2) You are traveling in the desert, and your position is represented by your coordinates (x, y) away from the city at the origin, measured in kilometers. The sand dunes around you can be modeled by the surface: h(x, y) = 80 sin (x) sin (y) + 100, where h represents the height in meters. a. Use your knowledge of trig functions to explain the shape of this surface, giving specific values and units for period, maximum, and minimum, and how you got them by inspecting the formula. You do not have to draw the surface. b. You are currently at the coordinate (-1, 2.5). In what direction should you walk to go uphill the fastest, and downhill the fastest?
2) You are traveling in the desert, and your position is represented by your coordinates (x, y) away from the city at the origin, measured in kilometers. The sand dunes around you can be modeled by the surface: h(x, y) = 80 sin (x) sin (y) + 100, where h represents the height in meters. a. Use your knowledge of trig functions to explain the shape of this surface, giving specific values and units for period, maximum, and minimum, and how you got them by inspecting the formula. You do not have to draw the surface. b. You are currently at the coordinate (-1, 2.5). In what direction should you walk to go uphill the fastest, and downhill the fastest?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement:
You are traveling in the desert, and your position is represented by your coordinates \((x, y)\) away from the city at the origin, measured in **kilometers**. The sand dunes around you can be modeled by the surface:
\[h(x, y) = 80 \sin \left( \frac{\pi}{2} x \right) \sin \left( \frac{\pi}{2} y \right) + 100,\]
where \(h\) represents the height in **meters**.
**Part a**: Use your knowledge of trigonometric functions to explain the shape of this surface, giving specific values and units for the period, maximum, and minimum, and how you got them by inspecting the formula. You do not have to draw the surface.
**Part b**: You are currently at the coordinate \((-1, 2.5)\). In what direction should you walk to go uphill the fastest, and downhill the fastest?
### Explanation:
**Part a:**
To understand the shape of the surface described by the equation \(h(x, y) = 80 \sin \left( \frac{\pi}{2} x \right) \sin \left( \frac{\pi}{2} y \right) + 100\), let's break down the components:
1. **Sine Function Characteristics:**
The sine function, \(\sin(\theta)\), oscillates between -1 and 1 with a period of \(2\pi\).
2. **Adjusted Sine Functions:**
- For \(\sin \left( \frac{\pi}{2} x \right)\):
- The input \(\frac{\pi}{2} x\) indicates the period is adjusted. The period \(T\) can be found using:
\[
\frac{\pi}{2} T = 2\pi \implies T = 4 \text{ kilometers}.
\]
- This function oscillates between -1 and 1 every 4 kilometers along the x-axis.
- Similarly, for \(\sin \left( \frac{\pi}{2} y \right)\):
- The period is also \(4 \text{ kilometers}\) along the y-axis.
3. **Combined Surface Shape:**
- The product \(\sin \left( \frac{\pi}{2} x \right](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1597ed32-e45d-4583-a76f-92703bb626d9%2F50bed2d9-a4f1-4c5e-aca6-4b3dbe8ea420%2Flaooxsj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
You are traveling in the desert, and your position is represented by your coordinates \((x, y)\) away from the city at the origin, measured in **kilometers**. The sand dunes around you can be modeled by the surface:
\[h(x, y) = 80 \sin \left( \frac{\pi}{2} x \right) \sin \left( \frac{\pi}{2} y \right) + 100,\]
where \(h\) represents the height in **meters**.
**Part a**: Use your knowledge of trigonometric functions to explain the shape of this surface, giving specific values and units for the period, maximum, and minimum, and how you got them by inspecting the formula. You do not have to draw the surface.
**Part b**: You are currently at the coordinate \((-1, 2.5)\). In what direction should you walk to go uphill the fastest, and downhill the fastest?
### Explanation:
**Part a:**
To understand the shape of the surface described by the equation \(h(x, y) = 80 \sin \left( \frac{\pi}{2} x \right) \sin \left( \frac{\pi}{2} y \right) + 100\), let's break down the components:
1. **Sine Function Characteristics:**
The sine function, \(\sin(\theta)\), oscillates between -1 and 1 with a period of \(2\pi\).
2. **Adjusted Sine Functions:**
- For \(\sin \left( \frac{\pi}{2} x \right)\):
- The input \(\frac{\pi}{2} x\) indicates the period is adjusted. The period \(T\) can be found using:
\[
\frac{\pi}{2} T = 2\pi \implies T = 4 \text{ kilometers}.
\]
- This function oscillates between -1 and 1 every 4 kilometers along the x-axis.
- Similarly, for \(\sin \left( \frac{\pi}{2} y \right)\):
- The period is also \(4 \text{ kilometers}\) along the y-axis.
3. **Combined Surface Shape:**
- The product \(\sin \left( \frac{\pi}{2} x \right
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