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)

UpStudy AI · Accepted Answer

Let's solve each part of the problem step by step.

### Part 1: Simplifying the Expression
1. **Expression**: $$ 3.2 \cdot 0.16 \sqrt{80} + 2 \sqrt{20} - \sqrt{125} $$

- First, simplify each square root:
     - $$ \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} $$
     - $$ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} $$
     - $$ \sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5} $$

- Substitute back into the expression:
     $$
     3.2 \cdot 0.16 \cdot 4\sqrt{5} + 2 \cdot 2\sqrt{5} - 5\sqrt{5}
     $$

- Calculate $$ 3.2 \cdot 0.16 \cdot 4 $$:
     $$
     3.2 \cdot 0.16 = 0.512 \quad \text{and} \quad 0.512 \cdot 4 = 2.048
     $$

- Now substitute:
     $$
     2.048\sqrt{5} + 4\sqrt{5} - 5\sqrt{5}
     $$

- Combine like terms:
     $$
     (2.048 + 4 - 5)\sqrt{5} = (6.048 - 5)\sqrt{5} = 1.048\sqrt{5}
     $$

### Part 2: Simplifying the Second Expression
2. **Expression**: $$ 3.2 \cdot 0.2 \sqrt{\frac{\sqrt{27}+\sqrt{48}+\sqrt{75}}{\sqrt{27}}} $$

- Simplify the square roots:
     - $$ \sqrt{27} = 3\sqrt{3} $$
     - $$ \sqrt{48} = 4\sqrt{3} $$
     - $$ \sqrt{75} = 5\sqrt{3} $$

- Substitute back:
     $$
     \sqrt{27} + \sqrt{48} + \sqrt{75} = 3\sqrt{3} + 4\sqrt{3} + 5\sqrt{3} = 12\sqrt{3}
     $$

- Now substitute into the expression:
     $$
     3.2 \cdot 0.2 \sqrt{\frac{12\sqrt{3}}{3\sqrt{3}}} = 3.2 \cdot 0.2 \sqrt{4} = 3.2 \cdot 0.2 \cdot 2
     $$

- Calculate:
     $$
     3.2 \cdot 0.2 = 0.64 \quad \text{and} \quad 0.64 \cdot 2 = 1.28
     $$

### Part 3: Area of a Ring
3. **Formula**: $$ A = \pi(R^2 - r^2) $$

- **Make $$ R $$ the subject**:
     $$
     R^2 = \frac{A}{\pi} + r^2
     $$
     $$
     R = \sqrt{\frac{A}{\pi} + r^2}
     $$

- **Determine $$ R $$ if $$ A = 120 \, \text{mm}^2 $$ and $$ r = 10 \, \text{mm} $$**:
     $$
     R = \sqrt{\frac{120}{\pi} + 10^2} = \sqrt{\frac{120}{\pi} + 100}
     $$

### Part 4: Arithmetic Sequence
4. **Calculate the 16th term**:
   - Given $$ T_5 = 6 $$ and $$ T_{12} = 41 $$:
     - The common difference $$ d $$ can be found:
     $$
     T_{12} - T_5 = 41 - 6 = 35 \quad \text{(7 terms)}
     $$
     $$
     d = \frac{35}{7} = 5
     $$

- Now find $$ T_1 $$:
     $$
     T_5 = T_1 + 4d \Rightarrow 6 = T_1 + 20 \Rightarrow T_1 = 6 - 20 = -14
     $$

- Now calculate $$ T_{16} $$:
     $$
     T_{16} = T_1 + 15d = -14 + 15 \cdot 5 = -14 + 75 = 61
     $$

### Part 5: Total Seats in the Theatre
5. **Seats in the theatre**:
   - The first row has 60 seats, the second has 68, and the third has 76. This is an arithmetic sequence with:
     - First term $$ a = 60 $$
     - Common difference $$ d = 8 $$
     - Number of terms $$ n = 22 $$

- Total seats:
     $$
     S_n = \frac{n}{2} \cdot (2a + (n-1)d) = \frac{22}{2} \cdot (2 \cdot 60 + 21 \cdot 8)
     $$
     $$
     = 11 \cdot (120 + 168) = 11 \cdot 288 = 3168
     $$

### Part 6: Position of the Term in Arithmetic Sequence
6. **Given $$ T_n = 64 $$**:
   - Use the formula:
     $$
     T_n = T_1 + (n-1)d \Rightarrow 64 = -24 + (n-1) \cdot 8
     $$
     $$
     64 + 24 = (n-1) \cdot 8 \Rightarrow 88 = (n-1) \cdot 8 \Rightarrow n-1 = 11 \Rightarrow n = 12
     $$

### Summary of Results
1. $$ 1.048\sqrt{5} $$
2. $$ 1.28 $$
3. $$ R = \sqrt{\frac{120}{\pi} + 100} $$
4. $$ T_{16} = 61 $$
5. Total seats = 3168
6. Position of the term = 12
1. $$ 1.048\sqrt{5} $$ 2. $$ 1.28 $$ 3. $$ R = \sqrt{\frac{120}{\pi} + 100} $$ 4. The 16th term is 61. 5. The theatre has a total of 3,168 seats. 6. The term is in the 12th position.

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