Question

Tomas is planning out a rail trail using a map with a marked grid. The head of the trail, which has an information kiosk, is located at (3, 0). He moves 1 unit east and 3 units north and places a pin at (4, 3) to represent the location of another information kiosk. He continues placing pins so that each pin is 1 unit east and 3 units north in relation to the previous pin. The map extends 30 units east and 40 units north to represent the land available for the rail trail to use. Where will the foot of the trail be if it also has an information kiosk and Tomas wants to make the trail as long as possible? Enter the coordinates in the boxes.

Answer 1

Kiosks only occur when the y-coordinate is a multiple of 3. The closest to the 40 units of northern land would be at 39. 39 divided by 3 would mean Tomas placed an additional 13 kiosks after the first. With the x-coordinate incrementing 1 unit east each time, the x-coordinate of the last kiosk endpoint is 3+13 or 16. Thus the foot of the trail ends at (16,39).

Answer 2

Since the head of the trail is located at (3, 0), the trail moves 3 units north with each move and the map extends to 40 units to the north, then the number of 3 units in 40 units is 13. 

Thus, he can move only thirteen 3 units to the north and hence, thirteen 1 unit to the east.

Thus, the x-coordinate of the foot of the trail is 3 + 13 = 16 and the y-coordinate of the foot of the trail is 0 + 3(13) = 39.

Therefore, the location of the foot of the trail is at (16, 39).