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/11 E.urcises 17 26. detem1i11e the trlues of r. if | Ch 1.4 - 22

Elementary Linear Algebra: A Matrix Approach | 2nd Edition | ISBN: 9780131871410 | Authors: Lawrence E. Spence ISBN: 9780131871410 187

Solution for problem 22 Chapter 1.4

Elementary Linear Algebra: A Matrix Approach | 2nd Edition

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Elementary Linear Algebra: A Matrix Approach | 2nd Edition | ISBN: 9780131871410 | Authors: Lawrence E. Spence

Elementary Linear Algebra: A Matrix Approach | 2nd Edition

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Problem 22

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. -2.rl + .Q =5

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Chapter 17: Physiology of the Kidneys Primary function of kidneys is in the regulation of the extracellular fluid (plasma and interstitial fluid) environment of the body • Accomplished through formation of urine o During this process, kidneys regulate § Volume of blood plasma à contribute significantly to the regulation of blood pressure § Concentration of waste products in blood § Concentration of electrolytes (Na, K, HCO3, etc.) § pH of plasma • Kidneys considered most potent acid-­‐base regulator Gross Anatomy of the Urinary System • Kidneys: lie on either side of the vertebral column below the diaphragm and liver o About the size of a fist o Coronal Section of kidney has 2 regions § Outer renal cortex – reddish brown and granular because of numerous capillaries § Deeper renal medulla – striped due to the presence of microscopic tubules and blood vessels • Composed of 8-­‐15 conical renal pyramids separated by renal columns o Cavity of the kidney is divided into several portions § Minor calyx – small depression that each pyramid projects into § Major calyx – formed by union of several minor calyces à major calyces join to form renal pelvis • Renal pelvis: cavity where urine produced in the kidneys is drained to o Urine then channeled from the kidneys to the urinary bladder through the ureters o Ureters: long ducts that undergo peristalsis (wave-­‐like contractions) § Pacemaker of these peristaltic waves is located in the renal calyces and pelvis, which contain smooth muscle – renal calyces and pelvis also undergo peristalsis to aid emptying of urine from kidney • Urinary bladder: storage sac for urine o Shape is determined by amount of urine it contains § Empty = pyramidal § As it fills = ovoid and bulges into abdomen o Drained inferiorly by urethra o Has a muscular wall – detrusor muscle Microscopic Anatomy of the Kidney • Nephron – functional unit of the kidney o Responsible of urine formation o Comprised of renal corpuscle and renal tubules § Renal corpuscle – glomerulus and Bowman’s capsule • Glomerular filtration § Renal tubules – proximal convoluted tubule, distant convoluted tubulues and Loop of Henle • Tubular secretion • Tubular absorption o 2 types: § Cortical – original in the outer 2/3 of the cortex • More numerous § Juxtamedullary – originate in the inner 1/3 of the cortex (next to the medulla) • Have longer nephron loops • Play an important role in kidney’s ability to produce a concentrated urine • Glomerular (Bowman’s) capsule: surrounds the glomerulus o Contains an inner visceral layer and outer parietal layer o Space between the two layers is continuous with the lumen of the tubule o Filtrate that enters glomerular capsule passes into lumen of the proximal convoluted tubule § Proximal convoluted tubule contains millions of microvilli to increase surface area for reabsorption § See absorption o Fluid passes from the proximal convoluted tubule to the nephron loop (Loop of Henle) § This fluid is carries into the medulla in the descending limb of the loop and returns to the cortex in the ascending limb o Distal convoluted tubule: coiled tubule in the cortex § See secretion § Shorter than proximal tubule § Relatively few microvilli § Terminates as it empties into a collecting duct o Collecting duct: receives fluid from the distal convoluted tubule of several nephrons § Fluid is drained by the collecting duct from the cortex to the medulla as the collecting duct passes through a renal pyramid § Fluid is now called urine à passes into minor calyx à funneled through renal pelvis out of the kidney into the ureter • Vasculature o Renal artery: where arterial blood enters kidney o Renal artery divides into interlobular arteries that pass between pyramids through the renal columns o Arcuate arteries branch from interlobular arteries at the boundary of the cortex and medulla o Afferent arterioles: microscopic arterioles formed from branching of Arcuate arteries § Deliver blood into glomeruli (capillary network that produce a blood filtrate that enters the urinary tubules) o Efferent arteriole: where remaining blood in the glomerulus leaves through § Delivers blood into another capillary network – peritubular capillaries surrounding the renal tubules § This blood is