To find the values of \( t \) for which the rocket’s height \( h(t) \) is 92 feet, we can set the height equation equal to 92:
\[ H = 188t - 16t^2 \]
We need to solve the equation:
\[ 188t - 16t^2 = 92 \]
Rearranging it gives:
\[ -16t^2 + 188t - 92 = 0 \]
To make it easier to work with, we can multiply the entire equation by -1:
\[ 16t^2 - 188t + 92 = 0 \]
Now we will use the quadratic formula to find the values of \( t \). The quadratic formula is:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, \( a = 16 \), \( b = -188 \), and \( c = 92 \).
Now, let's calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-188)^2 = 35344 \] \[ 4ac = 4 \cdot 16 \cdot 92 = 7392 \] \[ b^2 - 4ac = 35344 - 7392 = 27952 \]
Now we can find the square root of the discriminant:
\[ \sqrt{27952} \approx 167.2 \]
Now we can substitute back into the quadratic formula:
\[ t = \frac{-(-188) \pm 167.2}{2 \cdot 16} \]
Calculating the two possible values for \( t \):
-
For the plus sign: \[ t_1 = \frac{188 + 167.2}{32} = \frac{355.2}{32} \approx 11.1 \]
-
For the minus sign: \[ t_2 = \frac{188 - 167.2}{32} = \frac{20.8}{32} \approx 0.65 \]
Thus, the two values of \( t \) for which the rocket's height is 92 feet are approximately:
\[ t \approx 0.65 \text{ seconds and } t \approx 11.1 \text{ seconds.} \]
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