Question

The height of a horse is usually measured in hands instead of in feet, where 1 hand equals \(1 / 3 \mathrm{ft}\) (exactly). (a) How tall in centimeters is a horse of \(18.6\) hands? (b) What is the volume in cubic meters of a box measuring \(6 \times 2.5 \times 15\) hands?

Short answer

(a) The horse is about 188.98 cm tall. (b) The box's volume is approximately 0.2361 cubic meters.

Step-by-step solution

Convert Hands to Feet - Part (a)

To find the height of the horse in feet, we start by converting the height from hands to feet. We know that one hand is equivalent to \( \frac{1}{3} \) feet. Therefore, a horse that is \(18.6\) hands tall is \(18.6 \times \frac{1}{3} = 6.2\) feet tall.

Convert Feet to Centimeters - Part (a)

Next, we convert the height from feet to centimeters. We know that 1 foot is equal to 30.48 cm. So, \(6.2\) feet is \(6.2 \times 30.48 = 188.976\) cm. Hence, the horse is approximately \(188.98\) cm tall.

Convert Box Dimensions to Feet - Part (b)

The dimensions of the box are given as \(6\), \(2.5\), and \(15\) hands. We need to convert each dimension from hands to feet. Thus, each dimension in feet is: \(6 \times \frac{1}{3} = 2\) feet, \(2.5 \times \frac{1}{3} \approx 0.8333\) feet, and \(15 \times \frac{1}{3} = 5\) feet.

Calculate the Volume in Cubic Feet - Part (b)

Now, calculate the volume of the box in cubic feet. The formula for the volume of a rectangular prism is length \(\times\) width \(\times\) height. Therefore, the volume is \(2 \times 0.8333 \times 5 = 8.333\) cubic feet.

Convert Cubic Feet to Cubic Meters - Part (b)

Finally, we convert the volume from cubic feet to cubic meters. We know that 1 cubic foot is approximately equal to 0.0283168 cubic meters. Hence, the volume in cubic meters is \(8.333 \times 0.0283168 \approx 0.2361\) cubic meters.

Key concepts

Metric System

The metric system is a decimal-based system of measurement used worldwide. It's important because it simplifies calculations by using units of ten. For example, distances are often measured in meters, while smaller lengths might use centimeters or millimeters.
This system helps ensure consistency and accuracy in measurements, making it easier to convert between different units.

• For example, 1 meter equals 100 centimeters.
• 1 kilometer equals 1000 meters.

The metric system is universally accepted in the scientific community due to its simplicity and ease in conversion.

Hands Measurement

Measuring horses by "hands" is a tradition that dates back centuries. In this system, one hand is equivalent to exactly one-third of a foot. This quirky measurement comes from the average width of a human hand and has been standardized over time.
When converting hands to feet, simply divide the number of hands by 3 because 1 hand = 1/3 foot.

• Example: A horse that is 18.6 hands tall is 18.6 ÷ 3 = 6.2 feet tall.

Understanding hands is crucial for tasks related to equine care and competition, where this unit is commonly used.

Volume Calculation

Knowing how to calculate volume is essential in many real-world applications, like packaging and space planning. Volume calculation for a box involves using the formula: length × width × height.
When given dimensions in a nonstandard unit like hands, ensure first to convert these into a consistent unit.

• Example: For a box measuring 6 × 2.5 × 15 hands:
• Convert each measurement to feet.
• Calculate: 6 hands = 2 feet, 2.5 hands ≈ 0.8333 feet, and 15 hands = 5 feet.
• Volume = 2 × 0.8333 × 5 ≈ 8.333 cubic feet.

Accurate volume calculation is vital in fields like shipping and storage management.

Cubic Meters

Cubic meters are a metric unit used to measure volume, particularly in large spaces or volumes, such as air in a room or concrete for construction. They offer a standardized way to express volume globally.
To convert from cubic feet to cubic meters, remember: 1 cubic foot ≈ 0.0283168 cubic meters.

• Example: A box with a volume of 8.333 cubic feet:
• Convert using the factor: 8.333 × 0.0283168 ≈ 0.2361 cubic meters.

Cubic meters are preferred in the metric system for their ease of use and universal application.

Dimensional Analysis

Dimensional analysis is a method used to convert one type of measurement into another. This can be particularly useful when working with different unit systems, such as converting dimensions from hands to feet and then to metric units.
The key is to ensure that units cancel out appropriately during the calculation process.

• For instance, when converting hands to centimeters, first turn hands into feet, then into centimeters, ensuring consistency in units.
• Example: 18.6 hands × 1/3 (feet/hand) × 30.48 (cm/foot) = 188.98 cm tall.

Following these steps helps maintain accuracy and understand the relationships between different units.