Answered: Let i, j, k, n be positive integers.…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 32EQ

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A Precocious Child and Her Blocks A child has 64 blocks that are 1-inch cubes. She wants to arrange the blocks into a solid rectangle h blocks long and w blocks wide. There is a relationship between h and w that is determined by the restriction that all 64 blocks must go into the rectangle. A rectangle h blocks long and w blocks wide uses a total of h X w blocks. Thus, hw = 64. Applying some elementary algebra, we get the relationship that we need: w=64h(2.3) Use a formula to express the perimeter P in terms of h and w. Using Equation 2.3, find a formula that expresses the perimeter P in terms of the height only. How should the child arrange the blocks if she wants the perimeter to be the smallest possible? Do parts b and c again, this time assuming that the child has 60 blocks rather than 64 blocks. In this situation, the relationship between h and w is w = 60/h. Note: Be careful when you do part c. The child will not cut the blocks into pieces
Population Genetics-First Cousins This is a continuation of Exercise 5. Double first cousins are first cousins in two waysthat is, each of the parents of one is a sibling of a parent of the other. First cousins in general have six different grandparents whereas double first cousins have only four different grandparents. For mating between double first cousins, the proportions of the genotypes of the children can be accounted for in terms of those of parents, grandparents, and great-grandparent: 3=122+14+18. a. Find all solutions to the cubic equation above and Identify which is 1. b. After 5 generations, what proportion of the population will be homozygous? c. After 20 generations, what proportion of the population will be homozygous? 5. Population Genetics In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypesthat is, instances in which the two alleles carried at a particular site on an individuals chromosomes are both the same. For population in which blood-related individual mate, them is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. in general. the frequency of homozygous children from mating of blood-related parents is greater than that for children from unrelated parents Measured over a large number of generations, the proportion of heterozygous genotypesthat is, nonhomozygous genotypeschanges by a constant factor 1 from generation to generation. The factor 1 is a number between 0 and 1. If 1=0.75, for example then the proportion of heterozygous individuals in the population decreases by 25 in each generation In this case, after 10 generations, the proportion of heterozygous individuals in the population decreases by 94.37, since 0.7510=0.0563, or 5.63. In other words, 94.37 of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For mating between siblings 1 can be determined as the largest value of for which 2=12+14. This equation comes from carefully accounting for the genotypes for the present generation the 2 term in terms of those previous two generations represented by for the parents generation and by the constant term of the grandparents generation. a Find both solutions to the quadratic equation above and identify which is 1 use a horizontal span of 1 to 1 in this exercise and the following exercise. b After 5 generations, what proportion of the population will be homozygous? c After 20 generations, what proportion of the population will be homozygous?

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