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The problem in the image involves radioactive decay and uses the formula:

\[
A = A_0 \times 0.5^{t/H}
\]

Where:
- \( A_0 \) is the initial amount of the substance.
- \( t \) is the time elapsed.
- \( H \) is the half-life of the substance.
- \( A \) is the remaining amount after time \( t \).

Let's break down and solve the questions given:

### (a) Uranium 228
- The half-life \( H \) of Uranium 228 is 9.3 minutes.
- We start with \( A_0 = 5 \) grams of Uranium 228.

#### (i) How much uranium 228 is left after 2 minutes?
\[
A = 5 \times 0.5^{\frac{2}{9.3}}
\]

Let's calculate this.

#### (ii) How long will you have to wait until there are only 3 grams left?
We solve for \( t \) when \( A = 3 \) grams.
\[
3 = 5 \times 0.5^{\frac{t}{9.3}}
\]
First, divide both sides by 5:
\[
0.6 = 0.5^{\frac{t}{9.3}}
\]
Take the logarithm on both sides:
\[
\log(0.6) = \frac{t}{9.3} \log(0.5)
\]
Solve for \( t \).

### (b) Uranium 235
- The half-life \( H \) of Uranium 235 is 713 million years.
- We start with \( A_0 = 5 \) grams of Uranium 235.

#### (i) How much uranium 235 is left after 200 million years?
\[
A = 5 \times 0.5^{\frac{200}{713}}
\]

#### (ii) How long will you have to wait until there are only 3 grams left?
We solve for \( t \) when \( A = 3 \) grams.
\[
3 = 5 \times 0.5^{\frac{t}{713}}
\]

Let's calculate each of these. I will proceed with the calculations now.Here are the results for each part of the problem:

### (a) Uranium 228:
1. After 2 minutes, the amount of Uranium 228 left is approximately **4.31 grams**.
2. To have only 3 grams of Uranium 228 left, you will need to wait approximately **6.85 minutes**.

### (b) Uranium 235:
1. After 200 million years, the amount of Uranium 235 left is approximately **4.12 grams**.
2. To have only 3 grams of Uranium 235 left, you will need to wait approximately **525.46 million years**.

Let me know if you'd like further details or explanations. Here are some follow-up questions:

1. What is the physical meaning of half-life in a radioactive decay process?
2. How does the initial quantity of a substance affect the rate of decay?
3. Can the half-life of a substance change under different conditions?
4. How do logarithms help in solving decay-related time problems?
5. What are some practical applications of understanding radioactive decay in real life?

**Tip:** When solving exponential decay problems, taking the logarithm of both sides helps simplify expressions where the variable is in the exponent.

The half-life of a radioactive substance is given by the formula A = A0 * 0.5^(t/H), where A is the remaining amount after time t, A0 is the initial amount, and H is the half-life. Calculate the remaining amount of Uranium 228 and Uranium 235 after a specified time.

Learn how to calculate the remaining amount of Uranium 228 and Uranium 235 using the half-life formula. This problem covers key concepts of exponential decay and logarithmic equations. Suitable for high school students in Grades 11-12.

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