Question

Walking and rowing Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is 600 m from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x\). Find and graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x .\) Find and graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land to minimize the total time of her trip. What is that minimum time?

Short answer

Answer: Kelly should land her boat at point L on the shore (x meters from point P) to minimize her total trip time. The minimum total time can be found by analyzing the graph of the function \(T(x)\) and finding the minimum value of the function.

Step-by-step solution

Understand the Problem and Notation

First, let's try to understand the problem. Kelly is trying to reach the beach house by rowing to a point on the shore (x meters from P) and then jogging along the shore to the beach house. Let's denote the point where she will land as L.

Create a Function for the Total Length of Her Trip

We can use the Pythagorean theorem to find the total distance d(x) for Kelly's trip. Let's break her trip into two parts: 1. The distance she rows: This is the distance from the island to point L on shore. We know she is 200m from the shore (and perpendicular to it). Since L is x meters from P, we can use the Pythagorean theorem to find the distance: \((200^2 + x^2)^{1/2}\) 2. The distance she jogs: This is the distance from point L to the beach house, which is 600m from P. So, the distance she jogs is \((600 - x)\). 3. Now, let's add the distances to get the total length of her trip: \(d(x) = (200^2 + x^2)^{1/2} + (600 - x)\)

Find the Function for the Total Time of Her Trip

We know that Kelly can row at 2 m/s and jog at 4 m/s. Using the formula time = distance/speed, we can find the time for both parts of her trip: 1. Rowing time: \((200^2 + x^2)^{1/2} / 2\) 2. Jogging time: \((600 - x) / 4\) Adding these two times, we get the total time function for her trip: \(T(x) = (200^2 + x^2)^{1/2} / 2 + (600 - x) / 4\)

Graph the Functions d(x) and T(x)

Graph both functions d(x) and \(T(x)\) using a graphing calculator or online graphing tool. Note that the domain of x is from 0 to 600.

Determine Where Kelly Should Land

Analyze the graph of the function \(T(x)\) to find the point on the shore (x value) at which her total trip time is minimized. This x-value will be the point where she should land her boat to minimize her total trip time.

Find the Minimum Total Time

Based on the graph, find the minimum value of the function \(T(x)\), which represents the minimum time it will take Kelly to complete her trip.

Key concepts

Function Graphing

Function graphing is a valuable tool in calculus optimization. It helps visualize how different variables in a function interact and evolve. To find the best landing point for Kelly, graphing her trip's total time function, \(T(x)\), and distance function, \(d(x)\), gives a clear picture of how varying \(x\) impacts her journey.

• Total Distance Function, \(d(x)\): Visualizing \(d(x)\) helps understand the total distance she covers in rowing and jogging. The distance function combines two segments based on the Pythagorean theorem and simple arithmetic.
• Total Time Function, \(T(x)\): Viewing \(T(x)\) shows how the time component of her journey changes. This function uses both the rowing and jogging speeds to calculate total travel time.

Graphing these functions entails plotting \(x\) on the \(x\)-axis while representing corresponding distance or time on the \(y\)-axis. This visual method allows for straightforward identification of optimal solutions, such as finding minimum travel time or shortest path.

Pythagorean Theorem

The Pythagorean theorem is an essential tool in calculating distances in optimization problems. In Kelly's case, she rows from the island to the shore. Since her rowing path forms a right-angled triangle with the shore, the theorem fits perfectly. In the scenario given:

• Row Distance: The perpendicular distance from the island to the shore is \(200\, \text{m}\), and the horizontal distance from point \(P\) on shore is \(x\, \text{m}\).
• Applying the Theorem: Using \(a^2 + b^2 = c^2\), where \(a = 200\) and \(b = x\), we find the rowing distance as \((200^2 + x^2)^{1/2}\).

This calculation gives the precise rowing distance to any shore point \(L\) that Kelly might choose, allowing for a direct input into her total distance function \(d(x)\).

Time Function

The time function for Kelly’s trip illustrates how long her journey takes depending on where she lands the boat. The function \(T(x)\) splits into two parts: time rowing and time jogging.- **Rowing Time:** This time is derived by dividing the row distance by the rowing speed: \[ \frac{(200^2 + x^2)^{1/2}}{2} \]- **Jogging Time:** Similarly, the jogging duration is the remaining shore distance divided by jogging speed: \[ \frac{600 - x}{4} \]The sum of these times forms the total time function:\[ T(x) = \frac{(200^2 + x^2)^{1/2}}{2} + \frac{600 - x}{4} \]Understanding this function's implications helps find where Kelly spends the least amount of time, precisely guiding her landing point.

Distance Calculation

Calculating distance is a foundational concept in optimization problems like this one. To determine the total travel distance Kelly has to cover, one must consider both her water and land journeys.- **Rowing Segment:** Influenced by the Pythagorean theorem, as effectively explained above, this distance accounts for her rowing path: \[ \text{Rowing Distance} = (200^2 + x^2)^{1/2} \]- **Jogging Segment:** This part of her journey is simpler, being the straight run along the beach from point \(L\) to the house: \[ \text{Jogging Distance} = 600 - x \]The total distance, \(d(x)\), becomes a crucial function:\[ d(x) = (200^2 + x^2)^{1/2} + (600 - x) \]Understanding how both parts contribute allows for accurate evaluation and optimization of Kelly’s route, aiding in minimizing the trip's time.