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

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Question

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?

Answer

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 :

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

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Question

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) $m \widehat{CAB}$

Answer

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.

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Question

A giant clock has a minute hand that is six feet long. The time is now 3:50 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Answer

Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is

$C = 2 \pi r= 2 \pi \cdot 6 = 12 \pi$ feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus $\frac{50}{60 } = \frac{5}{6}$ of a fourth, so its tip has traveled this circumference $3\frac{5}{6}$ times.

This is

$3\frac{5}{6} \times 12 \pi = \frac{23}{6} \times 12 \pi = 46 \pi$ feet. This is

$46 \pi \times 12 = 552 \pi$ inches.

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Question

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

Answer

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.

The circumference of a circle seven feet in diameter is $\pi$ times this diameter, or $7 \pi$ feet.

The quantities are equal.

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Question

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?

Answer

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.

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Question

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

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

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

Answer

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

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.

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Question

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

Answer

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.

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Question

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

Answer

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.

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Question

In the above figure, express in terms of .

Answer

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

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

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

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

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Question

In the above diagram, radius .

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

Answer

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 $\frac{144}{360}$ :

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

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Question

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?

Answer

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?

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

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Question

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?

Answer

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 :

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

Compare your answer with the correct one above

Question

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) $m \widehat{CAB}$

Answer

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.

Compare your answer with the correct one above

Question

A giant clock has a minute hand that is six feet long. The time is now 3:50 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Answer

Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is

$C = 2 \pi r= 2 \pi \cdot 6 = 12 \pi$ feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus $\frac{50}{60 } = \frac{5}{6}$ of a fourth, so its tip has traveled this circumference $3\frac{5}{6}$ times.

This is

$3\frac{5}{6} \times 12 \pi = \frac{23}{6} \times 12 \pi = 46 \pi$ feet. This is

$46 \pi \times 12 = 552 \pi$ inches.

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Question

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

Answer

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.

The circumference of a circle seven feet in diameter is $\pi$ times this diameter, or $7 \pi$ feet.

The quantities are equal.

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Question

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?

Answer

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.

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Question

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

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

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

Answer

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

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.

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Question

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

Answer

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.

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Question

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

Answer

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.

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Question

In the above figure, express in terms of .

Answer

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

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

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

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

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