Question

In the figure below, the circle has center O and has radius 5. If the length of arc AB (shown in bold) is between 2n and 31, what is the lowest possible integer value of x. (72 doesn't work)

Answer 1

Given that

Radius = 5

`\begin{gathered} \text{Length of the arc is betw}e\text{n 2}\pi\text{ and 3}\pi \\ \text{Length of the arc = }(\theta)/(360)\text{ x 2}\pi r \\ \text{Where }\theta\text{ = x} \\ \text{For L = 2}\pi \\ 2\pi\text{ = }\frac{x}{360\text{ }}\text{ x 2}\pi\text{ x 5} \\ 2\pi\text{ = }\frac{x\cdot\text{ 10}\pi}{360} \\ \text{Cross multiply} \\ 360\cdot\text{ 2}\pi\text{ = 10}\pi x \\ \text{Isolate x} \\ x\text{ = }(720\pi)/(10\pi) \\ x=72^o \\ \text{For 3}\pi \\ 3\pi\text{ = }(x)/(360)\text{ x 2}\pi\cdot\text{ 5} \\ 3\pi\text{ = }(10\pi x)/(360) \\ \text{Cross multiply} \\ 3\pi\text{ x 360 = 10}\pi x \\ \text{Isolate x} \\ \text{x = }(1080\pi)/(10) \\ \text{x = 108}^o \end{gathered}`

Therefore, x can either be 72 or 108 degrees

The lowest possible integer is 72