Three players on a reality TV show are brought to the centre of a large, flat field.

Question

Three players on a reality TV show are brought to the centre of a large, flat field. Each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each contestant) the following three displacements: A = 72.4m, 32.0° east of north, B = 57.3m, 36.0° south of west, C = 17.8m due south, The three displacements lead to the point in the field where the keys to a new Porsche are buried. Two players start measuring immediately, but the winner first calculates where to go. What does she calculate?

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Answer

The winner calculated and understood that 129⁰ is the direction that will take her to the buried keys of the Porsche Car.

Explanation

In this scenario, we have three contestants armed with meter sticks, compasses, calculators, and shovels, tasked with finding the keys to a new Porsche buried in a large, flat field. Each contestant has been provided with three different displacements: A = 72.4 meters at 32.0 degrees east of north, B = 57.3 meters at 36.0 degrees south of west, and C = 17.8 meters due south.

The key to success in this challenge lies in calculating the net displacement to the location of the buried Porsche keys. The contestant who can first determine this net displacement will be the winner.

So, let us break down the displacements provided to each contestant:

Contestant A:

Displacement A = 72.4 meters at 32.0 degrees east of north

Contestant B:

Displacement B = 57.3 meters at 36.0 degrees south of west

Contestant C:

Displacement C = 17.8 meters due south

Now, we need to calculate the net displacement, which will guide us to the location of the Porsche keys. To achieve this, we’ll use vector addition to find the resultant displacement, which can be thought of as the straight-line distance from the starting point to the destination.

Calculations for Each Constants

Let us proceed with the calculations for each contestant:

We need to understand that the angles of the vectors, measured from the +x-axis toward the +y-axis, are (90⁰ – 32⁰) = 58⁰, (180⁰ + 36⁰) = 216⁰, and 270⁰.

Contestant A:

Contestant A’s displacement is 72.4 meters at 32.0 degrees east of north. To calculate the horizontal (east-west) and vertical (north-south) components of this displacement, we’ll use trigonometry.

Horizontal component (A_x) = 72.4 meters * cos(58.0 degrees) = 38.37 m
Vertical component (A_y) = 72.4 meters * sin(58.0 degrees)= 61.40 m

Contestant B:

Contestant B’s displacement is 57.3 meters at 36.0 degrees south of west. We’ll use trigonometry once again to find the horizontal and vertical components of this displacement. Therefore, the angle we are using in this case is (180⁰ + 36⁰) = 216⁰

Horizontal component (B_x) = 57.3 meters * cos(216.0 degrees) = -46.36 m
Vertical component (B_y) = 57.3 meters * sin(216.0 degrees) = -33.68 m (negative because it’s south)

Contestant C:

Contestant C’s displacement is straightforward, as it’s due south. This means the entire 17.8-meter displacement is along the vertical axis.

Vertical component (C_y) = -17.8 meters (negative because it’s south)

Finding What She Calculates

Now, it’s time to add up the horizontal and vertical components of the displacements for each contestant. The net displacement (D) can be found as:

D_x = A_x + B_x + C_x

Remember that C_x = 0.00

Therefore, D_x = A_x + B_x + C_x = 38.37 + (-46.36) + (0.00) = – 7.99 m

Additionally, D_y = A_y + B_y + C_y = 61.40 + (-33.68) + (-17.80) = 9.92 m

Once we have the net horizontal and vertical components, we can calculate the magnitude and direction of the net displacement, which will guide the winner to the buried Porsche keys.

To find the magnitude (D) of the net displacement:
D = √(D_x² + D_y²) = √((-7.99)² + 9.92²) = 12.7 m

To determine the direction (θ) of the net displacement:
θ = arctan(D_y / D_x) = arctan(9.92 / (-7.99) = 129⁰ = 39⁰ South of west

Therefore, we have now seen that the losers measured three angles and three distances (72.4 + 57.3 + 17.8) making a total of 147.5 meters, which is one meter at a time. At the same time, the winner was able to measure one angle and one shorter distance.

Finally, we can see that the contestant who performs these calculations first and identifies 129 degrees as the correct location and direction of the net displacement will emerge as the winner of this exciting challenge, claiming the keys to the new Porsche buried in the field.