the diameter of a tree was 20 in. during the following year, the circumference increased 2 in. about how…

Explanation:

Step1: Recall the formula for circumference

The formula for the circumference of a circle is (C = \pi d), where (C) is the circumference and (d) is the diameter. Differentiating both sides with respect to (d), we get (dC=\pi dd).

Step2: Solve for (dd) (change in diameter)

We know that (dC = 2) (the increase in circumference). Substituting (dC = 2) into (dC=\pi dd), we can solve for (dd). Rearranging the equation (dd=\frac{dC}{\pi}).

Step3: Recall the formula for area

The formula for the area of a circle is (A=\pi r^{2}=\frac{\pi d^{2}}{4}). Differentiating with respect to (d), we use the power rule ((x^{n})^\prime=nx^{n - 1}). So (dA=\frac{\pi}{4}\times2d;dd=\frac{\pi d}{2}dd).

Step4: Substitute (d = 20) and (dd=\frac{2}{\pi}) into the area - change formula

Substitute (d = 20) and (dd=\frac{2}{\pi}) into (dA=\frac{\pi d}{2}dd). Then (dA=\frac{\pi\times20}{2}\times\frac{2}{\pi}).

Answer:

The tree's diameter increased by (\frac{2}{\pi}) inches. The change in the tree's cross - sectional area is (20) square inches.