Question

find the circumference and arc lengths d = 32.00 cm

Explanation:

Step1: Find the circumference

The formula for the circumference \(C\) of a circle is \(C=\pi d\), where \(d\) is the diameter. Given \(d = 32.00\space cm\) and taking \(\pi\approx3.1416\), we have \(C=\pi\times32.00\).

$$C = 3.1416\times32.00=100.5312\space cm$$

Step2: General formula for arc length

The formula for the arc length \(L\) of a circle is \(L=\frac{\theta}{360^{\circ}}\times C\) (when \(\theta\) is the central angle in degrees). But since no central angle \(\theta\) is given for the arc length part in the problem statement (assuming a full - circle arc for the sake of showing the formula application with the given diameter, i.e., \(\theta = 360^{\circ}\)), for a full - circle arc (circumference is a special case of arc length when \(\theta=360^{\circ}\)), \(L = C\). If we assume a general \(\theta\), the formula remains \(L=\frac{\theta}{360^{\circ}}\times\pi d\)

Answer:

The circumference \(C = 100.53\space cm\) (rounded to two decimal places). If we assume a full - circle arc (i.e., \(\theta = 360^{\circ}\)), the arc length \(L = 100.53\space cm\). For a general central angle \(\theta\) (not given in the problem), the arc length formula is \(L=\frac{\theta}{360^{\circ}}\times\pi\times32.00\space cm\)