Problem 44 - Thomas Calculus

Question

The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? The tree's cross-section area?

Step-by-Step Solution

Answer

The diameter increased by \( \frac{2}{\pi} \) inches, and the area increased by \( \pi \left( \frac{10 + \frac{2}{\pi}}{2} \right)^2 - 25\pi \).

Step 1: Understand the Relationship Between Diameter and Circumference

The formula that relates the circumference (C) of a circle to its diameter (D) is given by the equation: \( C = \pi D \). Knowing the circumference increased by 2 inches allows us to determine the rate of change in diameter.

Step 2: Calculate the New Circumference

The initial circumference is \( C_1 = \pi \times 10 \). After a year, the new circumference \( C_2 \) is \( C_1 + 2 = \pi \times 10 + 2 \). Therefore, \( C_2 = 10 \pi + 2 \).

Step 3: Determine the New Diameter

Using the circumferential formula \( C = \pi D \), substitute the new circumference \( C_2 = 10\pi + 2 \) to find the new diameter \( D_2 \): \( D_2 = \frac{C_2}{\pi} = \frac{10\pi + 2}{\pi} = 10 + \frac{2}{\pi} \).

Step 4: Calculate the Diameter Increase

The increase in diameter \( \Delta D \) is the difference between the new and old diameters: \( \Delta D = D_2 - 10 = \frac{2}{\pi} \).

Step 5: Calculate the Cross-Sectional Area Increase

The area \( A \) of a circle is given by \( A = \pi \left( \frac{D}{2} \right)^2 \). The original area is \( A_1 = \pi \left( \frac{10}{2} \right)^2 = 25\pi \). The new area is \( A_2 = \pi \left( \frac{10 + \frac{2}{\pi}}{2} \right)^2 \). Calculate \( A_2 \) and find the increase \( \Delta A = A_2 - A_1 \).

Key Concepts

Circumference and Diameter Relationship

In the world of geometry, understanding the relationship between a circle's circumference and its diameter is key. This relationship is beautifully simple: the circumference, which is the distance around the circle, is always equal to the diameter of the circle multiplied by the mathematical constant \(\pi\), expressed in the formula as \(C = \pi D\). This formula shows how tightly connected the diameter and circumference are.

• If you know the diameter of a circle, simply multiply it by \(\pi\) (approximately 3.14159) to find the circumference.
• Conversely, if you have the circumference and need the diameter, divide the circumference by \(\pi\).

When we apply this concept to the problem of the tree, noticing that the circumference increased by 2 inches allows us to find out how much the diameter increased. The change in circumference directly relates to a change in diameter by dividing the increase by \(\pi\). Thus, if a circle's circumference grows, so does its diameter.

Circle Area Calculation

Calculating the area of a circle relies heavily on understanding your circle's diameter or radius. The area \(A\) is found using the formula \(A = \pi r^2\), where \(r\) is the radius of the circle. Alternatively, if you know the diameter \(D\), you can convert it to the radius by dividing it by 2, giving you the formula \(A = \pi \left( \frac{D}{2} \right)^2\).

• The area represents the number of square units that fit inside the circle.
• This formula helps calculate the space within the circle, which increases as the diameter increases.

In the given problem, we see the importance of this concept when determining how much the tree's cross-sectional area has increased reflecting the change in diameter. An increase in diameter results in a larger area because the circle gets wider.

Rate of Change

In calculus and real-life applications, the rate of change is a crucial concept. It describes how one quantity changes in relation to another. In our exercise with the tree, the focus is on how the diameter and area of a circle change when the circumference changes. The rate at which these changes occur is vital for predicting and understanding growth patterns.

• Calculating the rate of change for diameter involves finding the difference between the new and original diameter values, which is achieved using the relationship between circumference and diameter.
• For the area, once new values of diameter are determined, the rate of area change relies on calculating differences in circular areas with new and old diameters.

Understanding rates of change helps in various fields, as it sheds light on how and why changes occur, reflecting patterns over time or in response to certain actions.