To determine the slant height of the cone, we first need to establish the relationship between the surface area, radius, and slant height.
The total surface area \( A \) of a cone is given by the formula:
\[ A = \pi r l + \pi r^2 \]
where:
- \( r \) is the radius of the base,
- \( l \) is the slant height,
- \( \pi r l \) is the lateral surface area,
- \( \pi r^2 \) is the area of the base.
Given:
- Total surface area \( A = 1,496 , \text{cm}^2 \)
- Diameter \( d = 28 , \text{cm} \)
First, let's find the radius \( r \):
\[ r = \frac{d}{2} = \frac{28}{2} = 14 , \text{cm} \]
Now we substitute \( r \) into the surface area formula:
\[ 1,496 = \pi (14) l + \pi (14)^2 \]
Next, we calculate \( \pi (14)^2 \):
\[ \pi (14)^2 = \pi \times 196 = 196\pi \]
Substituting this back into the surface area equation gives us:
\[ 1,496 = 14\pi l + 196\pi \]
Next, we can factor out \( \pi \):
\[ 1,496 = \pi (14l + 196) \]
Now, we will divide both sides by \( \pi \):
\[ \frac{1,496}{\pi} = 14l + 196 \]
Substituting \( \pi \approx 3.14 \):
\[ \frac{1,496}{3.14} \approx 476.07 \]
Now, we can substitute back into the formula:
\[ 476.07 = 14l + 196 \]
To isolate \( 14l \), we subtract \( 196 \):
\[ 476.07 - 196 = 14l \]
Calculating the left-hand side:
\[ 280.07 = 14l \]
Now, we divide by 14 to find \( l \):
\[ l = \frac{280.07}{14} \approx 20.00 , \text{cm} \]
Therefore, the slant height of the cone is approximately \( 20.00 , \text{cm} \).