To find the height of the cylinder given its surface area, we can use the formula for the surface area of a cylinder:
\[ A = 2\pi r^2 + 2\pi rh \]
Where:
- \(A\) is the surface area
- \(r\) is the radius
- \(h\) is the height
Given:
- \(A = 3106.716\) square feet
- \(r = 17\)
First, we can substitute the given values into the surface area formula:
\[ 3106.716 = 2\pi (17^2) + 2\pi (17)h \]
Calculating \(2\pi(17^2)\):
\[ 17^2 = 289 \quad \text{so} \quad 2\pi(289) \approx 2 \times 3.14 \times 289 \approx 1811.56 \]
Now we substitute this back into the surface area equation:
\[ 3106.716 = 1811.56 + 2\pi(17)h \]
Next, isolate \(2\pi(17)h\):
\[ 3106.716 - 1811.56 = 2\pi(17)h \] \[ 1295.156 = 2\pi(17)h \]
Calculating \(2\pi(17)\):
\[ 2\pi(17) \approx 2 \times 3.14 \times 17 \approx 214.48 \]
Now we substitute this value back into the equation for height \(h\):
\[ 1295.156 = 214.48h \]
Now, solve for \(h\):
\[ h = \frac{1295.156}{214.48} \approx 6.04 \]
Thus, rounding to the nearest hundredth, the height of the cylinder is approximately:
\[ \boxed{6.04} \text{ feet.} \]