To find the value of [tex]\( p \)[/tex] given that 33 out of 100 organisms have short legs, we will follow these steps based on the Hardy-Weinberg equilibrium.
1. Understanding Hardy-Weinberg Equilibrium:
The Hardy-Weinberg equation is [tex]\( p^2 + 2pq + q^2 = 1 \)[/tex]. Here:
- [tex]\( p \)[/tex] represents the frequency of the dominant allele (legs).
- [tex]\( q \)[/tex] represents the frequency of the recessive allele (short legs).
- [tex]\( p^2 \)[/tex] is the proportion of the population that is homozygous dominant (having two dominant alleles).
- [tex]\( 2pq \)[/tex] is the proportion of the population that is heterozygous (having one dominant and one recessive allele).
- [tex]\( q^2 \)[/tex] is the proportion of the population that is homozygous recessive (having two recessive alleles).
2. Identifying the Given Information:
- Total number of organisms, [tex]\( N \)[/tex], is 100.
- Number of organisms with short legs, which corresponds to the homozygous recessive group ([tex]\( q^2 \)[/tex]), is 33.
3. Calculate [tex]\( q^2 \)[/tex]:
[tex]\[
q^2 = \frac{\text{Number of homozygous recessive organisms}}{\text{Total number of organisms}} = \frac{33}{100} = 0.33
\][/tex]
4. Determine [tex]\( q \)[/tex] by taking the square root of [tex]\( q^2 \)[/tex]:
[tex]\[
q = \sqrt{0.33} \approx 0.574
\][/tex]
5. Use the relationship [tex]\( p + q = 1 \)[/tex] to find [tex]\( p \)[/tex]:
[tex]\[
p = 1 - q = 1 - 0.574 \approx 0.426
\][/tex]
6. Final Value of [tex]\( p \)[/tex]:
[tex]\[
p \approx 0.426
\][/tex]
Therefore, the closest option to the calculated value of [tex]\( p \)[/tex] is option C:
[tex]\[
\boxed{0.43}
\][/tex]
