The Taco Cart Inquiry Problem Done by: Sonali Timmath, Ekjot Kaur Grewal, & Ompreet Kaur Sarang

Part 1 Guess who will reach the cart first: We hypothesize that Ben will reach the cart first. He reaches the half point mark when Mark reaches the road. Mark will have to walk a further distance on the road than Ben, who has to walk a shorter distance to the cart. In order to give a definitive answer, we need the speed on sand, on road and the dimensions. Speed on sand: 2 ft/s Distance to road ft Speed on road: 5 ft/s Distance to taco cart on road ft How does this extra information affect your initial guess? Now that we know that the speed on the road is much greater than the speed on sand, this changes our initial guess. Mark gets to travel much faster once he gets onto the road, which means he will reach the taco cart faster than Ben, who is still at his slower speed of 2 ft/s.

Part 1- Continued It takes Mark 4 minutes and seconds to reach the taco cart. It takes Ben 5 minutes and seconds to reach the taco cart.

Part 2 Determine the exact position of the cart on the road such that both Ben and Mark will reach it at the exact same time: First, we need an equation. The variables are defined in the diagram.

Part 2 Graphing the equation on Desmos, we find that when a= 310.1ft, Ben and Mark will get to the taco cart at the same time.

Part 2 Find the distance each person will have to walk to arrive at the taco cart at the same time. Distance Mark has to walk: ft on sand, ft on road Distance Ben has to walk: ft on sand Time Mark has to walk: seconds Time Ben has to walk: seconds

Part 3 Determine an optimum path to the cart that results in the fastest time. Numerically: Create 10 or more possible paths to the cart and calculate the times. Determine an interval of time that contains the fastest time. Graphically: Set up an algebraic expression and determine the solution graphically. Make sure to clearly indicate the meaning of each variable used in the expression. Comment on the accuracy, or lack of, your previous Numerical solution compared to the graphical. Algebraically: Use Calculus to determine a solution, showing all steps. Verify all your answers using Wolfram. Illustrate the optimum path with a clearly labelled diagram.

Part 3- Numerically Example of one of our calculations in the table on the next page (for m=50):

Part 3- Numerically The interval where the minimum (fastest time) is located: m= [125, 175] Values of m (ft) Values of x (ft) Values of y (ft) Total time (sec)

Part 3- Graphically The Equation: The variables are defined in the diagram.

Part 3- Graphically t(m)= The answer graphically is m= , t= seconds This is much more accurate than the numerical method.

Part 3- Algebraically We found the derivative of the algebraic expression found in the graphical method:

Part 3- Algebraically Through the sign chart of the derivative, we were able to find the optimum time to the cart by determining the minimum time....and here we can see Ekjot hard at work, doing her favourite thing in the world. Creating sign charts. :)

Using Wolfram to check answers. To prove our answers we used wolfram to find the derivative of the original function f(x). In addition were able to find where the derivative has a zero, which was used to determine the minimum of f(x) and the optimum time to the taco cart. Original f(X) f’(x)

Diagram of the Optimum Path The red line is the optimum path to the taco cart. m = 142.1ft420.5 ft ft ft ft Tacos! Ben and Mark

Part 4: Reflection What parts of this Inquiry Problem did you enjoy the most and the least? Why? Our favourite part of the inquiry was using the skills we had learned in finding derivatives in functions to figure out the information we needed to find things like the optimum time. Our least favourite part of the Inquiry problem was finding the solutions algebraically, because the numbers were not nice whole numbers. What skills/strategies did this problem help you to develop? How will these skills benefit your learning, current and future? This problem forced us to think critically and make use of all the information we were given. We used the skills we learned, like optimization to find the best possible choice, in this case, the path to the taco cart, as found in part 4. These problem solving skills will help us understand the importance of what we learn, and how math is applied everywhere in the real world. What features of Desmos and Wolfram Alpha did you find the most useful? These softwares were very helpful because they allowed us to easily create, visualize, and analyze functions as we solved the questions. Whenever we were unsure about a calculated answer, we used Wolfram to ensure the equation was correct, for example finding the derivative of f(x) and finding the zero of the derivative in order to find the optimum path with the shortest time.