To determine the surface area of a cone, we can use the following formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
where \( r \) is the radius of the base, \( l \) is the slant height, and \( \pi \) is a constant (approximately 3.14 in this case).
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Calculate the radius \( r \): The diameter of the base is given as 14 ft. Therefore, the radius \( r \) is: \[ r = \frac{\text{diameter}}{2} = \frac{14}{2} = 7 \text{ ft} \]
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Identifying the slant height \( l \): The slant height \( l \) is given as 19 ft.
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Calculate the surface area:
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First, compute the area of the base: \[ \pi r^2 = \pi (7)^2 = \pi \times 49 = 3.14 \times 49 = 153.86 \text{ ft}^2 \]
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Next, compute the lateral surface area: \[ \pi r l = \pi \times 7 \times 19 = 3.14 \times 7 \times 19 \]
First, calculate \( 7 \times 19 \): \[ 7 \times 19 = 133 \] Now, compute: \[ \pi r l = 3.14 \times 133 = 417.62 \text{ ft}^2 \]
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Combine the areas: Now, add the area of the base and the lateral surface area: \[ \text{Surface Area} = \pi r^2 + \pi r l = 153.86 + 417.62 = 571.48 \text{ ft}^2 \]
Therefore, the surface area of the cone is approximately:
\[ \boxed{571.48} \text{ ft}^2 \]