Question:

The figure shows the shape of tunnel entrance. If the curved portion is 3/4 of a circle and the base of the entrance is 12 feet across, what is the perimeter, in feet, of the curved portion of the entrance?

1. \(9π\)
2. \(12π\)
3. \(9π\sqrt{2}\)
4. \(18π\)
5. \(9π\sqrt{2}\)

Correct Answer: C
Solution and Explanation:

Approach Solution 1:

This is a GMAT problem-solving question in which you have to use the details given in the question to solve the problem. The problem in this category are coming from different areas of math topics. This one particularly is from circles.
The option is given in such a way that it is difficult to guess the correct answer. The candidates need to know the right approach to get the required answer. Only one of the given five choices is correct.

The candidates should have a good understanding of circles and trigonometry to solve this problem.
We can observe from the given data in the question that,
the base makes an isosceles right angle triangle of 45, 45,90 degrees with the center of the circle
Let r be the radius of the circle.
Base corresponds to the side opposite to 90 deg
--> √2 r = 12
--> r = \(6\sqrt{2}\)

Now we got the radius of the circle
Perimeter of 3/4th of the circle = 3/4∗2π∗6√2
= 9π√2

C is the correct choice.

Approach Solution 2:

This is a GMAT problem-solving question in which you have to use the details given in the question to solve the problem. The problem in this category are coming from different areas of math topics. This one particularly is from circles and basic trigonometry.
The option is given in such a way that it is difficult to guess the correct answer. The candidates need to know the right approach to get the required answer. Only one of the given five choices is correct.

The candidates should have a good understanding of circles and trigonometry to solve this problem.
Glue the base's endpoints to the center. The angle covered is 360 * 3/4, or 270 degrees, because the curved part represents 3/4 of the circle.

The base's angle with the center is 360 - 270 = 90 degrees.

Also, take note that the circle's base is a chord.
Drop a perpendicular onto the base from the circle's center. It will cut through both the base and the angle occupied by the center.
Now, using trigonometry, we will discover that
r sin 45 = (12 / 2)
The value of sin45 degree is 1 / sqrt2
(you can use trigonometric tables to get the values of sin and cos of different angles.)

r = 6 sqrt 2
Now we have the radius of the circle.
We are asked to find out the perimeter of the curved part.
=> perimeter of the curved part = (3/4) * 2 pi * r

= (9 * pi * sqrt 2)
This is given in choice C.

C is the correct choice.

Approach Solution 3:

To find the perimeter of the curved portion of the entrance, we need to find the circumference of the circle that the curved portion is 3/4 of. If the base of the entrance is 12 feet across, then the diameter of the circle is 12 feet. The circumference of a circle can be found using the formula C = πd, where C is the circumference and d is the diameter.
Substituting d = 12, we get C = π * 12 = 37.7 (approximately).
Since the curved portion is 3/4 of the circle, the circumference of the curved portion is (3/4) * 37.7 = 28.3 (approximately).
So the perimeter of the curved portion of the entrance is = 9π√2

C is the correct choice.

“The figure shows the shape of the tunnel entrance. If the curved portion" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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