Evaluate the following :
(i) The sum of length , breadth and depth of a cuboid is 19 cm , and the length of its diagonal is 11cm, then find the surface area of the cuboid.
(ii) If the radii of two circular cylinders are in the ratio 3:4, and the their heights are in the ratio 6:5 , then find the ratio of their curved surface areas.

Options

A. `{:("(i)","(ii)"),(180 cm^(2) ,9:11):}`
B. `{:("(i)","(ii)"),(240 cm^(2) ,9:10):}`
C. `{:("(i)","(ii)"),(120 cm^(2) ,8:5):}`
D. `{:("(i)","(ii)"),(321 cm^(2) ,7:10):}`

Explanation

### Step-by-Step Solution:

#### Part (i): Finding the Surface Area of the Cuboid

1. **Given Information**:
- The sum of length (L), breadth (B), and height (H) of the cuboid is:
\[ L + B + H = 19 \text{ cm} \]
- The length of the diagonal (D) of the cuboid is:
\[ D = 11 \text{ cm} \]

2. **Using the Diagonal Formula**:
The formula for the diagonal of a cuboid is given by:
\[ D = \sqrt{L^2 + B^2 + H^2} \]
Squaring both sides gives:
\[ D^2 = L^2 + B^2 + H^2 \]
Substituting the value of D:
\[ 11^2 = L^2 + B^2 + H^2 \implies 121 = L^2 + B^2 + H^2 \]

3. **Using the Identity**:
We know:
\[ (L + B + H)^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Substituting \(L + B + H = 19\):
\[ 19^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
This gives:
\[ 361 = 121 + 2(LB + BH + HL) \]

4. **Solving for \(LB + BH + HL\)**:
Rearranging the equation:
\[ 361 - 121 = 2(LB + BH + HL) \implies 240 = 2(LB + BH + HL) \]
Dividing by 2:
\[ LB + BH + HL = 120 \]

5. **Calculating Surface Area**:
The surface area (SA) of a cuboid is given by:
\[ SA = 2(LB + BH + HL) \]
Substituting the value we found:
\[ SA = 2 \times 120 = 240 \text{ cm}^2 \]

#### Part (ii): Finding the Ratio of Curved Surface Areas of Two Cylinders

1. **Given Ratios**:
- The ratio of the radii of two cylinders is:
\[ R_1 : R_2 = 3 : 4 \]
- The ratio of their heights is:
\[ H_1 : H_2 = 6 : 5 \]

2. **Curved Surface Area Formula**:
The curved surface area (CSA) of a cylinder is given by:
\[ CSA = 2\pi R H \]
Therefore, for the two cylinders:
- For Cylinder 1:
\[ CSA_1 = 2\pi R_1 H_1 \]
- For Cylinder 2:
\[ CSA_2 = 2\pi R_2 H_2 \]

3. **Finding the Ratio of Curved Surface Areas**:
The ratio of their curved surface areas is:
\[ \frac{CSA_1}{CSA_2} = \frac{R_1 H_1}{R_2 H_2} \]
Substituting the ratios:
\[ \frac{CSA_1}{CSA_2} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20} = \frac{9}{10} \]

### Final Answers:
- (i) The surface area of the cuboid is **240 cm²**.
- (ii) The ratio of the curved surface areas of the two cylinders is **9:10**.