Final answer:
The tree's cross-sectional area increases by (2 / (2π))^2π square inches.
Step-by-step explanation:
The diameter of a tree is related to its circumference through the formula C = πd, where C is the circumference and d is the diameter. In this case, the tree's diameter increased by 2 inches, which means the increase in circumference is also 2 inches. Since the relationship between diameter and circumference is linear, we can say that the ratio of the increase in circumference to the increase in diameter is constant. Therefore, the increase in diameter can be found using the equation:
change in diameter = (increase in circumference) / π
Substituting the given values, we get:
change in diameter = 2 / π inches
This is the amount by which the tree's diameter increased. To find the increase in the tree's cross-sectional area, we can use the formula for the area of a circle:
area = πr^2, where r is the radius of the circle. Since the radius is half the diameter, the increase in the tree's cross-sectional area is given by:
change in area = π(change in diameter / 2)^2
Substituting the previously calculated change in diameter, we have:
change in area = π(2 / (2π))^2 square inches
Simplifying, we get:
change in area = (2 / (2π))^2π square inches
Therefore, the tree's cross-sectional area increases by (2 / (2π))^2π square inches.