32 Simplify without using calculator: $$ 3.2 .16 \sqrt{80}+2 \sqrt{20}-\sqrt{125} $$ (3) $$ 3.2 .2 \sqrt{\frac{\sqrt{27}+\sqrt{48}+\sqrt{75}}{\sqrt{27}}} $$ (4) [13] QUESTION 4 4.1 The formula for the area of a ring is given as $$ A=\pi\left(R^{2}-r^{2} ight) $$ 4.1.1 Make $$ R $$ the subject of the formula for the given equation. (3) 4.1.2 Determine the value of R if $$ A=120 \mathrm{~mm}^{2} $$ and $$ r=10 \mathrm{~mm} $$. (2) 4.2 Calculate the $$ 16^{ ext {th }} $$ term of an arithmetic sequence if the $$ 5^{ ext {th }} $$ term is 6 and $$ 12^{ ext {th }} $$ term is 41 . (6) 4.3 A theatre has 60 seats in the first row, 68 seats in the second row and 76 seats in the third row and so forth in the increasing pattern. If the theatre has 22 rows, how many seats are in the theatre? (3) 4.4 The value of the of a certain term $$ \left(T_{n} ight) $$, of an arithmetic sequence is 64 . Calculate the position of the term if the first term is -24 and constant difference is 8 . (3)

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32 Simplify without using calculator: $$ 3.2 .16 \sqrt{80}+2 \sqrt{20}-\sqrt{125} $$ (3) $$ 3.2 .2 \sqrt{\frac{\sqrt{27}+\sqrt{48}+\sqrt{75}}{\sqrt{27}}} $$ (4) [13] QUESTION 4 4.1 The formula for the area of a ring is given as $$ A=\pi\left(R^{2}-r^{2}ight) $$ 4.1.1 Make $$ R $$ the subject of the formula for the given equation. (3) 4.1.2 Determine the value of R if $$ A=120 \mathrm{~mm}^{2} $$ and $$ r=10 \mathrm{~mm} $$. (2) 4.2 Calculate the $$ 16^{	ext {th }} $$ term of an arithmetic sequence if the $$ 5^{	ext {th }} $$ term is 6 and $$ 12^{	ext {th }} $$ term is 41 . (6) 4.3 A theatre has 60 seats in the first row, 68 seats in the second row and 76 seats in the third row and so forth in the increasing pattern. If the theatre has 22 rows, how many seats are in the theatre? (3) 4.4 The value of the of a certain term $$ \left(T_{n}ight) $$, of an arithmetic sequence is 64 . Calculate the position of the term if the first term is -24 and constant difference is 8 . (3)

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Let's solve the problems step by step.

### Problem 1: Simplify $$ 3.2 \cdot 16 \sqrt{80} + 2 \sqrt{20} - \sqrt{125} $$

1. **Calculate $$ \sqrt{80} $$**:
   $$
   \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}
   $$

2. **Calculate $$ \sqrt{20} $$**:
   $$
   \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
   $$

3. **Calculate $$ \sqrt{125} $$**:
   $$
   \sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}
   $$

4. **Substitute back into the expression**:
   $$
   3.2 \cdot 16 \cdot 4\sqrt{5} + 2 \cdot 2\sqrt{5} - 5\sqrt{5}
   $$

5. **Calculate $$ 3.2 \cdot 16 \cdot 4 $$**:
   $$
   3.2 \cdot 16 = 51.2 \quad \text{and} \quad 51.2 \cdot 4 = 204.8
   $$

6. **Combine the terms**:
   $$
   204.8\sqrt{5} + 4\sqrt{5} - 5\sqrt{5} = (204.8 + 4 - 5)\sqrt{5} = 203.8\sqrt{5}
   $$

So, the simplified expression is $$ 203.8\sqrt{5} $$.

### Problem 2: Simplify $$ 3.2 \cdot 2 \sqrt{\frac{\sqrt{27}+\sqrt{48}+\sqrt{75}}{\sqrt{27}}} $$

1. **Calculate $$ \sqrt{27} $$**:
   $$
   \sqrt{27} = 3\sqrt{3}
   $$

2. **Calculate $$ \sqrt{48} $$**:
   $$
   \sqrt{48} = 4\sqrt{3}
   $$

3. **Calculate $$ \sqrt{75} $$**:
   $$
   \sqrt{75} = 5\sqrt{3}
   $$

4. **Combine the square roots**:
   $$
   \sqrt{27} + \sqrt{48} + \sqrt{75} = 3\sqrt{3} + 4\sqrt{3} + 5\sqrt{3} = 12\sqrt{3}
   $$

5. **Substitute back into the expression**:
   $$
   3.2 \cdot 2 \sqrt{\frac{12\sqrt{3}}{3\sqrt{3}}} = 3.2 \cdot 2 \sqrt{4} = 3.2 \cdot 2 \cdot 2 = 12.8
   $$

So, the simplified expression is $$ 12.8 $$.