drained into veins that parallel the course of the arteries in the kidney à interlobular, arcuate, and interlobar veins § Interlobar veins leave kidney as a single renal vein à empties into inferior vena cava o Arrangement of blood vessels is unique § Only one in which capillary bed is drained by an arteriole instead of a venule and delivered to a second capillary bed downstream Glomerular Filtration • Urine formation begins with the filtration of plasma from glomerular capillaries into Bowman’s capsule à glomerular filtration – results in formation of filtrate • Process utilizes a pressure gradient • Endothelial cells of the glomerular capillaries have large pores (fenestrae) o Causes glomerular capillaries to be 100-­‐400x more permeable to plasma water and dissolved solutes than skeletal muscle capillaries o Pores are still small enough to prevent entry of RBC, WBC and platelets • Before fluid in blood plasma can enter the interior of the glomerular capsule, it must pass through 3 layers of selective filters o Fluid entering is referred to as filtrate – will become modified as it passed through the different segments of the nephron tubules to become urine o Capillary fenestrae – first potential filtration barrier § Large enough to allow proteins to pass but are surrounded by charges that may present some barriers to plasma proteins o Glomerular basement membrane – second potential barrier § Layer of collagen and proteoglycans lying immediately outside the endothelium § Most resists fluid flow into the capsule lumen § Offer some barrier to plasma proteins o Slit Diaphragm – third potential barrier § Visceral layer of glomerular capsule is made up of podocytes § Unique epithelial cells with a bulbous cell body, primary processes and thousands of foot processes § Processes are attached to the basement membrane § Narrow slits between adjacent foot processes provide passageways for molecules entering the interior of the glomerular capsule as glomerular filtrate § Slit diaphragm – links interdigitating foot processes and presents the last potential filtration barrier Forces Involved in Filtration • Filtration occurs due to opposing forces o Important elements § Blood pressure § Protein concentration in plasma compared to filtrate (not many proteins in filtrate) • Glomerular filtration rate (GFR) o Volume filtered from the glomeruli into Bowman’s capsule per unit time § Average ~115-­‐125mL/min § Consider average blood volume is 5.5L à total blood volume is filtered every 40 minutes • Osmotic pressure established by presence of protein in plasma and not really in filtrate o Greater colloid osmotic pressure of plasma promotes the osmotic return of filtered water o Net filtration pressure of about 10 mmHg – since glomerular capillaries are extremely permeable and have a large surface area, this modest filtration pressure produces an extraordinary large volume of filtrate Regulation of GFR • Vasoconstriction or vasodilation of afferent arterioles affects rate of blood flow into glomerulus • Observe variety of intrinsic and extrinsic mechanisms to ensure GFR is high enough to eliminate wastes and regulate blood pressure – but not too high as to cause excessive water loss o Sympathetic nerve endings § Vasoconstriction occurs in afferent arterioles § Helps to preserve blood volume and divert blood to the muscles and heart o Renal auto regulation § Observe relatively constant GFR with fluctuation BP • Afferent arterioles dilate when BP falls below average and constrict when BP is above average • Changes in efferent arterioles are of secondary importance § Myogenic regulation • Smooth muscle of afferent arterioles sensitive to stretch – result in vasoconstriction when stretched § Result of effects of locally produced chemicals on the afferent arterioles • Tubuloglomerular feedback o Macula densa: specialized cells that act as the sensor – part of a larger functional unit (Juxtaglomerular apparatus) • Increased delivery of NaCl and H2O to the distal tubule causes macula densa to release chemical signal causing constriction of the afferent arteriole Renal Reabsorption • Most salt and water filtered return to blood via reabsorption o Defined as the return of filtered molecules from the renal tubules to the blood o Proximal convoluted tubule facilitates reabsorption • Obligatory water loss – minimal volume needed to ensure excretion of metabolic wastes o ~400 mL of urine a day • Observe that transport of water is passive, occurring via osmosis à therefore a concentration gradient must be established between filtrate and blood that favors osmotic return o Filtrate is iso-­‐osmotic to plasma, so we must use interstitial fluid o In some cases, active transport of solute concentrations is used to facilitate osmosis • Process begins in the epithelial cells of the proximal convoluted tubule o Epithelial cells are joined by apically located tight junctions – create regions for exchange • Sodium drives reabsorption o Sodium concentration in filtrate and