Question

Use the information at the right. Use the Table feature on a graphing calculator (or spreadsheet software) as indicated. The General Sherman tree, a sequoia tree in Sequoia National Park, has the greatest volume of any tree in the world. The distance around the tree (the circumference) measures 102.6 feet. Show how to find the distance straight through the center of the tree (the diameter) without measuring it directly. Then find this value to the nearest tenth of a foot.

Short answer

The diameter of the tree, to the nearest tenth of a foot, is approximately 32.7 feet.

Step-by-step solution

Understanding the Problem

We know that the circumference of the tree is 102.6 feet. We are asked to find the diameter of the tree, which is the distance straight through the center of the tree. We will use the formula for the circumference of a circle \(C = πd\) to find the value of \(d\).

Rearrange the Formula

We need to solve for \(d\) in the formula \(C = πd\). To isolate \(d\), divide both sides of the equation by \(π\) to get \(d = C/π\).

Substitute the given Circumference

We know that \(C = 102.6\) feet. So, substitute \(C\) with 102.6 in the equation to get \(d = 102.6/π\).

Calculate the Diameter

Perform the division to find the diameter of the tree \(d\).

Round to the nearest tenth

Round off the answer you obtained in step 4 to the nearest tenth of a foot.

Key concepts

Circle Geometry

In our daily world, circle geometry is present in various forms, from the rings on our fingers to the wheels of our bikes. A circle is a round plane figure whose boundary (the circumference) is equidistant from its center. The distance from one edge to another, passing through the center, is known as the diameter. Understanding circle geometry is crucial when you’re working with circles in real-life situations like finding the size of a tree as in our exercise.

Circle geometry not only involves the understanding of terms like radius, diameter, and circumference but also involves recognizing the relationships between these measurements. For instance, the diameter is always twice the length of the radius, which can be expressed algebraically as the equation: \( d = 2r \). Similarly, the circumference of a circle, or the distance around it, can be linked to the diameter through the mathematical constant \( \(pi\) \), with the formula \( C = \(pi\)d \). This relationship is pivotal in solving problems involving circular objects where direct measurement may not be possible.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions and equations to solve for unknown variables. It is a cornerstone of algebra and enables us to find the value of variables that represent real-world quantities.

In the context of our example with determining the diameter of a tree, algebraic manipulation plays a vital role. With the circumference of the General Sherman tree known, the task is to extract the diameter from the circumference. The original formula links circumference and diameter via \( C = \(pi\)d \). To find the diameter, algebraic manipulation requires us to 'undo' the multiplication of \( \(pi\) \) with the diameter. This is achieved by dividing both sides of the equation by \( \(pi\) \), that is \( d = \frac{C}{\(pi\)} \). Such manipulation gives us a viable method to extract the diameter, demonstrating how algebra provides tools to address real-world problems efficiently.

Calculating Diameter

Calculating the diameter of a circle when its circumference is known is a direct application of the formulas taught in geometry. The diameter can be obtained by taking the circumference and dividing it by \( \(pi\) \). While it sounds straightforward, there are crucial points to remember during calculation, especially dealing with the precision of measurement and rounding.

For the General Sherman tree, with a circumference of 102.6 feet, the diameter (\( d \)) is calculated as \( d = \frac{102.6}{\(pi\)} \) feet. Performing this calculation on a calculator can provide a very long decimal value. As per the example, we require the diameter to the nearest tenth of a foot, which ensures the answer is practical and suitable for reporting purposes. This step of rounding is often crucial in real-world problems when a degree of precision is specified.

Mathematical Formulas

Mathematical formulas are the lifeblood of solving geometry problems efficiently. These formulas are derived from the properties of figures and act as tools to unravel measurements and relationships between them. In the case of circle geometry, a solid grasp of the relevant formulas is indispensable.

The two key formulas in circle geometry are \( C = \(pi\)d \) and \( A = \(pi\)r^2 \), with \( C \) representing the circumference, \( A \) the area, \( d \) the diameter, and \( r \) the radius of the circle. When dealing with exercises like finding the diameter from the circumference, the correct application of these formulas is essential. This approach underscores the power of mathematical formulas: with them, we can find unknown quantities and apply these concepts to solve real-life challenges, from architecture to conservation, enhancing our understanding and ability to interact with the world around us.