Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

The part of a tree normally sawed into lumber is the trunk, a solid shaped approximately like a right circular cylinder. If the radius of the trunk of a certain tree is growing \(\frac{1}{2}\) inch per year and the height is increasing 8 inches per year, how fast is the volume increasing when the radius is 20 inches and the height is 400 inches? Express your answer in board feet per year (1 board foot \(=1\) inch by 12 inches by 12 inches).

Step-by-Step Solution

Answer

The volume increases at approximately \(243.27\pi\) board feet per year.

Step 1: Understand the Problem

We need to determine the rate at which the volume of a cylinder (the tree trunk) is increasing with respect to time. The formula for the volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Step 2: Apply the Chain Rule for Differentiation

To find the rate of change of the volume \( \frac{dV}{dt} \), differentiate the volume formula with respect to time \( t \). This results in the equation \( \frac{dV}{dt} = \pi (2r \frac{dr}{dt} h + r^2 \frac{dh}{dt}) \).

Step 3: Substitute Known Values

The problem gives \( \frac{dr}{dt} = \frac{1}{2} \) inches/year, \( \frac{dh}{dt} = 8 \) inches/year, \( r = 20 \) inches, and \( h = 400 \) inches. Substitute these values into the differential equation: \[ \frac{dV}{dt} = \pi (2 \times 20 \times \frac{1}{2} \times 400 + 20^2 \times 8) \].

Step 4: Simplify the Expression

Calculate each part of the expression: \( 2 \times 20 \times \frac{1}{2} \times 400 = 8000 \) and \( 20^2 \times 8 = 3200 \). Thus, \( \frac{dV}{dt} = \pi (8000 + 3200) = \pi \times 11200 \).

Step 5: Convert Cubic Inches to Board Feet

Since 1 board foot equals 144 cubic inches (= 1 inch by 12 inches by 12 inches), divide \( \pi \times 11200 \) cubic inches/year by 144 to convert to board feet per year. This results in \( \frac{\pi \times 11200}{144} \approx 243.27\pi \) board feet per year.

Key Concepts

Cylinder Volume

The volume of a cylinder is a fundamental concept often needed in geometry and calculus-related problems. A right circular cylinder, like a tree trunk, has a straightforward formula for volume calculation:

• The formula is given by: \( V = \pi r^2 h \)
• Where \( V \) is the volume, \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.

This formula allows us to compute how much space the cylinder occupies. For students solving word problems involving cylinders, identifying the radius and height is crucial.
Understanding the relationship between these dimensions and how they affect the overall volume is key. When dealing with exercises like the tree trunk problem, focusing on the dimensions' rate of change is essential to determine how fast the volume itself is changing over time.

Calculus Derivatives

Derivatives are a cornerstone of calculus, providing a powerful tool for understanding how quantities change. In the context of this exercise, we are concerned with how the volume of the cylinder changes as time progresses.A derivative can be thought of as the "rate of change" or "slope" of a function at a given point. To find out how quickly the volume of the tree trunk is changing, we need to take the derivative of the cylinder's volume function with respect to time \( t \). This requires us to modify the existing formula for volume to incorporate not just space, but time.
By taking the derivative \( \frac{dV}{dt} \), we are essentially computing how small changes in radius and height over time contribute to changes in volume. This step leads us to apply a specific technique called the "Chain Rule" for further differentiation.

Chain Rule Differentiation

The chain rule is an essential method in calculus used to differentiate complex functions, especially when they involve multiple variables like radius and height in the cylinder volume problem.Whenever a function is dependent on other functions that change over time, the chain rule allows us to differentiate the outer function with respect to an independent variable by also considering the inner derivatives. For the volume of a cylinder, which is dependent on both radius \( r \) and height \( h \), the derivative with respect to time \( t \) involves the chain rule:

• Apply the chain rule: \( \frac{dV}{dt} = \pi (2r \frac{dr}{dt} h + r^2 \frac{dh}{dt}) \)
• This expression accounts for simultaneous changes in \( r \) and \( h \) influencing the rate of change in volume.

By using the known rates of change for the radius and height, we substitute them into our derivative to find the precise rate at which volume is increasing.

Measurement Conversion

Understanding measurement conversion is vital for translating real-world units into usable data in mathematical problems.In our exercise, we initially calculate the rate of change in volume in cubic inches per year. However, the final result must be expressed in board feet per year because that's a common lumber measurement. The conversion from cubic inches to board feet follows this principle:

• 1 board foot = 144 cubic inches
• To convert \( \pi \times 11200 \) cubic inches per year to board feet, simply divide by 144.

Therefore, the final conversion gives \( \frac{\pi \times 11200}{144} \), which translates our findings from mathematically derived units to practical, everyday measurements for understanding and utilization in the intended industry.