To determine the surface area of a cone, we use the formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
where:
- \( r \) is the radius of the base,
- \( l \) is the slant height.
First, we need to find the radius of the base. The diameter of the base is given as 14 ft, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{14 \text{ ft}}{2} = 7 \text{ ft} \]
Next, we know the slant height \( l \) is 19 ft.
Now we can substitute \( r \) and \( l \) into the surface area formula:
- Calculate the area of the base:
\[ \pi r^2 = 3.14 \times (7 \text{ ft})^2 = 3.14 \times 49 \text{ ft}^2 = 153.86 \text{ ft}^2 \]
- Calculate the lateral surface area:
\[ \pi r l = 3.14 \times 7 \text{ ft} \times 19 \text{ ft} = 3.14 \times 133 \text{ ft}^2 = 417.82 \text{ ft}^2 \]
- Now, add both areas together to find the total surface area:
\[ \text{Surface Area} = 153.86 \text{ ft}^2 + 417.82 \text{ ft}^2 = 571.68 \text{ ft}^2 \]
Finally, rounding to the nearest hundredth, the surface area of the cone is:
\[ \boxed{571.68} \text{ ft}^2 \]