The image presents a series of biology questions related to DNA, RNA, and cellular processes. I will solve the mathematical problems. Question VIII. concerns a single strand of a DNA molecule containing $3 \times 10^4$ nucleotides. Question 9 asks: If a DNA molecule undergoes two rounds of replication, how many free nucleotides are required? Question 10 asks: In a DNA molecule, the number of cytosine nucleotides is 14% of the total number of nucleotides. Calculate the number of nucleotides of each type in the DNA molecule. Question 11 asks: If this DNA molecule is used as a template, how many $mRNA$ molecules are needed, assuming each $mRNA$ molecule contains an average of $10^4$ nucleotides?

1. Problem Description

The image presents a series of biology questions related to DNA, RNA, and cellular processes. I will solve the mathematical problems.

Question VIII. concerns a single strand of a DNA molecule containing $3 \times 10^4$ nucleotides.

Question 9 asks: If a DNA molecule undergoes two rounds of replication, how many free nucleotides are required?

Question 10 asks: In a DNA molecule, the number of cytosine nucleotides is 14% of the total number of nucleotides. Calculate the number of nucleotides of each type in the DNA molecule.

Question 11 asks: If this DNA molecule is used as a template, how many $mRNA$ molecules are needed, assuming each $mRNA$ molecule contains an average of $10^4$ nucleotides?

2. Solution Steps

Question 9:

In the first replication, the number of nucleotides needed is equal to the number of nucleotides in the original strand, which is $3 \times 10^4$. The complementary strand also needs $3 \times 10^4$ nucleotides. Therefore, the number of nucleotides needed is $2 \times (3 \times 10^4) = 6 \times 10^4$. This produces two copies of the DNA, so the copy number becomes $2$.

In the second replication, each of these two copies replicates. Thus, $2 \times (3 \times 10^4) = 6 \times 10^4$ are needed per replication. Since there are two replications occurring at the same time, in this case, we need to add $6 \times 10^4$ twice. Thus, $2 \times (3 \times 10^4) = 6 \times 10^4$. The total needed for this second round is $6 \times 10^4 + 6 \times 10^4 = 12 \times 10^4$.

The total amount of nucleotides needed is $6 \times 10^4 + 12 \times 10^4 = 18 \times 10^4 = 1.8 \times 10^5$.

Question 10:

The single strand contains $3 \times 10^4$ nucleotides. Since DNA is a double helix, the total number of nucleotides in the DNA molecule is $2 \times (3 \times 10^4) = 6 \times 10^4$.

Cytosine (C) accounts for 14% of the total nucleotides, so the number of C nucleotides is $0.14 \times (6 \times 10^4) = 0.84 \times 10^4 = 8.4 \times 10^3$.

Since the amount of C equals the amount of Guanine (G), the number of G nucleotides is $8.4 \times 10^3$.

The remaining nucleotides are Adenine (A) and Thymine (T). They account for $100\% - 14\% - 14\% = 72\%$ of the total number of nucleotides. So $72 \% / 2 = 36\%$ each.

$0.36 \times (6 \times 10^4) = 2.16 \times 10^4$ so the number of A and T nucleotides is $2.16 \times 10^4$ respectively.

Question 11:

The single strand has $3 \times 10^4$ nucleotides, thus the corresponding mRNA strand can have at most that many, meaning we are dealing with strands with length approximately $3 \times 10^4$.

If each $mRNA$ molecule has $10^4$ nucleotides, then the number of mRNA molecules needed is

$(3 \times 10^4) / (1 \times 10^4) = 3$.

3. Final Answer

Question 9: $1.8 \times 10^5$

Question 10: C = $8.4 \times 10^3$, G = $8.4 \times 10^3$, A = $2.16 \times 10^4$, T = $2.16 \times 10^4$

Question 11: 3