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

A giant clock has a minute hand four and one-half feet in length. Since noon, the tip of the minute hand has traveled $110 \pi$ feet. Which of the following is true of the time right now?

The time is between 12:00 midnight and 12:30 AM.

The time is between 11:30 PM and 12:00 midnight.

The time is between 11:00 PM and 11:30 PM.

The time is between 12:30 AM and 1:00 AM.

The time is between 1:00 AM and 1:30 AM.

Explanation

Every hour, the tip of the minute hand travels the circumference of a circle, which here is

$C = 2 \pi r = 2 \pi \times 4 \frac{1}{2} = 9 \pi$ feet.

The minute hand has traveled $110 \pi$ feet since noon, so it has traveled the circumference of the circle

$\frac{110 \pi}{9 \pi} = \frac{110 }{9 } =12 \frac{2}{9}$ times.

Since $12 \leq 12 \frac{2}{9} \leq 12 \frac{1}{2}$ , between 12 and $12 \frac{1}{2}$ hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

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.

Acute triangle $\bigtriangleup ABC$ is inscribed in a circle. Which is the greater quantity?

(a) $m \overarc {ABC} - m \overarc {ACB}$

(b) $m \overarc {A B} - m \overarc {A C}$

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

It is impossible to determine which is greater from the information given

Explanation

Examine the figure below, which shows $\bigtriangleup ABC$ inscribed in a circle.

Inscribed angle

By the Arc Addition Principle,

$m \overarc {ABC} = m \overarc {AB}+ m \overarc {BC}$

and

$m \overarc {ACB} = m \overarc {AC}+ m \overarc {BC}$

Consequently,

$m \overarc {ABC} - m \overarc {ACB}$

$= (m \overarc {AB}+ m \overarc {BC} ) - ( m \overarc {AC}+ m \overarc {BC})$

$= m \overarc {AB}+ m \overarc {BC} - m \overarc {AC}- m \overarc {BC}$

$= m \overarc {A B} - m \overarc {A C}$

The two quantities are equal.

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 $\frac{144}{360}$ :

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

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. $\angle B \cong \angle C$

Which is the greater quantity?

(a) $m \widehat{ABC}$

(b) $m \widehat{CAB}$

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Explanation

To compare $m \widehat{ABC}$ and $m \widehat{CAB}$ , we note that

$m \widehat{ABC} = m \widehat{AB} + m \widehat{BC}$

and

$m \widehat{CAB} = m \widehat{AB} + m \widehat{AC}$

We need to be able to compare $m \widehat{BC}$ and $m \widehat{AC}$ . If we know which of the intercepting angles is the greater, then we know which of the arcs is greater. The intercepting angles are $\angle A , \angle B$ , respectively. However, we are not given this relationship.

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?

$4.54\pi in$

$3.01\pi in$

Explanation

To begin, let's recall our formula for length of an arc.

$Arc=\frac{\theta}{360^{\circ}}2 \pi r$

Now, just plug in and simplify

$Arc=\frac{65^{\circ}}{360^{\circ}}2 \pi (4in)=\frac{13^{\circ}}{9^{\circ}}\pi in\approx 4.54in$

So, our answer is 4.54in

The clock at the town square has a minute hand eight feet long. How far has its tip traveled since noon if it is now 12:58 PM?

$48.6 ; \textrm{ft}$

$50.2 ; \textrm{ft}$

$47.2 ; \textrm{ft}$

$48.2 ; \textrm{ft}$

$49.4 ; \textrm{ft}$

Explanation

This question is asking for the length of an arc corresponding to $\frac{58}{60}$ of a circle with radius eight feet. The question can be answered by evaluating for :

$\frac{58}{60} \cdot 2 \pi r=\frac{58}{60} \cdot 2 \pi \cdot 8 \approx 48.6$

A giant clock has a minute hand seven feet long. Which is the greater quantity?

(A) The distance traveled by the tip of the minute hand between 1:30 PM and 2:00 PM

(B) The circumference of a circle seven feet in diameter

(A) and (B) are equal

(A) is greater

(B) is greater

It is impossible to determine which is greater from the information given

Explanation

The tip of a minute hand travels a circle whose radius is equal to the length of that minute hand, which, in this question, is seven feet long. The circumference of this circle is $2 \pi$ times the radius, or $2 \pi \times 7 = 14\pi$ feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or $\frac{1}{2} \times 14 \pi = 7 \pi$ feet.