### Problem 4.1: Make $$ R $$ the subject of the formula $$ A = \pi(R^2 - r^2) $$

1. **Divide both sides by $$ \pi $$**:
   $$
   \frac{A}{\pi} = R^2 - r^2
   $$

2. **Add $$ r^2 $$ to both sides**:
   $$
   R^2 = \frac{A}{\pi} + r^2
   $$

3. **Take the square root**:
   $$
   R = \sqrt{\frac{A}{\pi} + r^2}
   $$

### Problem 4.1.2: Determine the value of $$ R $$ if $$ A = 120 \, \text{mm}^2 $$ and $$ r = 10 \, \text{mm} $$

1. **Substitute the values into the formula**:
   $$
   R = \sqrt{\frac{120}{\pi} + 10^2} = \sqrt{\frac{120}{\pi} + 100}
   $$

### Problem 4.2: Calculate the $$ 16^{th} $$ term of an arithmetic sequence if the $$ 5^{th} $$ term is 6 and the $$ 12^{th} $$ term is 41

1. **Let the first term be $$ a $$ and the common difference be $$ d $$**:
   $$
   T_5 = a + 4d = 6 \quad (1)
   $$
   $$
   T_{12} = a + 11d = 41 \quad (2)
   $$

2. **Subtract equation (1) from (2)**:
   $$
   (a + 11d) - (a + 4d) = 41 - 6
   $$
   $$
   7d = 35 \implies d = 5
   $$

3. **Substitute $$ d $$ back into equation (1)**:
   $$
   a + 4(5) = 6 \implies a + 20 = 6 \implies a = -14
   $$

4. **Calculate the $$ 16^{th} $$ term**:
   $$
   T_{16} = a + 15d = -14 + 15(5) = -14 + 75 = 61
   $$

### Problem 4.3: Calculate the total number of seats in the theatre

1. **Identify the pattern**:
   - First row: 60 seats
   - Second row: 68 seats
   - Third row: 76 seats

This is an arithmetic sequence where:
   - First term $$ a = 60 $$
   - Common difference $$ d = 8 $$
   - Number of rows $$ n = 22 $$

2. **Calculate the total number of seats**:
   $$
   T_n = a + (n-1)d = 60 + (22-1) \cdot 8 = 60 + 168 = 228
   $$

3. **Sum of the arithmetic series**:
   $$
   S_n = \frac{n}{2} \cdot (a + T_n) = \frac{22}{2} \cdot (60 + 228) = 11 \cdot 288 = 3168
   $$

### Problem 4.4: Calculate the position of the term $$ T_n = 64 $$

1. **Use the formula for the $$ n^{th} $$ term**:
   $$
   T_n = a + (n-1)d
   $$
   $$
   64 = -24 +
**Solutions:** 1. **Simplify $$ 3.2 \cdot 16 \sqrt{80} + 2 \sqrt{20} - \sqrt{125} $$:** \[ 203.8\sqrt{5} $$ 2. **Simplify $$ 3.2 \cdot 2 \sqrt{\frac{\sqrt{27}+\sqrt{48}+\sqrt{75}}{\sqrt{27}}} $$:** $$ 12.8 $$ 3. **Make $$ R $$ the subject of the formula $$ A = \pi(R^2 - r^2) $$:** $$ R = \sqrt{\frac{A}{\pi} + r^2} $$ 4. **Determine the value of $$ R $$ when $$ A = 120 \, \text{mm}^2 $$ and $$ r = 10 \, \text{mm} $$:** $$ R = \sqrt{\frac{120}{\pi} + 100} $$ 5. **Calculate the $$ 16^{th} $$ term of the arithmetic sequence:** $$ T_{16} = 61 $$ 6. **Calculate the total number of seats in the theatre:** $$ 3168 \, \text{seats} $$ 7. **Calculate the position of the term $$ T_n = 64 $$:** $$ n = 10 $$

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