ISEE Upper Level Quantitative Reasoning - How to find the length of an arc

A giant clock has a minute hand six feet long. How far, in inches, did the tip move between noon and 1:20 PM?

$192 \pi \textrm{ in}$

$48 \pi \textrm{ in}$

$384 \pi \textrm{ in}$

$768 \pi \textrm{ in}$

$96\pi \textrm{ in}$

Explanation

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius 6 feet. This circumference is $2\pi r = 2 \pi \times 6 = 12\pi$ feet. One hour and twenty minutes is $1 \frac{1}{3}$ hours, so the tip of the hand moved $1 \frac{1}{3} \times 12 \pi = 16 \pi$ feet, or $16 \pi \times 12 = 192 \pi$ inches.

Intercepted

In the above diagram, radius .

Give the length of $\overarc {AB}$ .

$\frac{48 \pi }{5}$

$\frac{24 \pi }{5}$

$\frac{288\pi }{5}$

$\frac{72 \pi }{5}$

Explanation

The circumference of a circle is $2 \pi$ multiplied by its radius , so

$C = 2 \pi \cdot OA = 2 \pi \cdot 12 = 24 \pi$ .

$\angle ACB$ , being an inscribed angle of the circle, intercepts an arc $\overarc {AB}$ with twice its measure:

$m \overarc {AB} = 2 \cdot m\angle ACB = 2 \cdot 72 ^{\circ } = 144 ^{\circ }$

The length of $\overarc {AB}$ is the circumference multiplied by $\frac{144}{360}$ :

$\frac{144}{360} \times 24 \pi = \frac{48 \pi }{5}$ .

A giant clock has a minute hand three feet long. How far, in inches, did the tip move between noon and 12:20 PM?

$24\pi \textrm{ in}$

It is impossible to tell from the information given

$12\pi \textrm{ in}$

$18\pi \textrm{ in}$

$36\pi \textrm{ in}$

Explanation

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius feet. This circumference is $2\pi r = 2 \pi \times 3 = 6\pi$ feet. minutes is one-third of an hour, so the tip of the minute hand moves $\frac{1}{3} \times 6\pi = 2\pi$ feet, or $2\pi \times 12 = 24\pi$ inches.

Inscribed

In the above figure, express in terms of .

$y = x- \frac{27}{4}$

$y = x- \frac{1}{4}$

$y = \frac{1}{2}x - 10$

$y = \frac{1}{2}x - \frac{3}{2}$

Explanation

The measure of an arc - $\overarc{AB}$ - intercepted by an inscribed angle - $\angle ACB$ - is twice the measure of that angle, so

$m \overarc{AB} = 2 \cdot m \angle ACB$

$\frac{4y}{4} = \frac{4x- 27}{4}$

$y = x- \frac{27}{4}$

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of $65^{\circ}$ . If the screen has a radius of 4 inches, what is the length of the arc of the highlighted wedge?