plasma are equal but lower in the cytoplasm of the epithelial cells § Low NA+ concentration due to low permeability and Na/K pump moving Na into interstitial fluid § A potential difference is created across the proximal tubule epithelial cell wall à electrically favors Cl-­‐ movement from tubular fluid to interstitial fluid o An increase in osmotic pressure of interstitial fluid surround proximal tubule cells creates an osmotic gradient between interstitial fluid and tubular filtrate o The proximal tubule is permeable to water so water moves out by osmosis • As water moves, solute concentration in tubular fluid increases, if mechanisms in place then solute moves o Thus solutes follow solvents – explains how the passive absorption of these solutes occur o Also observe use of Na gradient for co-­‐transport of solute • Transport maximum is limited – if concentration exceeds saturation, there is a loss of urine • Obligatory water reabsorption: water is moved through osmosis by settling up sodium gradient • Fluid components are kept Osmotically balanced due to properties of the renal tubule o ~85% of original filtrate is reabsorbed in early regions leaving ~15% to enter the distal convoluted tubule and collecting ducts § 15% x GFR of 180L/day = 27L/day § This is still an excessive amount so it must be reabsorbed to varying degrees to in accordance with the needs of the body § This is fixed by hormones that act on the distal tubule and collecting duct Countercurrent Multiplier System and Countercurrent Exchange • For organism survival with limited water intake, a mechanism must be in place to produce a concentrated (hyperosmotic) urine • Human kidney can produce a maximum urinary concentration of 1400mosm/L o Almost 5x the plasma osmolality o Occurs in medullary collecting ducts § Presence of ADH allows the reclaim of water o How does the medullary interstitial fluid become so concentration § Functions of the loop of Henle in the Juxtamedullary nephrons, vasa recta and trapping of urea in kidney medulla • Ascending Limb of the Loop of Henle: o Divided into two regions: § Thin segment near the tip of the loop § Thick segment carries the filtrate into the distal convoluted tubule in the renal cortex o NaCl is actively extruded from the thick segment § Na moves passively down its concentration gradient from the filtrate into the cells à powers inward secondary active transport of K+ and Cl-­‐ § Na is then transported into the interstitial fluid via the Na/K pump • Cl-­‐ passively follows the Na+ due to the electrical attraction • K+ passively diffuses back into the filtrate o The ascending limb is not permeable to water so water cannot follow the flow of salt as seen in the proximal tubule à by the time the filtrate reaches the distal tubule the filtrate is very dilute and the interstitial fluid is hypertonic to it • Descending Limb of the Loop of Henle o Does not actively transport salt and is impermeable to the passive transport of it o Unlike the ascending limb, the descending limb is permeable to water § Allows it to release water Osmotically § Increases the salt concentration arriving in the ascending limb à increases salt transport by the ascending limb so that the NaCl concentration of the interstitial fluid is multiplied • Countercurrent Multiplication o Positive feedback mechanism is created by the interactions of the proximal and distal tubule effecting the concentrations of filtrate and interstitial fluid § The more salt extruded by the ascending limb, the more concentrated the fluid that is delivered to it from the descending limb will be § Positive feedback mechanism that multiplies the concentration of the interstitial fluid and descending limb fluid is called the countercurrent multiplier system • Steps of the mechanism: Start with isosmotic fluid leaving descending and reaching ascending à NaCl is pumped out actively à NaCl trapped in interstitial fluid by vasa recta (blood vessels) o NaCl pumped out of ascending limb causes interstitial fluid to be hypertonic o The hypertonic interstitial fluid causes the descending limb to release water through osmosis à causes filtrate to be somewhat hypertonic when it reaches back to the ascending limb o The now higher NaCl concentration in the ascending limb allows it to pump out more NaCl than it did before because more NaCl is available to carriers à causes interstitial fluid to become even more concentrated o The increased concentration of interstitial fluid causes even more water to be drawl out of the descending limb à causes filtrate to be even more hypertonic when it reaches back to the ascending limb o The progression repeats to a higher extend each time until the maximum concentration is reached in the inner medulla – the maximum value is determined by the capacity of the active transport pumps working along the lengths of the thick segments of the ascending limbs • What does the countercurrent multiplier accomplish o Increases concentration of renal interstitial fluid from 