ISEE Upper Level Quantitative Reasoning - How to find circumference

The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.

1 foot, 11 inches

2 feet

2 feet, 1 inch

4 feet 2 inches

3 feet 10 inches

Explanation

The circumference of a circle can be determined by multiplying its radius by $2 \pi$ , so the circumferences of the two smallest circles are

$C_{1} = 2 \cdot 2 \pi = 4 \pi$

and

$C_{2} = 2 \cdot 5 \pi = 10 \pi$

The circumferences form an arithmetic sequence with common difference

$C_{2}- C_{1}= 10 \pi - 4 \pi = 6 \pi$

The circumference of a circle can therefore be found using the formula

$C_{n} = C_{1} + (n-1)d$

where $C_{1} = 4 \pi$ and $d = 6 \pi$ ; we are looking for that of the th smallest circle, so

$C_{8} = 4 \pi + (8-1)6 \pi$

$= 4 \pi + 7 \cdot 6 \pi$

$= 4 \pi + 42 \pi$

$= 46 \pi$

Since the radius of a circle is the circumference of the circle divided by $2 \pi$ , the radius of this eighth circle is

$r = \frac{46 \pi}{2 \pi} = 23$ inches, or 1 foot 11 inches.

Find the circumference of a circle with a radius of 4cm.

$8 \pi \text{ cm}$

$4\pi \text{ cm}$

$12\pi \text{cm}$

$2\pi \text{ cm}$

$16\pi \text{ cm}$

Explanation

To find the circumference of a circle, we will use the following formula:

$C = 2 \cdot \pi \cdot r$

where r is the radius of the circle.

Now, we know the radius of the circle is 4cm.

Knowing this, we can substitute into the formula. We get

$C = 2 \cdot \pi \cdot 4\text{cm}$

$C = 2\pi \cdot 4\text{cm}$

$C = 8\pi \text{ cm}$

Track

The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure. Quinnella wants to run around the track for one and a half miles. If Quinnella starts at point C and runs counterclockwise, which of the following is closest to the point at which she will stop running?

(Assume the five points are evenly spaced)

Between Points B and C

Between Points C and D

Between Points D and E

Between Points E and A

Between Points A and B

Explanation

A circle of radius 600 feet will have a circumference of

$C = 2 \pi r \approx 2 \cdot 3.14 \cdot 600 = 3,768$ feet.

Quinnella will run one and a half miles, or

$5,280 \times 1\frac{1}{2} = 7,920$ feet,

which is about times the circumference of the circle.

Quinnella will run around the track twice, returning to Point C; she will not quite make it to Point B a third time, since that is one-fifth of the track, or 0.2. The correct response is that she will be between Points B and C.

Track

The track at Simon Bolivar High School is a perfect circle of radius 500 feet, and is shown in the above figure. Manuel starts at point C, runs around the track counterclockwise three times, and continues to run clockwise until he makes it to point D. Which of the following comes closest to the number of miles Manuel has run?

$2\frac{1}{4} \textup{ mi}$

$2\frac{1}{2} \textup{ mi}$

$2\frac{3}{4} \textup{ mi}$

$2 \textup{ mi}$

$3 \textup{ mi}$

Explanation

The circumference of a circle with radius 500 feet is

$C = 2 \pi r \approx 2 \cdot 3.14 \cdot 500 = 3,140$ feet.

Manuel runs this distance three times, then he runs from Point C to D, which is about four-fifths of this distance. Therefore, Manuel's run will be about

$3,140 \times 3 \frac{4}{5} = 11,932$ feet.

Divide by 5,280 to convert to miles:

$11,932 \div 5,280 \approx 2.26$ ,

making $2\frac{1}{4}$ miles the response closest to the actual running distance.

Track

The track at Truman High School is shown above; it is comprised of a square and a semicircle.

Veronica begins at Point A, runs three times around the track counterclockwise, and continues until she reaches Point B. Which of the following comes closest to the distance Veronica runs?

$1\frac{1}{2} \textup{ mi}$

$1 \textup{ mi}$

$1\frac{1}{4} \textup{ mi}$

$1\frac{3}{4} \textup{ mi}$

$2 \textup{ mi}$

Explanation

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is about

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 500 = 785$ feet.

The distance around the track is about

feet.

Veronica runs around the track three complete times, for a distance of about

$3 \times 2,285 = 6,855$ feet.

She then runs from Point A to Point E, which is another 500 feet; Point E to Point D, which is yet another 500 feet, and, finally Point D to Point B, for a final 785 feet. The total distance Veronica runs is about

feet.

Divide by 5,280 to convert to miles:

$8,640 \div5,280 \approx 1.6$

The closest answer is $1 \frac{1}{2}$ miles.

Track

The track at James Buchanan High School is shown above; it is comprised of a square and a semicircle.

Diane wants to run two miles. If she begins at Point A and begins running counterclockwise, when she is finished, which of the five points will she be closest to?

Explanation

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 400 = 628$ feet.

The distance around the track is about

feet.

Diane wants to run two miles, or

$5,280 \times 2 = 10, 560$ feet.

She will make about

$10,560 \div 1,828 \approx 5.78$

circuits around the track.

Equivalently, she will run the track 5 complete times for a total of about

$5 \times 1,828 = 9,140$ feet,

so she will have

feet to go.

She is running counterclockwise, so she will proceed from Point A to Point D, running another 800 feet, leaving

feet.

She will almost, but not quite, finish the 628 feet from Point D to Point B.

The correct response is Point B.

Track

The track at Monroe Elementary School is a perfect circle of radius 400 feet, and is shown in the above figure.

Evan and his younger brother Mike both start running from Point A. Evan runs counterclockwise, running once around the track and then on to Point E; Mike runs clockwise, meeting Evan at Point E and stopping.

Which of the following is the greater quantity?

(a) Twice Mike's average speed.

(b) Evan's average speed.

(Assume the five points are evenly spaced)

(b) is greater

(a) is greater

(a) and (b) are equal

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

Explanation

It is not actually necessary to know the radius or length of the track if we know the points are equally spaced. Evan runs once around the track counterclockwise and then on to Point E, which is the next point after A; this means he runs around the track $1\frac{1}{5} = \frac{6}{5}$ times. Mike runs around the track clockwise from Point A to Point E, in the same time, meaning he runs around the track $\frac{4}{5}$ times.

Therefore, Evan's speed is $\frac{ \frac{6}{5}}{\frac{4}{5}} = \frac{3}{2}$ times Mike's speed. As a result, Twice Mike's speed would be greater than Evan's speed, making (b) the greater.

Let $\pi = 3.14$

Find the circumference of a circle with a radius of 8in.

$50.24\text{in}$

$25.12\text{in}$

$200.96\text{in}$

$56.52\text{in}$

$43.96\text{in}$

Explanation

To find the circumference of a circle, we will use the following formula:

$C = 2\pi r$

where r is the radius of the circle.

Now, we know the radius of the circle is 8in.

We also know that $\pi = 3.14$ .

Knowing all of this, we can substitute into the formula. We get

$C = 2 \cdot 3.14 \cdot 8\text{in}$

$C = 16\text{in} \cdot 3.14$

$C = 50.24\text{in}$

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the circumference of the crater?

$26 \pi m$

Explanation

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .