2. Adapted from Edwards and Penney Section 12.4, exercise 72 Imagine that you are standing on a hillside somewhere near the equator, and there is a nearby lighthouse. If we use the lighthouse as a landmark, we can set up a useful coordinate system. Let r represent horizontal distance due east from the lighthouse, and y represent horizontal distance due north from the lighthouse, both measured in metres. So the negative r axis points due west. Within 100m of the lighthouse, the altitude of the ground (in metres above sea level) is quite accuately described by the multivariate function. H(x, y) = 200+ You are at the point where = 40m and y = 25m, with the positive z-axis to the east and the positive y-axis to the north. So you are at height H(40,25) = 196.7m. 1 250 (-2r² + 3ry-y²)+sin(x - y). ƏH (a) Calculate . If you head due east, would you initially be ascending or de- Әх scending? At what angle (in degrees) from the horizontal? Show your work. If you head due south, would you initially be ascending or dy descending? At what angle (in degrees) from the horizontal? Show your work. ƏH (b) Calculate
2. Adapted from Edwards and Penney Section 12.4, exercise 72 Imagine that you are standing on a hillside somewhere near the equator, and there is a nearby lighthouse. If we use the lighthouse as a landmark, we can set up a useful coordinate system. Let r represent horizontal distance due east from the lighthouse, and y represent horizontal distance due north from the lighthouse, both measured in metres. So the negative r axis points due west. Within 100m of the lighthouse, the altitude of the ground (in metres above sea level) is quite accuately described by the multivariate function. H(x, y) = 200+ You are at the point where = 40m and y = 25m, with the positive z-axis to the east and the positive y-axis to the north. So you are at height H(40,25) = 196.7m. 1 250 (-2r² + 3ry-y²)+sin(x - y). ƏH (a) Calculate . If you head due east, would you initially be ascending or de- Әх scending? At what angle (in degrees) from the horizontal? Show your work. If you head due south, would you initially be ascending or dy descending? At what angle (in degrees) from the horizontal? Show your work. ƏH (b) Calculate
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2. Adapted from Edwards and Penney Section 12.4, exercise 72
Imagine that you are standing on a hillside somewhere near the equator, and there
is a nearby lighthouse. If we use the lighthouse as a landmark, we can set up a
useful coordinate system. Let z represent horizontal distance due east from the
lighthouse, and y represent horizontal distance due north from the lighthouse, both
measured in metres. So the negative r axis points due west. Within 100m of the
lighthouse, the altitude of the ground (in metres above sea level) is quite accuately
described by the multivariate function.
H(x, y) = 200+
(-2r²+3ry-y²)+sin(x - y).
250
You are at the point where = 40m and y = 25m, with the positive z-axis to the
east and the positive y-axis to the north. So you are at height H(40.25) = 196.7m.
ƏH
(a) Calculate
If you head due east, would you initially be ascending or de-
əx
scending? At what angle (in degrees) from the horizontal? Show your work.
(b) Calculate If you head due south, would you initially be ascending or
ƏH
dy
descending? At what angle (in degrees) from the horizontal? Show your work.
(c) Use MATLAB and the surf command to plot H(x, y) on the domain
I € [-100, 100], y [-100,100]. Label your axes, and include your plot in
your answers.
(d) Zoom out, and now use MATLAB and the surf command to plot H(x, y) on
the domain z € [-1000,1000], [-1000, 1000]. List at least two reasons
why H(x, y) might not be a good model for a hillside at this larger scale.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff248105c-fea1-4a4f-a936-0565e5757746%2F561c7976-09a6-4b63-9612-8414b1d6fbde%2Fyvndug_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Adapted from Edwards and Penney Section 12.4, exercise 72
Imagine that you are standing on a hillside somewhere near the equator, and there
is a nearby lighthouse. If we use the lighthouse as a landmark, we can set up a
useful coordinate system. Let z represent horizontal distance due east from the
lighthouse, and y represent horizontal distance due north from the lighthouse, both
measured in metres. So the negative r axis points due west. Within 100m of the
lighthouse, the altitude of the ground (in metres above sea level) is quite accuately
described by the multivariate function.
H(x, y) = 200+
(-2r²+3ry-y²)+sin(x - y).
250
You are at the point where = 40m and y = 25m, with the positive z-axis to the
east and the positive y-axis to the north. So you are at height H(40.25) = 196.7m.
ƏH
(a) Calculate
If you head due east, would you initially be ascending or de-
əx
scending? At what angle (in degrees) from the horizontal? Show your work.
(b) Calculate If you head due south, would you initially be ascending or
ƏH
dy
descending? At what angle (in degrees) from the horizontal? Show your work.
(c) Use MATLAB and the surf command to plot H(x, y) on the domain
I € [-100, 100], y [-100,100]. Label your axes, and include your plot in
your answers.
(d) Zoom out, and now use MATLAB and the surf command to plot H(x, y) on
the domain z € [-1000,1000], [-1000, 1000]. List at least two reasons
why H(x, y) might not be a good model for a hillside at this larger scale.
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