To find the height of the water tower from the ground to the vertex of the cone, we will add the height of the cylinder to the height of the cone.
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Height of the cylinder: The height of the cylinder is given as 6 ft.
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Height of the cone: To find the height of the cone, we can use the slant height and the radius of the base of the cone.
The diameter of the cylinder (which is the same as the diameter of the base of the cone) is 6 ft. Therefore, the radius \( r \) of the cone is: \[ r = \frac{diameter}{2} = \frac{6 , \text{ft}}{2} = 3 , \text{ft} \]
The slant height \( l \) of the cone is given as 4 ft. We can use the Pythagorean theorem to find the height \( h \) of the cone: \[ l^2 = h^2 + r^2 \] Substituting the known values: \[ 4^2 = h^2 + 3^2 \] \[ 16 = h^2 + 9 \] \[ h^2 = 16 - 9 = 7 \] \[ h = \sqrt{7} \approx 2.6457513110645906 , \text{ft} \]
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Height of the tower: The total height of the tower from the ground to the vertex of the cone is: \[ \text{Total height} = \text{Height of cylinder} + \text{Height of cone} + \text{Height off the ground} \] \[ \text{Total height} = 6 , \text{ft} + h + 8 , \text{ft} \] \[ \text{Total height} = 6 + 2.6457513110645906 + 8 \approx 16.6457513110645906 \]
Finally, rounding to the nearest tenth: \[ \text{Total height} \approx 16.6 , \text{ft} \]
Thus, the answer is 16.6 ft.
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