333 feet

Step-by-step solution

Part (a): Convert 19.2 hands to feet

To convert the height of the horse in hands to feet, we need to multiply the number of hands (19.2) by the conversion factor (1/3 ft per hand): \(19.2 \text{ hands} \times \frac{1}{3} \text{ ft/hand} = 6.4 \text{ ft}\) So, the horse is 6.4 feet tall.

Part (b): Convert 17.8 hands to meters

First, we need to convert the horse height from hands to feet: \(17.8 \text{ hands} \times \frac{1}{3} \text{ ft/hand} = 5.9333 \text{ ft}\) Then, we will convert the height in feet to meters using the conversion factor (1 ft = 0.3048 meters): \(5.9333 \text{ ft} \times 0.3048 \text{ m/ft} = 1.808 \text{ m}\) So, the horse is approximately 1.808 meters tall.

Part (c): Calculate the minimum height of the trailer in feet

First, we need to determine the height of the horse in feet: \(20.5 \text{ hands} \times \frac{1}{3} \text{ ft/hand} = 6.8333 \text{ ft}\) Next, we need to add the required clearance of 3.0 feet to the horse's height: \(6.8333 \text{ ft} + 3.0 \text{ ft} = 9.8333 \text{ ft}\) So, the minimum height of the trailer should be approximately 9.8333 feet.

Key concepts

Measurement Systems

Measurement systems are essential for understanding and communicating the size or scale of something. Throughout history, different systems have been used worldwide, each with unique units suitable for various practical applications. In this context, we're exploring a measurement unit known as "hands."

The use of hands to measure the height of horses is a tradition that dates back to ancient times. One hand is defined as exactly one-third of a foot, which is standardized in modern measurements. This allows for clear communication between people even if they're using different local systems because the conversion to other units is fixed.

Understanding measurement systems not only aids in comprehending the size and dimensions of objects but also enhances our ability to translate those measurements into different scales. This exercise exemplifies conversion between hands, feet, and meters, emphasizing the importance of universal measurement systems.

Length Conversion

Length conversion involves changing a measurement from one unit to another. It's crucial in fields where precise measurements are necessary, such as construction, science, and everyday problem solving. Effective conversion requires understanding the relationship between the units.

In the problem, we encounter conversions from hands to feet and feet to meters. It's important to understand the two main steps:

• Converting hands to feet: Multiply the number of hands by \( \frac{1}{3} \) because one hand equals \( \frac{1}{3} \) foot.
• Converting feet to meters: Since one foot equals approximately 0.3048 meters, you multiply the number of feet by this factor.

These conversions help ensure accurate calculation, maintaining consistency and clarity when different measurement systems are involved. Length conversion helps seamlessly integrate different systems into a single, understandable framework.

Mathematical Problem Solving

Mathematical problem solving involves applying mathematical concepts to find solutions to practical problems. It often requires a step-by-step approach to resolve each part logically and sequentially.

We can see this process in the exercise through three distinct steps:

• Identifying the need for conversion: Recognizing which unit needs to be converted and to which unit is the first step.
• Performing accurate calculations: Calculations involve multiplying the given number by the conversion factor, ensuring precision.
• Applying results to real-world scenarios: For instance, determining the minimum height of a trailer involves adding the horse’s height to the required clearance.

This exercise is a good example of breaking down a problem to tackle conversions and applying mathematical logic effectively. Problem-solving skills help enhance critical thinking and enable a structured approach to solving diverse mathematical queries.