Problem 71 - Elementary Algebra

Question

The diameter at basal height of a tree is the diameter about \(3 \mathrm{ft}\) above the ground. The approximate shape of the trunk of a tree above basal height is a cone. The formula for the volume of a cone is \(V=\frac{\pi r^{2} h}{3}\), where \(r\) is the radius and \(h\) is the height. Measured from its basal height, a tree is \(50 \mathrm{ft}\) tall. The diameter of its trunk is \(2.5 \mathrm{ft}\). Find the approximate volume of lumber in cubic feet in this trunk. ( \(\pi \approx 3.14\).) Round to the nearest whole number. (Source: G. John Smith; www.math.bcit.ca, 1997)

Step-by-Step Solution

Answer

The approximate volume of lumber in the trunk is 82 cubic feet.

Step 1: Determine the radius

Given the diameter of the tree trunk is 2.5 ft. The radius is half of the diameter. Therefore, the radius (r) is: \[ r = \frac{2.5}{2} = 1.25 \text{ ft} \]

Step 2: Use the formula for the volume of a cone

The formula for the volume of a cone is \[ V = \frac{\pi r^{2} h}{3} \]. Plug in the radius (r) and the height (h) into the formula. Here, \[ r = 1.25 \text{ ft} \text{ and } h = 50 \text{ ft} \]

Step 3: Calculate the volume

Substitute the values for radius and height, and use \( \pi \approx 3.14 \): \[ V = \frac{\pi \times (1.25)^{2} \times 50}{3} \approx \frac{3.14 \times 1.5625 \times 50}{3} \approx \frac{3.14 \times 78.125}{3} \approx \frac{245.3125}{3} \approx 81.77 \text{ cubic feet} \]

Step 4: Round to the nearest whole number

The volume calculated in the previous step is approximately 81.77 cubic feet. Rounding it to the nearest whole number gives 82 cubic feet.

Key Concepts

Cone Volume Formula

The cone volume formula is essential for calculating the volume of a cone. The formula is given by \( V = \frac{\pi r^{2} h}{3} \). Here, \( V \) represents the volume, \( r \) is the radius of the base, and \( h \) is the height of the cone. A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex. This formula utilizes the radius and height to determine the space enclosed within the cone.
To use the formula effectively:

• Identify the radius of the base.
• Identify the height from the base to the apex.
• Plug both values into the formula along with pi.

The step-by-step calculation simplifies the complex process, making it easier to understand and apply in various real-world scenarios.

Basal Height

In the context of trees, basal height is a term often used in forestry to describe the height above the ground where the diameter of the tree is measured. Typically, it is measured about three feet above the ground surface. This measurement is crucial because tree trunks can taper differently depending on the species and environmental factors.
Basal height provides a standardized point of reference for measuring tree diameter, which can be used to calculate the volume of lumber. It ensures consistent and reliable measurements, especially when comparing different trees.
Using basal height as a reference point, we can approximate the shape of a tree trunk above this height as that of a cone, simplifying the process of volume approximation.

Volume Calculation Steps

Understanding the steps involved in volume calculation is vital. Here's a detailed look:
Step 1: Determine the radius. Given that the diameter is 2.5 feet, the radius \( r \) is half of the diameter:
\[ r = \frac{2.5}{2} = 1.25 \text{ ft} \]
Step 2: Use the cone volume formula \( V = \frac{\pi r^{2} h}{3} \).
Step 3: Plug in the values for radius \( r \) and height \( h \) into the formula. Here, \( r = 1.25 \text{ ft} \text{ and } h = 50 \text{ ft} \).
Step 4: Perform the calculations:
\[ V = \frac{\pi \times (1.25)^{2} \times 50}{3} \approx \frac{3.14 \times 1.5625 \times 50}{3} \approx \frac{3.14 \times 78.125}{3} \approx \frac{245.3125}{3} \approx 81.77 \text{ cubic feet} \]
Following these steps ensures accuracy in calculations and helps students understand the progression from identifying measurements to obtaining the final volume.

Radius and Diameter Relationship

The relationship between radius and diameter is fundamental in geometry. The diameter of a circle or the base of a cone is always twice the radius. Mathematically, this relationship is expressed as \( d = 2r \), where \( d \) represents the diameter and \( r \) represents the radius.
Understanding this relationship helps in easily converting between these two measurements:

• If you know the diameter, divide by 2 to get the radius.
• If you know the radius, multiply by 2 to get the diameter.

In our example, the diameter of the tree trunk is given as 2.5 feet, so:
\[ r = \frac{2.5}{2} = 1.25 \text{ ft} \] Expertly navigating this relationship allows for accurate inputs into formulas requiring either measurement.

Rounding Numbers

Rounding numbers is a common practice in mathematics to simplify results and make them easier to interpret. Rounding involves reducing the number of digits in a figure while keeping its value close to the original. This is particularly useful when dealing with decimals.
In our example, we calculated the volume as approximately 81.77 cubic feet. The nearest whole number is 82, making this the rounded volume.
Rules for Rounding:

• If the digit after the place you're rounding to is 5 or more, round up.
• If the digit is less than 5, round down.

By rounding numbers, particularly in measurements and calculations, we ensure the results are practical and easy to use, without losing significant accuracy.