Question

1. if a man is 160 cm tall and a woman is 150 cm tall, what is the tallest height that their child could attain?

hint: start by filling in the childs alleles and work backwards. enter one h or h in each blank.
fathers alleles: h h h
mothers alleles: h h h
childs alleles: h h h h h h

Explanation:

Step1: Recall the formula for height prediction (polygenic inheritance)

For human height, a common polygenic model assumes additive effects of alleles. Let's assume that each \( H \) allele contributes a certain height increment and each \( h \) allele contributes a base or smaller increment. From the given alleles, father has 2 \( H \) and 1 \( h \), mother has 3 \( h \), child has 2 \( H \) and 4 \( h \)? Wait, no, looking at the alleles: Father's alleles: H, H, h (so 2 H, 1 h); Mother's alleles: h, h, h (3 h); Child's alleles: H, H, h, h, h, h (2 H, 4 h)? Wait, maybe the formula is based on a model where the base height for a person with all \( h \) alleles is, say, for mother (all h) is 150 cm, and each \( H \) adds a certain amount. Let's find the contribution of each \( H \).

Father's height: 160 cm, with 2 H and 1 h. Mother's height: 150 cm, with 3 h. Let's let the base height (all h) be \( B \), and each \( H \) adds \( x \) cm.

For mother: \( B = 150 \) (since 3 h, maybe the base is for 3 h? Wait, maybe the number of alleles: let's count the number of H and h. Wait, the child has 6 alleles? Wait, maybe it's a model where height is determined by the number of H alleles, with each H contributing. Let's re - examine.

Wait, the father has alleles H, H, h (so 2 H, 1 h), height 160. Mother has h, h, h (3 h), height 150. Let's assume that the base height (when all alleles are h) is calculated from the mother. If mother has 3 h and height 150, then per h - related base? Wait, maybe the formula for the child's height in a polygenic model (simplified) is:

\( \text{Child's height} = \frac{\text{Father's height} + \text{Mother's height}}{2}+ \text{adjustment for H alleles} \)

But with the allele counts. Alternatively, let's find the contribution of each H.

Let’s suppose that the height is determined by the number of H alleles. Let the height for a person with \( n \) H alleles be \( H(n)=B + n\times x \), where \( B \) is the base height (when \( n = 0 \)) and \( x \) is the contribution per H.

Mother has \( n = 0 \) H (all h), height 150, so \( B = 150 \).

Father has \( n = 2 \) H, height 160. So \( 150+2x = 160 \), solving for \( x \): \( 2x=160 - 150=10 \), so \( x = 5 \).

Now, the child has 2 H alleles (from the child's alleles: H, H, h, h, h, h - count the Hs: 2 H). Wait, no, wait the child's alleles: H, H, h, h, h, h. So number of H is 2, number of h is 4? Wait, maybe the number of H alleles is 2. Wait, but let's check the father: father has 2 H, height 160, mother has 0 H, height 150, x = 5 per H.

Now, the child's number of H alleles: from the child's alleles, we have 2 H. Wait, but let's use the formula for the tallest possible? Wait, no, the child's alleles are given as H, H, h, h, h, h. Wait, maybe I made a mistake. Wait, the hint says to work backwards. Wait, maybe the tallest height the child could attain is when we consider the maximum number of H alleles the child could inherit. Wait, the father can pass on at most 2 H (since he has 2 H and 1 h), and the mother can pass on 0 H (since she has 3 h). Wait, but the child's alleles are given as 2 H and 4 h. Wait, maybe the formula is:

\( \text{Child's height}=\frac{\text{Father's height}+\text{Mother's height}}{2}+\text{correction for H alleles} \)

Wait, another approach: The mid - parental height is \( \frac{160 + 150}{2}=155 \). But with the H alleles, each H adds some height. Wait, from father: 160 - 155 = 5, which is the contribution of his extra H (he has 2 H, mother has 0 H, so the difference between father and mid - parental is 5, which is 1 H's contribution? Wait, no.

Wait, let's go back to the allele counts. Father: 2 H, 1 h; Mother: 3 h; Child: 2 H, 4 h. Wait, maybe the number of H alleles in the child is 2 (same as father's H count). Wait, but we need the tallest height. Wait, maybe the tallest height is when the child inherits the maximum number of H alleles from the father. The father has 2 H, so the child can inherit at most 2 H.

Now, let's calculate the height. Let's assume that each H allele contributes 5 cm (from father: 160 - 150 = 10, and he has 2 H, so 10/2 = 5 per H). Mother's height is 150 (all h), father's height is 150+2*5 = 160, which matches.

Now, the child has 2 H alleles. So the child's height is \( 150+2\times5=160 \)? No, that can't be. Wait, mid - parental height is (160 + 150)/2 = 155. Then, for each H allele, the adjustment. Wait, maybe the formula for the child's height in a polygenic model (for height) is:

\( \text{Height of son}=\frac{\text{Father's height}+\text{Mother's height}}{2}+ 6.3 \) (but this is for sons, and daughters have a different formula, but the problem doesn't specify gender). Wait, but the given alleles suggest a different model.

Wait, let's re - evaluate the allele counts. Father: 2 H, 1 h; Mother: 3 h; Child: 2 H, 4 h. Wait, maybe the total number of alleles is 6 (2 from father, 4 from mother? No, usually in human genetics, we have 2 alleles per gene, but this is a simplified model.

