A solid right circular cylinder has radius r and height 5r. A solid right circular cone is carved out from one end of the base of cylinder. If base radius of cone is r and height is `2 sqrt2 r` then, find the ratio between total surface area of cone to the total surface area of remaining part of cylinder.

Options

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

Answer

To solve the problem, we need to find the ratio between the total surface area of the cone and the total surface area of the remaining part of the cylinder after the cone has been carved out. Let's break this down step by step.

### Step 1: Calculate the Total Surface Area of the Cone
The total surface area (TSA) of a cone is given by the formula:
\[ \text{TSA}_{\text{cone}} = \pi r l + \pi r^2 \]
where \( r \) is the radius of the base and \( l \) is the slant height of the cone.

1. **Find the slant height \( l \)**:
The slant height can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
where \( h \) is the height of the cone. Given \( h = 2\sqrt{2}r \) and \( r = r \):
\[ l = \sqrt{(2\sqrt{2}r)^2 + r^2} = \sqrt{8r^2 + r^2} = \sqrt{9r^2} = 3r \]

2. **Substituting \( l \) into the TSA formula**:
\[ \text{TSA}_{\text{cone}} = \pi r (3r) + \pi r^2 = 3\pi r^2 + \pi r^2 = 4\pi r^2 \]

### Step 2: Calculate the Total Surface Area of the Remaining Part of the Cylinder
The total surface area of a cylinder is given by:
\[ \text{TSA}_{\text{cylinder}} = 2\pi r h + 2\pi r^2 \]
where \( h \) is the height of the cylinder.

1. **Substituting the values**:
The height of the cylinder is \( 5r \):
\[ \text{TSA}_{\text{cylinder}} = 2\pi r (5r) + 2\pi r^2 = 10\pi r^2 + 2\pi r^2 = 12\pi r^2 \]

2. **Adjusting for the carved-out cone**:
The remaining surface area after carving out the cone consists of:
- The curved surface area of the cylinder.
- The area of the top circular base of the cylinder (which is now open).
- The area of the base of the cone (which is also open).

Therefore, the total surface area of the remaining part of the cylinder is:
\[ \text{TSA}_{\text{remaining}} = \text{Curved Surface Area of Cylinder} + \text{Area of Top Base} - \text{Area of Base of Cone} \]
\[ = 10\pi r^2 + 0 - \pi r^2 = 10\pi r^2 - \pi r^2 = 9\pi r^2 \]

### Step 3: Find the Ratio of the Total Surface Areas
Now, we can find the ratio of the total surface area of the cone to the total surface area of the remaining part of the cylinder:
\[ \text{Required Ratio} = \frac{\text{TSA}_{\text{cone}}}{\text{TSA}_{\text{remaining}}} = \frac{4\pi r^2}{9\pi r^2} = \frac{4}{9} \]

### Final Answer
The ratio between the total surface area of the cone to the total surface area of the remaining part of the cylinder is:
\[ \text{Ratio} = 4:9 \]