In the given questions two quantities are given one as quantity I and another as quantity II you have to determine relationship between two quantities choose appropriate option
The respsetive ratio between height and radius of right circular conical structure (A) is 3:1. The volume of cone (A) is 1078 cubic meter
quantityI:measure of slant height of the given cone (A)
quantityII:measure of slant height if another cone B whose radius and height are 14 m and 16m respectively

Options

A. `IltII`
B. `IgtII`
C. `IleII`
D. `IgeII`

Answer

To solve the problem, we need to find the slant heights of two cones, A and B, and then compare them.

### Step 1: Understand the dimensions of Cone A
The ratio of height to radius for Cone A is given as 3:1.
- Let the radius \( r_A = x \) meters.
- Then the height \( h_A = 3x \) meters.

### Step 2: Use the volume formula for Cone A
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
For Cone A, we know:
\[ V_A = 1078 \text{ cubic meters} \]
Substituting the values:
\[ 1078 = \frac{1}{3} \pi (x^2)(3x) \]
This simplifies to:
\[ 1078 = \pi x^3 \]
Using \( \pi \approx \frac{22}{7} \):
\[ 1078 = \frac{22}{7} x^3 \]
Multiplying both sides by \( 7 \):
\[ 7 \times 1078 = 22 x^3 \]
Calculating:
\[ 7556 = 22 x^3 \]
Dividing by 22:
\[ x^3 = \frac{7556}{22} = 343 \]
Taking the cube root:
\[ x = \sqrt[3]{343} = 7 \text{ meters} \]
Thus, the radius \( r_A = 7 \) meters and the height \( h_A = 3 \times 7 = 21 \) meters.

### Step 3: Calculate the slant height of Cone A
The slant height \( l_A \) of a cone can be calculated using the Pythagorean theorem:
\[ l_A = \sqrt{r_A^2 + h_A^2} \]
Substituting the values:
\[ l_A = \sqrt{7^2 + 21^2} = \sqrt{49 + 441} = \sqrt{490} = 7\sqrt{10} \text{ meters} \]

### Step 4: Calculate the slant height of Cone B
For Cone B, the radius \( r_B = 14 \) meters and the height \( h_B = 16 \) meters.
Using the same formula for slant height:
\[ l_B = \sqrt{r_B^2 + h_B^2} \]
Substituting the values:
\[ l_B = \sqrt{14^2 + 16^2} = \sqrt{196 + 256} = \sqrt{452} \text{ meters} \]

### Step 5: Compare the slant heights
Now we need to compare \( l_A \) and \( l_B \):
- \( l_A = 7\sqrt{10} \)
- \( l_B = \sqrt{452} \)

Calculating \( \sqrt{452} \):
\[ \sqrt{452} = \sqrt{4 \times 113} = 2\sqrt{113} \]
Now we need to compare \( 7\sqrt{10} \) and \( 2\sqrt{113} \).

Calculating the approximate values:
- \( \sqrt{10} \approx 3.16 \) so \( 7\sqrt{10} \approx 22.12 \)
- \( \sqrt{113} \approx 10.63 \) so \( 2\sqrt{113} \approx 21.26 \)

Since \( 22.12 > 21.26 \), we conclude:
\[ l_A > l_B \]

### Conclusion
**Quantity I (slant height of Cone A) is greater than Quantity II (slant height of Cone B).**