Problem 66 - Thinking Mathematically

Question

India Jones is standing on a large rock in the middle of a square pool filled with hungry, man-eating piranhas. The edge of the pool is 20 feet away from the rock. India's mom wants to rescue her son, but she is standing on the edge of the pool with only two planks, each \(19 \frac{1}{2}\) feet long. How can India be rescued using the two planks?

Step-by-Step Solution

Answer

The combined length of the two planks (\(19 \frac{1}{2} + 19 \frac{1}{2} = 39\) feet) is longer than the length of the pool's diagonal (\(20\sqrt{2} \approx 28.28\) feet). Therefore, the two planks can be laid from one corner of the square pool to the opposite corner (along the pool's diagonal), passing over the rock. Hence, India can be rescued.

Step 1: Determine the Length of the Diagonal

Firstly, assume the pool as a square with side length of 20 feet. Therefore, calculate the length of the diagonal using Pythagoras' Theorem because it's a square, so its two perpendicular sides would form a right-angled triangle with the diagonal: \(\sqrt{(20^2) + (20^2)} = \sqrt{800} = 20\sqrt{2}\approx 28.28\) feet.

Step 2: Comparing the lengths

The planks are each \(19 \frac{1}{2}\) feet long. Therefore, if you add the lengths together an approximate total of 39 feet is achieved.

Step 3: Formulate rescue plan

Since the total length of the two planks (39 feet) is greater than the length of the pool's diagonal (approximately 28.28 feet), this indicates that in the worst-case scenario, the two planks would be long enough to stretch from one corner of the pool to another, passing over the rock India is standing on, thus allowing India's mom to rescue him by laying the planks from one corner of the pool to the opposite corner going over the rock.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates to right-angled triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula: \[ c^2 = a^2 + b^2 \]where \( c \) represents the hypotenuse, and \( a \) and \( b \) are the other two sides. In the context of India's dilemma, the pool is square-shaped, meaning each side is equal in length. The diagonal of this square pool acts as the hypotenuse of a right-angled triangle where each side of the square is a cathetus.

• For a square pool with sides of 20 feet, the length of the diagonal \( d \) would be calculated with: \[ d = \sqrt{20^2 + 20^2} = \sqrt{800} = 20\sqrt{2} \text{ feet} \]
• This results in an approximate diagonal length of 28.28 feet.

Understanding this concept helps in figuring out rescue strategies, like determining the reach needed by the planks.

Geometric Shapes

Geometric shapes, particularly squares and triangles, play a crucial role in many mathematical problems. A square, in particular, is a regular quadrilateral, meaning all four sides are equal, and all angles are 90 degrees. In geometry, understanding the properties of squares aids in calculating distances such as diagonals.

• For instance, a square's diagonal essentially divides the square into two congruent right-angled triangles.
• This property is leveraged often in calculations to determine necessary lengths or distances.

In India's scenario, the square pool's shape guides the rescue plan. The sides and the diagonal are essential for calculating whether the planks available are sufficient in length to reach India from the edge of the pool.

Rescue Strategy

When confronted with a situation requiring problem-solving prowess, a logical and strategic approach becomes essential. In the scenario of rescuing India, the key is understanding how to use available resources effectively. Here the two planks, each measuring \(19 \frac{1}{2}\) feet in length, serve as the tools for rescue.

• By combining the lengths of the planks, a total length of approximately 39 feet is available.
• This total length significantly exceeds the diagonal length of the square pool, calculated earlier to be about 28.28 feet.
• The strategy involves positioning the two planks from one corner of the pool, across its diagonal, to the opposite corner.

With the correct placement, these planks can bridge the square pool safely, spanning over the rock within, thereby facilitating India's mom in executing a successful rescue without entering the perilous waters. Understanding the interplay between plank length and pool geometry is crucial to devising such an effective rescue plan.