What is the approximate volume of a tree trunk if the trunk is 13 feet tall with a circumference of 4.5 feet?
Recall that the circumference of a circle is equal to pi times its diameter.
A. $33.09 ft ^3$
B. $27.14 ft ^3$
C. $20.95 ft ^3$
D. $17.16 ft ^3$
Answer (2)
The approximate volume of the tree trunk is calculated using the formula for the volume of a cylinder. The calculated volume is about 20.95 cubic feet, making the correct choice option C. 20.95 ft³.
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Assume the tree trunk is a cylinder and recall the volume formula: V = π r 2 h .
Calculate the radius using the circumference: r = 2 π C = 2 π 4.5 .
Determine the base area: A = π r 2 = π ( 2 π 4.5 ) 2 .
Compute the volume: V = A ∗ h = π ( 2 π 4.5 ) 2 ∗ 13 ≈ 20.95 f t 3 . The final answer is 20.95 f t 3 .
Explanation
Problem Analysis and Assumptions We are given a tree trunk that is 13 feet tall with a circumference of 4.5 feet. We need to find the approximate volume of the tree trunk. To do this, we will assume the tree trunk is a cylinder. The formula for the volume of a cylinder is V = π r 2 h , where r is the radius and h is the height.
Finding the Radius First, we need to find the radius of the tree trunk. We know the circumference is 4.5 feet. The formula for the circumference of a circle is C = 2 π r . We can rearrange this formula to solve for the radius: r = 2 π C . Substituting the given circumference, we have r = 2 π 4.5 .
Calculating the Base Area Now that we have the radius, we can calculate the area of the base of the tree trunk, which is a circle. The formula for the area of a circle is A = π r 2 . Substituting the radius we found, we have A = π ( 2 π 4.5 ) 2 .
Calculating the Volume Next, we can calculate the volume of the tree trunk using the formula V = A ∗ h . We know the height is 13 feet. Substituting the area we found and the height, we have V = π ( 2 π 4.5 ) 2 ∗ 13 .
Numerical Calculation Now, let's calculate the numerical value of the volume. V = π ( 2 π 4.5 ) 2 ∗ 13 ≈ 20.948769384470722 f t 3 .
Final Answer Finally, we need to choose the closest answer from the given options. The calculated volume is approximately 20.95 f t 3 . Therefore, the correct answer is C. 20.95 f t 3 .
Examples
Understanding the volume of tree trunks is crucial in forestry for estimating timber yield. Foresters use these calculations to determine how much wood can be harvested from a given area. This helps in sustainable forest management, ensuring that forests are not over-harvested and can continue to provide valuable resources and ecological benefits. By calculating the volume of individual trees, foresters can also assess the overall health and growth rate of a forest, which is important for long-term planning and conservation efforts.
