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# Euclidean and Rectilinear Distance Problem: The ODU Bookstore

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The ODU bookstore has decided to promote its new opening by having a shopping spree. The winner of the shopping spree will have to abide by the following rules. They will have 10 minutes to grab as many items as possible from the store. They must place a shopping cart in one spot and leave it there for the entire duration of the spree. Also they can only pick up one item at a time from the shelves and bring it to the cart. The following table provides a list of all the items in the store (22), the X,Y coordinates of each item, the cost of each item, and the quantity available. It is estimated that a person can travel 20 feet in one second, therefore if an item is 100 fee away it will take a person 5 second to retrieve the item and 5 seconds to return it to a cart for a total of 10 seconds. The shopping cart must be placed inside the store.

A. Assuming the shopping cart must be placed at X,Y coordinates (0,) and assuming Euclidean distances, what is the maximum profit a contestant can achieve from the shopping spree?

B. Assuming the shopping cart must be placed at X,Y coordinates (0,)) and assuming Rectilinear distances, what is the maximum profit a contestant can achieve?

C. Assuming you can place the shopping cart anywhere and assuming Euclidean distances, what is the maximum profit a contestant can achieve? Where should the contestant place the shopping cart?

C. Assuming you can place the shopping cart anywhere and assuming Rectilinear distances, what is the maximum profit a contestant can achieve. Where should the contestant place the shopping cart?

#### Solution Preview

Let's first define Eucliden and rectilinear distance. Assume that Xi and Yi are the coordinates of product i, and Xc and Yc are the coordinates of the cart. Then, the distance from the cart to the product is defined as follows:
Euclidean: sqrt( (Xi - Xc)^2 + (Yi - Yc)^2) [sqrt() means 'square root of']
Rectilinear: |Xi - Xc| + |Yi - Yc| [| | means 'absolute value of']

For ease of notation, the above equations will be defined as Euclidean(Xi, Xc, Yi, Yc), and Rectilinear(Xi, Xc, Yi, Yc)

The problem to solve is very similar for all four points of this question. Let's call Ni to the number of items of product i that the person will grab. Then, the time spent grabbing these items (assuming Euclidean distances), measured in seconds, will be:
Ni * Euclidean(Xi, Xc, Yi, Yc) * 2 / 20

The rationale for this formula should be clear. The total distance to get one item and then go back to the cart to store it, is twice the distance to the item [Euclidean(Xi, Xc, Yi, Yc) * 2]. Now, the person moves at 20 ft per second. Therefore, we divide that formula by 20. Finally, we multiply it by Ni, because this trip must be repeated once for each unit the person gets.

Clearly, the formula is ...

#### Solution Summary

Euclidean and rectilinear distance problems. The ODU Bookstore is examined.

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