### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Sectors

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:**

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 #1 : How To Find The Length Of An Arc

Note: Figure NOT drawn to scale

Refer to the above figure.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

**Correct answer:**

It is impossible to tell from the information given

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 #2 : 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:**

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 #3 : 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:**

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

(A) is greater

(B) is greater

(A) and (B) are equal

**Correct answer:**

(A) and (B) are equal

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 #1 : 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 11:00 PM and 11:30 PM.

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

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

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

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

**Correct answer:**

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

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 #2 : How To Find The Length Of An Arc

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

(a)

(b)

**Possible Answers:**

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

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

By the Arc Addition Principle,

and

Consequently,

The two quantities are equal.

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