Wait, the key is that the mother has all h (3 h) and height 150, father has 2 H and 1 h and height 160. So the difference in height between father and mother is 10 cm, which is due to 2 H alleles (since father has 2 H more than mother in terms of H count: mother has 0 H, father has 2 H). So each H contributes \( \frac{160 - 150}{2}=5 \) cm.

Now, the child has 2 H alleles (from the child's alleles: H, H, h, h, h, h). So the child's height is the base height (mother's height, 150) plus the contribution from H alleles. The child has 2 H, so contribution is 2*5 = 10. So 150+10 = 160? No, that's the father's height. Wait, maybe the formula is different.

Wait, another way: The mid - parental height is (160 + 150)/2 = 155. Then, for each H allele, we add a certain amount. If the child has 2 H alleles, and the father has 2 H alleles, mother has 0.

Wait, maybe the tallest height the child can attain is calculated as follows:

First, find the average of the parents' height: \( \frac{160 + 150}{2}=155 \)

Then, since the child has 2 H alleles (same as the father), and each H allele contributes to the upper end. Wait, the formula for the range of a child's height is usually mid - parental height ± 10 cm (for boys) or ± 8 cm (for girls), but this is a rough estimate. But with the allele model, let's use the allele contribution.

Since each H adds 5 cm (as calculated from father and mother: father has 2 H, 160; mother has 0 H, 150, 160 - 150 = 10, 10/2 = 5), and the child has 2 H, the child's height is 150+2*5 = 160? But that's the father's height. Wait, maybe the child can inherit the maximum number of H from the father. The father has 2 H, so the child can get 2 H, and with the mother's h, the height is calculated as:

Let’s define the height as a function of the number of H alleles. Let \( H(n)=150 + 5n \), where \( n \) is the number of H alleles.

For mother, \( n = 0 \), \( H(0)=150 \)

For father, \( n = 2 \), \( H(2)=150 + 10 = 160 \)

For child, \( n = 2 \), \( H(2)=150+10 = 160 \). But wait, is there a way to get a taller height? The father can only pass on 2 H (since he has 2 H), so the child can't get more than 2 H. So the tallest height the child can attain is 160 cm? Wait, no, maybe my model is wrong.

Wait, another approach: The formula for the child's height in a polygenic inheritance model (simplified) is:

\( \text{Child's height}=\frac{\text{Father's height}+\text{Mother's height}}{2}+\text{adjustment} \)

The adjustment for a son is + 6.3 cm, for a daughter is - 6.3 cm, but the problem doesn't specify gender. But with the allele counts, let's see.

Wait, the father's height is 160, mother's is 150. Mid - parental height is (160 + 150)/2 = 155. If we consider the allele contribution, and the child has 2 H (same as father), the height would be 155+(2 - 1)*5? No, this is getting confusing. Wait, going back to the allele counts:

Father: 2 H, 1 h (total 3 alleles)

Mother: 3 h (total 3 alleles)

Child: 2 H, 4 h (total 6 alleles)

Wait, maybe the height is determined by the ratio of H to total alleles. Father's H ratio: 2/3, height 160. Mother's H ratio: 0/3, height 150.

Let’s find the height per H ratio. The difference in ratio between father and mother is 2/3 - 0 = 2/3, and the difference in height is 10. So per unit ratio, height change is 10/(2/3)=15.

Child's H ratio: 2/6 = 1/3.

So child's height: 150+(1/3)*15 = 155? No, that doesn't match.

Wait, I think I made a mistake in the allele count interpretation. Let's look at the given alleles again. The father has 3 alleles: H, H, h (so 2 H, 1 h). The mother has 3 alleles: h, h, h (3 h). The child has 6 alleles: H, H, h, h, h, h (2 H, 4 h). So the child has inherited 2 H from the father (since father has 2 H) and 0 H from the mother (since mother has 0 H), and 1 h from the father and 3 h from the mother, totaling 4 h.

Now, let's assume that each H allele contributes 5 cm (as before: father has 2 H, 160 - 150 = 10, 10/2 = 5). Mother's height is 150 (all h). So the child's height is 150+2*5 = 160. But wait, the mid - parental height is 155, and if we consider that the child can have a height range, but the question is about the tallest height. Since the father can pass on at most 2 H, the child can have at most 2 H, so the tallest height is 150+2*5 = 160. But wait, let's check with the mid - parental height formula. The mid - parental height for a boy is (father's height + mother's height)/2+6.3, for a girl is (father's height + mother's height)/2 - 6.3.

For a boy: (160 + 150)/2+6.3 = 155+6.3 = 161.3

For a girl: (160 + 150)/2 - 6.3 = 155 - 6.3 = 148.7

But this contradicts the allele model. However, the problem gives alleles, so we should use the allele - based model.

Since father has 2 H and height 160, mother has 0 H and height 150, each H adds 5 cm. Child has 2 H, so height is 150+2*5 = 160. But wait, maybe the child can have a taller height? No, because the father can only pass on 2 H (he has 2 H), so the child can't get more than 2 H.

Wait, another way: The formula for the child's height when considering polygenic inheritance with the given alleles:

Let’s define the height as \( H = 150+5n \), where \( n \) is the number of H alleles.

- For mother (\( n = 0 \)): \( H = 150+0 = 150 \)
- For father (\( n = 2 \)): \( H = 150 + 2\times5=160 \)
- For child (\( n = 2 \)): \( H = 150+2\times5 = 160 \)

Answer:

The tallest height the child could attain is \(\boldsymbol{160}\) cm.