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# Use a software package to generate a phase portrait for

ISBN: 9781111427412 173

## Solution for problem 10.77 Chapter 10

Advanced Engineering Mathematics | 7th Edition

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Advanced Engineering Mathematics | 7th Edition

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Problem 10.77

Use a software package to generate a phase portrait for each of the following predator/prey models. (a) x= x 0.5x y, y= 2x y 1.2y (b) x= 3x 1.5x y, y= x y 1.6y (c) x= 1.6x 2.1x y, y= 1.9x y 0.4y (d) x= 1.8 0.2x y, y= 3.1x y 0.4y 13. Ge

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Mantle plumes and hot spots under the crust ­isolated magic volcanoes like the Hawaiian Islands, usually relate to hot spot activity. Hot plumes may originate in the moderate to deep mantle and aren't linked to interactions on the edges of crystal plates. Plate tectonic settings Mantle plumes ("hot spots") ­magma reaches the surface quickly (oceanic crust, faults) ­> mafic (basalt) ­magma reaches the surface slowly (melts a lot of continental crust) ­> felsic (rhyolite) Plate tectonics and intermediate rocks What and why: water from subducted ocean crust lowers melting point in mantle ­mantle melts mix with crust, producing more intermediate compositions Where: convergent plate boundaries (ocean and ocean continent collisions­­subduction zones) Plate tectonics and felsic rocks What: melting sedimentary/metasedimentary rocks (felsic­rich in Si) Why: energy added to crust by mafic /intermediate melts from below ­water from dehydrating sediments lowers melting points of rocks Where: the crust is thick: continent­­continent collisions Plate Setting Magma Formation Spreading Decompression melting Rifting Decompression melting Subduction Flux melting Hot Spot Decompression melting Plate Setting Magma Composition Spreading Mafic (Basalt) Rifting Usually mafic (Basalt) Subduction Intermediate (Andesite) Hot Spot Usually mafic (basalt) How igneous rocks appear in nature *Complex arrays of dikes are common in mountain belts. Dikes can carry magma upward through the crust *Dikes are a good relative age indicator ­The dike is always younger than the rock it cuts through Igneous Intrusions: Plutons *Ultimately, large volumes of magma that pond together and crystallizes to form a pluton. Classically, considered to be balloon shaped, these may have complex forms that relate to the regional stresses in the crust. *The cores of many of the great mountain belts are massive pluton and complexes like Sierra Nevada in Eastern California. **Batholith is frequently used for these bodies. *Likely that these bodies were deep pluming of volcanoes active long ago but later eroded away, revealing the plutonic roots of the system. Intrusive Igneous Rocks *Dome shaped ­Laccoliths: Dome parallel to lower rock layer *Irregular ‘blobs’ ­Pluton ­Batholith: huge mass composed of many plutons *Less dense magmas rise through the crust to produce volcanoes: denser magmas cool in the subsurface *Rising magmas slowly cool ­Viscosity increases ­Density Increases *Intrusions form as magma solidifies beneath the surface CH. 12 Volcanic Product *When magma reaches the Earth’s surface we call it lava *Aside from lava, volcanoes may eject a variety of common products ­Volcanic gases ­pyroclastic debris ­Lahars Lava Flows *Smoothly oozing low viscosity basalt flows, commonly form ropy structures called Pahoehoe (Hawaiian word) *Basalt lava that has lost much of this gas will be more viscous. This lava will form rough, blocky flows called aa. Lava Tubes *Lava tubes: Conduits for basaltic lava ­A cooled crust forms on top of a basalt flow ­A conduit­ a lava tube­ develops in the flow ­Tubes prevent cooling, facilitating flow for miles ­Lava tubes become caves that can transmit water Columnar Jointing *Thick, solidified flows from vertical fractures with hexagonal cross­section Pillow Basalts *Round blobs of basalt cooled in water *Underwater, basalt cools instantly forming a pillow *The pillow surface is cracked, quenched glass *Lava pressure ruptures pillow to form the next blob *The process repeats to form a mound of pillow basalts *Common on the mid­ocean ridge *Basaltic lava has low viscosity and can flow long distances *Andesitic lava is too viscous to flow far and tends to break up as it flows *Felsic lava is so viscous that it may pile up in a dome­shaped mass Volcanic Materials 1) Lava flows 2) Pyroclastic debris a. Tephra b. Ash Falls c. Pyroclastic Flows 3) Volcanic Gas 4) Lahars Pyroclastic Material: Tephra *Deposits of volcanic debris: fragments of material or hot magma ejected from vent *Lapilli *Bombs Pyroclastic *Larger chunks of lava, from grape to practically car size are called bombs *Note that on impact these DO NOT explode in flaming carnage ”Fire fountaining” *Produces abundant pyroclastic material. Including Tiny glass beads (Pele’s Tears) Ash Falls *Volcanic ash bits of volcanic glass less than 2mm across *Occurs in ALL eruptions *Major eruptions can cause global cooling *Ash reflects solar radiation Pyroclastic Flow *Avalanche of steam and hot ash that move downslope because it is too dense to rise. *Hot enough to glow in the dark (200­450 degrees Celsius) *Speeds of 50­300 Km/hr. *Kills everything in its path Volcanic Gases *Much more gas can be dissolved in a liquid at high pressure than at low pressure *Liquids that are rich in gaseous components will tend to degas (lose their gas components) as they decompress *Here on Earth, decompression occurs as magma travels from deep sources to shallow regions (like the surface) *Gases separate within magma *As magma rises, gas expands *Pressure builds *Explosive eruption of gas can not escape Gas Related Textures *Gas leaving decompressing magma may not be able to successfully escape. *Gas trapped in rapidly cooling lava forms vesicles or void spaces in the rock *Extreme examples are pumice and scoria *light­weight Volcanic Gases *CO2 usually about 10% by volume * Heavier than air so it collects in low areas *Seeps into lakes and collects at the bottom overturn can produce a suffocating cloud *Some volcanoes produce H2S or SO2 in significant amounts *Toxic to plants *Causes acid rain *Increase indicates impending eruption Lahars (Mud flows) *Eruption heats ice on the volcano, which mixes with ash from previous eruptions *Can also be triggered by heavy rain *Slurry of water, ash and loose rock flows down the mountain *Consistency of cement *80­100 km/hr. (highway speeds) *Follows river valleys *Can occur without an eruption, especially if a mountain is glaciated *Can travel tens of miles from the mountain Shield Volcano *Low silica, basaltic lava, low viscosity, flows readily *gentle slope *Non­explosive in general *lava fountains *fissure eruptions *Lava tubes Hazards *basaltic lava flow *Some ash *Volcanic Gas Cinder Cones *Low to moderate silica lava *steep slopes, generally symmetrical *Dominantly pyroclastic material Hazards *Tephra *Ash *Gas (minor) *Basaltic lava flow Columbia River Flood Basalts *this is the largest large igneous province in North America, extruded about 15 million years ago Composite Volcanoes (Stratovolcanoes) *Moderate to high silica lava; high viscosity *Steep slopes, generally symmetrical *layered lava flows and pyroclastic material *EXPLOSIVE! Hazards *ash flow (heavy) *Pyroclastic flow *Lava flow *Lahars *Gas­ acid rain Volcanic Dome *Rhyolite *Small to moderate size *Steep sides covered with angular rubble *Eruption is minor ­slow, quiet ­over months/years ­single eruption per dome ­slow flow of viscous lava solidifies to rubble *often occur on stratovolcanoes Hazards *Lava flow (localized) *Some ash *Collapse of dome can trigger eruption, landslide Caldera *Collapsed remains of giant stratovolcanoes, usually on continents *explosive eruptions of usually rhyolite in huge amounts *Pyroclastic flow over tens of thousands of square kilometers *Collapses to form caldera after major eruption CH. 13 Weathering *Physical and/or chemical alteration of rocks and minerals at the Earth’s surface *Most rocks and minerals are formed at high temperatures and pressures *In equilibrium at the temperature and pressure of formation (stable) *Unstable (metastable) at surface temperature and pressure Types of Weathering *Physical Weathering: the disintegration or disaggregation of rocks by breaking *Chemical weathering: The decomposition of rocks and minerals as chemical reactions alter them to minerals stable at surface conditions *Relative importance varies from environment to environment *Warm, humid­chemical weathering dominates *Cold, dry­ Physical weathering dominates Physical Weathering *Physical breakage of rocks into smaller pieces without changing the chemical composition i.e. * Frost wedging (ice wedging) *Root wedging *Salt wedging *Jointing *Thermal expansion FROST WEDGING *Freeze­thaw cycles *water expands by 9% when it freezes *The stress of expansion breaks the rock *Ice melts and the water percolates deeper into the newly expanded cracks Similar Weathering Processes *Same things happen with ­plant roots (root wedging) ­growing plants widen cracks ­root tip pressures can be very high ­Dissolved minerals (salt wedging) ­Minerals precipitate along fractures and enlarge them ROOT WEDGING *As roots grow they exert a force on the surrounding rock SALT WEDGING *Occurs in arid climates and coastal areas *Salt dissolved in water precipitates *Growing salt crystals exert a force on the surrounding rock JOINTING: set of parallel breaks (fractures) Formation of Joints: *Release confining pressure (unloading) when deep rocks are brought to surface *From rock deformation (tectonic stresses) during mountain building *From heating (thermal expansion) outer layer will expand and separate from the rock spalling. Chemical Weathering *Minerals are destroyed or altered by chemical reactions *Forms stable minerals from unstable/metastable precursors *Common chemical reactions ­dissolution ­Hydrolysis ­Oxidation ­Hydration: absorption of water by clay minerals **All require water DISSOLUTION: *Some minerals are soluble in water *i.e. Halite (NaCl) and other ‘salts’ *Minerals dissolve into constituent ions *ions carried away by the