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

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #1 : How To Find The Length Of An Arc

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?

Possible Answers:

Correct answer:

Explanation:

This question is asking for the length of an arc corresponding to  of a circle with radius eight feet. The question can be answered by evaluating for :

 

 

Example Question #2 : How To Find The Length Of An Arc

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. 

Which is the greater quantity? 

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

To compare  and , we note that

and 

We need to be able to compare  and . If we know which of the intercepting angles is the greater, then we know which of the arcs is greater. The intercepting angles are , respectively. However, we are not given this relationship. 

Example Question #3 : How To Find The Length Of An Arc

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?

Possible Answers:

The correct answer is not among these choices.

Correct answer:

Explanation:

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

 feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus  of a fourth, so its tip has traveled this circumference  times. 

This is

 feet. This is 

 inches.

Example Question #4 : How To Find The Length Of An Arc

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

Possible Answers:

(A) and (B) are equal

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

(B) is greater

(A) is greater

Correct answer:

(A) and (B) are equal

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  times the radius, or  feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or  feet.

 

The circumference of a circle seven feet in diameter is  times this diameter, or  feet.

The quantities are equal.

Example Question #2 : 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  feet. Which of the following is true of the time right now?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

 feet.

The minute hand has traveled  feet since noon, so it has traveled the circumference of the circle

 times.

Since , between 12 and  hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

Example Question #3 : How To Find The Length Of An Arc

Acute triangle  is inscribed in a circle. Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(a) and (b) are equal

Explanation:

Examine the figure below, which shows  inscribed in a circle.

Inscribed angle

By the Arc Addition Principle,

and

Consequently,

The two quantities are equal. 

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