Question

the diameter of a tree was 20 in. during the following year, the circumference increased 2 in. about how much did the tree’s diameter increase? about how much did the tree’s cross - sectional area change? the tree’s diameter increased by (type an exact answer, using π as needed.)

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.