300 mOsm in the cortex to 1200 mOsm in the inner medulla – the hypertonicity of the renal medulla is critical because it serves as the driving force of water reabsorption through the collecting ducts • Vasa recta: o Vessels that parallel the nephron loop o Have urea transporters and aquaporins in the plasma membrane § Allows for them to gain salt and urea and lose water o Countercurrent exchange: mechanism allowing vasa recta to maintain hypertonicity § Salt and other dissolved solutes found in high concentration in the interstitial fluid diffuse into descending vasa recta § The same solutes passively diffuse out of the ascending vasa recta and back into the interstitial fluid à completes countercurrent exchange § This constant circulation allows for the solute to be trapped in the medulla o Net action of the vasa recta is to remove water from the interstitial fluid of the renal medulla Effects of Urea § Urea functions as an osmotically active molecule o Trapped within the medullary interstitial fluid because of constant recycling – supports this regions high osmolality § Presence of NaCl and urea make interstitial fluid very hypertonic, creating an environment so that water can leave via osmosis from the collecting ducts Collecting Duct and ADH: § The collecting duct is impermeable to NaCl but has aquaporins allowing water to flow § Since the interstitial fluid is hypertonic, water flows out of the collecting duct via osmosis o The water does not dilute the surrounding interstitial fluid because it is transported back to general circulation in the vascular system § The force driving osmosis is created by the countercurrent multiplier system o Concentration gradient is thus kept relatively constant but can change based on variations in permeability to water made by regulating the number of aquaporins § Aquaporins – water channels in the plasma membrane of the collecting duct epithelial cells § When plasma osmolality increases by as little as 1%, the anterior pituitary secretes arginine vasopressin – functions as antidiuretic hormone (ADH) o In response, cAMP is produced and after a series of reactions inserts aquaporins on the plasma membrane o Increased number of aquaporins increases the collecting ducts permeability to water and allows for increased water reabsorption à facultative water reabsorption o Decreased release of ADH decreases the number of aquaporins in the membrane, decreasing water permeability, decreasing water reabsorption and causing for more dilute urine excretion in larger volumes § ADH release is stimulated by Osmoreceptors in the hypothalamus due to change in blood plasma osmolality § In cases of extreme dehydration increasing amounts of ADH will be released o Urine excretion will decrease until it reaches the obligatory water loss ~400 mL/day o Decrease in urine excretion is limited to this value because urine cannot become more concentrated than the interstitial fluid § Note: even in the complete absence of ADH some water will still be reabsorbed through the collecting ducts Renal Plasma Clearance § Renal clearance – ability of kidneys to remove molecules from blood plasma by excreting urine – “clearing” blood of particular solutes o Clearing occurs due to glomerular filtration and tubular secretion o Reabsorption decreases renal clearance § Excretion rate = (filtrate rate + secretion rate) – reabsorption rate o After filtration, if a solute is neither secreted or reabsorbed, the excretion rate = filtration rate o Under that assumption, the glomerular filtration rate can be determined and used to assess the health of the kidneys § Glomerular filtration rate = volume of blood plasma filtered per minute § While most substances produced in the body are always either secreted or reabsorbed, insulin is not à rate of insulin filtration is exactly equal to rate of excretion of it o GFR = (V x U) / P § V = rate of urine formation § U = concentration of substance in urine § P = concentration of substance in plasma § Renal plasma clearance is the volume of plasma from which a substance is completely removed in one minute by excretion in the urine o When the substance is neither reabsorbed nor secreted, GFR = renal plasma clearance o Renal plasma clearance = GFR = (VU)/P o If a solute is reabsorbed, the renal plasma clearance of a substance must be less than the GFR o If a solute is secreted, the renal plasma clearance will be less than the GFR § Glucose and amino acids are easily filtered from blood through Bowman’s capsule but are not found in excreted urine à indicates they must be completely reabsorbed o Reabsorption of these occurs on the proximal convoluted tubule via secondary active transport of carrier proteins o Carrier proteins exhibit property of saturation § One carrier = 1 molecule at a take § Transport maximum: exists when the transported molecule is present in such high concentrations that all carrier proteins are occupied and the transport rate reaches a maximum Renal Control of Electrolytes § Kidneys