Direction: Read the following Caselet DI carefully and answer the following questions. 3 cubes P, Q, and R are cut from the cubes A, B, and C respectively.
The length of the side of cube P is given by the equation x - 12x + 36 = 0 and the length of the side of cube A is 2 cm more than cube P.
The length of the side of cube Q is given by the equation y + 21y - 46 = 0 and the length of the side of cube B is 3 cm more than cube Q.
The length of the side of cube R is given by the equation z - 10z + 25 = 0 and the length of the side of cube C is 4 cm more than cube R.
Here x, y, and z are in centimeter.
What is the ratio of total surface area of cube P to total surface area of cube C?

Options

A. `2:3`
B. `5:7`
C. `4:9`
D. `8:3`

Answer

To solve the problem step by step, we will first find the lengths of the sides of cubes P, Q, and R using the given equations, and then calculate the total surface areas of cubes P and C to find the required ratio.

### Step 1: Solve for the length of the side of cube P
The equation for cube P is:
\[ x^2 - 12x + 36 = 0 \]

This can be factored as:
\[ (x - 6)^2 = 0 \]

Thus, we find:
\[ x = 6 \text{ cm} \]

### Step 2: Find the length of the side of cube A
The length of the side of cube A is 2 cm more than cube P:
\[ \text{Length of cube A} = x + 2 = 6 + 2 = 8 \text{ cm} \]

### Step 3: Solve for the length of the side of cube Q
The equation for cube Q is:
\[ y^2 + 21y - 46 = 0 \]

Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 21, c = -46 \).

Calculating the discriminant:
\[ b^2 - 4ac = 21^2 - 4 \cdot 1 \cdot (-46) = 441 + 184 = 625 \]

Now, substituting into the quadratic formula:
\[ y = \frac{-21 \pm \sqrt{625}}{2 \cdot 1} = \frac{-21 \pm 25}{2} \]

Calculating the two possible values:
1. \( y = \frac{4}{2} = 2 \text{ cm} \)
2. \( y = \frac{-46}{2} = -23 \text{ cm} \) (not valid since length cannot be negative)

Thus, the length of cube Q is:
\[ y = 2 \text{ cm} \]

### Step 4: Find the length of the side of cube B
The length of the side of cube B is 3 cm more than cube Q:
\[ \text{Length of cube B} = y + 3 = 2 + 3 = 5 \text{ cm} \]

### Step 5: Solve for the length of the side of cube R
The equation for cube R is:
\[ z^2 - 10z + 25 = 0 \]

This can be factored as:
\[ (z - 5)^2 = 0 \]

Thus, we find:
\[ z = 5 \text{ cm} \]

### Step 6: Find the length of the side of cube C
The length of the side of cube C is 4 cm more than cube R:
\[ \text{Length of cube C} = z + 4 = 5 + 4 = 9 \text{ cm} \]

### Step 7: Calculate the total surface area of cube P
The total surface area (TSA) of a cube is given by:
\[ \text{TSA} = 6 \times (\text{side})^2 \]

For cube P:
\[ \text{TSA of P} = 6 \times (6)^2 = 6 \times 36 = 216 \text{ cm}^2 \]

### Step 8: Calculate the total surface area of cube C
For cube C:
\[ \text{TSA of C} = 6 \times (9)^2 = 6 \times 81 = 486 \text{ cm}^2 \]

### Step 9: Find the ratio of the total surface area of cube P to cube C
The ratio is given by:
\[ \text{Ratio} = \frac{\text{TSA of P}}{\text{TSA of C}} = \frac{216}{486} \]

Simplifying this ratio:
\[ \frac{216 \div 54}{486 \div 54} = \frac{4}{9} \]

### Final Answer:
The ratio of the total surface area of cube P to the total surface area of cube C is:
\[ \frac{4}{9} \]