X、Y 和 Z 位于一个以 O 为圆心、半径为 $4$ cm 的圆上。角 XOY = $2.5$ 弧度。 [图形如以下所述]

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完整解法

1. Substitute the radius and central angle into the sector area formula

For the minor sector, $r=4$ cm and $\theta=2.5$ radians.

$\text{Area}=\frac{1}{2}r^2\theta=\frac{1}{2}(4)^2(2.5)=20$

答案

$20\text{ cm}^2$

1. Subtract the minor central angle from one full revolution to get the reflex angle

Reflex angle $XOY = 2\pi - 2.5$ radians.

答案

$2\pi - 2.5$ radians

1. Use the reflex angle to find the length of the major arc XZY

From part (b)(i), the reflex angle $XOY$ is $2\pi-2.5$ radians.

Arc length $=r\theta$

$=4(2\pi-2.5)$

$=8\pi-10$

So the length of arc $XZY$ is $(8\pi-10)\text{ cm}$ .

答案

$(8\pi-10)\text{ cm}$

1. Relate the major arc length to the circumference of the cone base

From part (b)(ii), arc length of $XZY = 8\pi-10$ cm.

This becomes the circumference of the cone base, so
$2\pi r=8\pi-10.$

2. Solve for the base radius of the cone

$r=\frac{8\pi-10}{2\pi}$
$r=4-\frac{5}{\pi}$

答案

$4-\dfrac{5}{\pi}\text{ cm}$

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