assist with regulation of several electrolytes by matching urinary excretion with dietary intake o 90% of filtered Na and K are reabsorbed by the proximal convoluted tubule § Occurs at a constant rate § Not subjected to hormonal regulation but in regard to body needs § Role of aldosterone in Na and K balance – modifies o Regulation can occur via the mineralocorticoid aldosterone § Na reabsorption • Excrete ~30mg a day in urine • Na concentration can diminish to 0 in the presence of aldosterone • Induces synthesis of all channels and pumps in collecting duct § Filtration relies on blood pressure and radius of afferent/efferent arteriole § Potassium secretion o Occurs in distal convoluted tubule and collecting ducts § Aldosterone-­‐dependent potassium secretion • Potassium rich mean increases blood potassium and stimulated adrenal cortex to release aldosterone – results in potassium secretion into filtrate § Also can occur via aldosterone-­‐independent potassium secretion • Potassium rich meal causes insertion of potassium channels into plasma membrane of cortical collecting duct cells due to concurrent elevation in blood potassium concentration o Another mechanism – sensory: § With increased sodium due to increase flow in filtrate, it is “sensed” by a primary cilium § So as flow “bends” the cilium it activated potassium channels leading to increased potassium secretion § Explains how some diuretics can cause low blood potassium levels Control of aldosterone secretion § Aldosterone promotes Na retention and K secretion – thus with a rise in blood K levels aldosterone secretion is increased (direct mechanism) § However, with Na the process is more complex and typically involves a fall in blood volume along with activation of the renin-­‐angiotensin-­‐aldosterone system (indirect mechanism) Atrial Natriuretic Peptide § With an increased blood volume, observe salt secretion and water loss which is due in part to inhibition of aldosterone release § However, natriuretic peptides are involved in long-­‐term Na and water balance (also blood volume and pressure) o Natriuretic peptides – serve as a counter-­‐regulatory system for the renin-­‐angiotensin-­‐aldosterone system § Produces in cells in the atria and other areas such as the brain § Cause an increase in GFR § Produces: • Natriuesis – increased sodium excretion • Diuresis – increased fluid in excretion § K sparing: since Na is secreted, spare K will be taken up Relationship Between Na, K and H § Blood K concentration indirectly affects blood H concentration § Observe that as extracellular H+ concentration increases in the collecting duct, some H+ moves into the cell and causes cellular K to diffuse outward into extracellular fluid o Reestablishes proper ratio of these ions in extracellular fluid with same occurring in distal convoluted tubule cells § As Na moves, K and or H are secreted to maintain charge balance o In acidosis – observe an increase in blood K concentration (not secreted) o In alkalosis – observe a decrease in blood K concentration (secreted) o Hyperkalemia (too much K in blood) leads to acidosis because when K is secreted, H is retained Renal Acid-­‐Base Regulation § Kidneys assist with blood pH regulation by excreting H+ and reabsorbing bicarbonate § H+ enters filtrate by glomerular filtration and secretion into renal tubules à the acidification of urine § Tubule cells are impermeable to bicarbonate so its absorption is indirect § During acidosis, proximal convoluted tubule can make more bicarbonate via glutamine metabolism o Bicarbonate goes into blood and formed NH3 buffers urine § During alkalosis, less bicarbonate is reabsorbed due to less H in filtrate o Excretion assists with compensation for alkalotic condition Urinary Buffers § As blood pH falls below 7.35, urine pH typically falls below 5.5 o Note that the nephron cannot produce a urine below 4.5 o To excrete more H+ it must be buffered § Bicarbonate cannot accomplish this because it is mostly reabsorbed § Alternatives: phosphates and ammonia

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Chapter 1.4, Problem 22 is Solved
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Textbook: Elementary Linear Algebra: A Matrix Approach
Edition: 2
Author: Lawrence E. Spence
ISBN: 9780131871410

This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. This full solution covers the following key subjects: . This expansive textbook survival guide covers 34 chapters, and 2741 solutions. The answer to “/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. -2.rl + .Q =5” is broken down into a number of easy to follow steps, and 25 words. The full step-by-step solution to problem: 22 from chapter: 1.4 was answered by , our top Math solution expert on 12/27/17, 07:57PM. Since the solution to 22 from 1.4 chapter was answered, more than 245 students have viewed the full step-by-step answer.

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/11 E.urcises 17 26. detem1i11e the trlues of r. if | Ch 1.4 - 22