The Hiker and the Submarine Diego Guaman(Group Leader) Brian Dawson Alex Flores Tobi Losotu Zahidah West Jorge Reyes Andy Cuevas Nyasia Walter Stephen Dronov Ivan Baltazar

You are planning a 20 mile hike in the Cascades You are planning a 20 mile hike in the Cascades. The guidebook provides you with a chart, plotting elevation above the trailhead (in feet) As a function of the distance hiked (in miles). This plot is shown and is modeled by the function: e(x)=125x2 - 6.25x3

Using the graph of e(x), explain in words how the tangent lines to the graph relate to the difficulty of the hike. Using the graph and the function of e(x), the tangent lines would relate the difficulty of the hike by displaying the various elevations the hiker would reach during their assent and dessent. (Zahidah)

(a) Using the graph of e(x), explain in words how the tangent lines to the graph relate to the difficulty of the hike The graph shows the as more miles are hiked, the elevation rises, making the journey more difficult until the hiker reaches the zero tangent line. After that, the elevation plummets after the zero tangent line when more miles are hiked. -Jorge Reyes

B. What can you say about the slope of the tangent line at the point of highest elevation on the graph? At the highest elevation we could see that they we have reached to a point that we can handle. Which is between 10 to 15 miles. After that we start descending to stop the 20 mile hike. At the highest point of elevation on the graph, the elevation rises then descend. Along the way of hiking the hiker is hiking up at 5 to 10 miles, then at a point that the hiker reaches highest elevation the hiker stops then begins to hike back down.

C) Find the instantaneous rate of change of elevation as a function of x = (Distance hiked). Include units in your answer. Diego: Lim f(x+h) – f(x) h->0 h Lim 125(x+h)2 – 6.25(x+h)3 – (125x2 – 6.25x3) *Factor h->0 h Lim 125(x2+2xh+h2) – 6.25(x3+3x2h+3xh2+h3) – 125x2 + 6.25x3 h->0 h Lim 125x2 + 250xh + 125h2 – 6.25x3 – 18.75x2h – 18.75xh2 – 6.25h3 – 125x2 + 6 . 2 5x3 h->0 h Lim 250x + 125h – 18.75x2 – 18.75xh – 6.25h2 h->0 e’(x) = 250x – 18.75x2 x=20 e’(20) = 250(20) – 18.75(20)2 e’(20) = -2500 feet/mile

Brian: Power Rule: d/dx (xn) = nxn-1 d/dx e(x) = 125x2 – 6 Brian: Power Rule: d/dx (xn) = nxn-1 d/dx e(x) = 125x2 – 6.25x3 e’(x) = 125 d/dx (x2) – 6.25 d/dx (x3) e’(x) = 125(2) x2-1 – 6.25(3) x3-1 e’(x) = 250x – 18.75x2 x = 20 e’(x) = 250x – 18.75x2 e’(20) = 250(20) – 18.75(20)2 e’(20) = -2500 feet/mile

D) Determine how far the hike will have traveled when he reaches the highest point on the trail.  The highest point on the trail is where the slope is zero: E'(x)=250x – 18.75x2 =0 X(250 –18.75x)= o Solutions X=0 (this is the initial point) 250 – 18.75x=0 -> x = 250/18.75 = 13.33 miles Andy Cuevas

D) 8,000 = 125 (13.501^2) - 6.25 (13.501^3) 8,000 estimated answer 22,784.62513 - 15380.76119 Nyasia

E. What is the elevation of the highest point on the trail? Stephen: e(x)= 125x2 – 6.25x3 e^1 = 2(125)x – 3(6.25) x2 x(250 – 18.75x)=0 250 = 18.75x x= 13.3 125(13.3)2 – 6.25(13.3)3 = 7,407ft The elevation of the highest point on the trail is 7,047ft

(e) What is the elevation of the of the highest point on the trail? Ivan: According to question (d): x= 13.333 By plugging in x into the function e(x)= = 7407ft We get that the elevation of the highest point on the trail is 7047ft