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Problem 378

# The circumference of a tree trunk is $$6.6 \mathrm{ft}$$. a) What is the diameter of the trunk? b) What is the area of a cross-section of the tree trunk?

Expert verified
a) The diameter of the tree trunk is approximately $$2.1 \mathrm{ft}$$. b) The area of a cross-section of the tree trunk is approximately $$3.463 \mathrm{ft}^2$$.
See the step by step solution

## Step 1: a) Finding the Diameter of the Trunk

Given the circumference of the tree trunk, we can determine the diameter using the formula: Circumference = $$\pi d$$ Here, circumference = $$6.6 \mathrm{ft}$$ and $$\pi$$ is approximately equal to $$3.14159$$. So, $$6.6 = 3.14159d$$. Now, we need to find the value for d (diameter) by dividing the circumference by $$\pi$$. d = $$\frac{6.6}{3.14159}$$ d = $$2.1 \mathrm{ft}$$ The diameter of the tree trunk is approximately $$2.1 \mathrm{ft}$$.

## Step 2: b) Finding the Area of a Cross-section of the Tree Trunk

To find the area of a cross-section of the tree trunk, we need to use the formula: Area = $$\pi r^2$$ First, we need to determine the radius (r) of the tree trunk. The radius is half the diameter; thus, let's divide the diameter by 2. r = $$\frac{2.1}{2}$$ r = $$1.05 \mathrm{ft}$$ Now, we can plug the value of r into the formula: Area = $$3.14159 (1.05) ^ 2$$ Area = $$\approx 3.463 \mathrm{ft}^2$$ The area of a cross-section of the tree trunk is approximately $$3.463 \mathrm{ft}^2$$.

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