water (leaching) **Water will readily dissolve minerals that have ionic bonds ACID HYDROLYSIS *Reactions to acid *source of acids ­Rainwater is naturally acidic (ph=5.6) *Decaying organic matter and plant roots produce weak acids *Anthropogenic sources such as mine drainage *H replaces cations (+) in the minerals structure *Promotes dissolution of mineral CaCO3+H2CO3=Ca+2+2HCO3­ Calcite ions carried away in solution ACID HYDROLYSIS *Important in weathering of silicates­feldspar, mica, amphiboles, pyroxenes *New “secondary” minerals (clay minerals) created by this process OXIDATION *Reaction with cations (+) and oxygen *The oxidation reaction works faster if oxygen is dissolved in water *Works in combination with hydrolysis and dissolution *Reaction with cations (+) and oxygen *The oxidation reaction works faster if oxygen is dissolved in water *Works in combination with hydrolysis and dissolution Chemical Weathering *Weathering of an iron rich silicate mineral *Amphibole and weak acid (rain) ­ Fe oxide + clay mineral +ions (carried away in solution RATES OF WEATHERING *The rate of weathering increases as surface area increases *Joints or fractures in the rocks increase surface area *Act as passageways for water to enter rock *Chemical influences chemical weathering ­Temperature ­10 degrees Celsius increase in temperature doubles the reaction time *rainfall *More water (humid climate) increases the rate of weathering *Resistance to weathering varies by composition *Fe bearing minerals are easily weathered by hydrolysis, dissolutions and oxidation *Quartz (SiO2) is very resistant to weathering DIFFERENTIAL WEATHERING *Caused by variations in weathering rate of different kinds of rock *occurs over a broad range of scales *Influences landscapes *Corners rounded during weathering *Decomposition most rapid at corners *Spheroidal weathering *Some layers are more resistant than others *Differential weathering produces different slopes So rocks weather, then what *Weathered material stays in place  soil forms *Weathered material is removed (eroded) transported to a new location where it accumulates (deposited) Soil *Soil is rock and sediment that has been modified by physical, chemical and biological processes over time so that it’s capable of sustaining plant growth *Soil forming processes operate slowly, so it requires long time periods to form soil Soil Consists of: ­Minerals ­Organic matter ­Water and gases (filling voids between particles) Soil Forming Factors: ­Parent rock ­Climate ­Slope Steepness ­Drainage ­Vegetation ­time PARENT ROCK: *Provides minerals for a soil *Different minerals react differently to weathering process *Rate of formation influenced by type of parent rock CLIMATE: *Primary control on soil formation *Chemical weathering depends on heat and moisture *Reaction rate increases as temperature increases *amount of water influences leaching of upper layers *Controls amount and kind of weathering **In warm humid climates chemical weathering is faster than physical weathering **In cold, dry climates, physical weathering is father than chemical weathering SLOPE STEEPNESS *Soils develop best on low slopes *Erosion removes material on steep slopes *Orientation of Slope (North or South facing) influences moisture level and vegetation DRAINAGE: *Soils formed on poorly drained areas are richer in organic material *Poor drainage increases acidity from decay of plants ­Accelerates chemical weathering ­Provides abundant organic material VEGETATION *Plants add organic material to soil *Root holes help water movement *Organic acids (decaying plants, animal wastes) are good dissolution/hydrolysis agents TIME *It takes time to form soil *Young soils are thinner than older soils *Difficult to replace soil that’s been removed or altered by human activity *Rates of weathering and soil formation vary through time *Steep slopes become more gradual *soil thickens and insulates parent rock from climatic extremes *Climatic conditions change SO THEN WHAT *Weathered material stays in place­ soil forms *Weathered material is removed (eroded)­ transported to a new location where it is deposited and may eventually form a sedimentary rock.

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##### ISBN: 9781111427412

Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Since the solution to 10.77 from 10 chapter was answered, more than 244 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. This full solution covers the following key subjects: . This expansive textbook survival guide covers 23 chapters, and 1643 solutions. The answer to “Use a software package to generate a phase portrait for each of the following predator/prey models. (a) x= x 0.5x y, y= 2x y 1.2y (b) x= 3x 1.5x y, y= x y 1.6y (c) x= 1.6x 2.1x y, y= 1.9x y 0.4y (d) x= 1.8 0.2x y, y= 3.1x y 0.4y 13. Ge” is broken down into a number of easy to follow steps, and 54 words. The full step-by-step solution to problem: 10.77 from chapter: 10 was answered by , our top Math solution expert on 12/23/17, 04:48PM.

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Use a software package to generate a phase portrait for