To determine whether the contractor should use the parabolic model \( y = 0.0015x^2 - 0.5x + 50 \), we need to find the lowest point (the vertex) of the parabolic equation, as this will indicate the height of the bridge at its lowest point.
The vertex of a parabola described by the equation \( y = ax^2 + bx + c \) occurs at the x-value given by:
\[ x = -\frac{b}{2a} \]
In our equation, \( a = 0.0015 \) and \( b = -0.5 \). Plugging in these values:
\[ x = -\frac{-0.5}{2 \cdot 0.0015} = \frac{0.5}{0.003} \approx 166.67 \text{ meters} \]
Now, we need to evaluate the height \( y \) at this x-value to find the lowest point of the bridge. We substitute \( x \) back into the original equation:
\[ y = 0.0015(166.67)^2 - 0.5(166.67) + 50 \]
Calculating \( 166.67^2 \):
\[ 166.67^2 \approx 27778.89 \]
Calculating \( 0.0015 \times 27778.89 \):
\[ 0.0015 \times 27778.89 \approx 41.67 \]
Calculating \( -0.5 \times 166.67 \):
\[ -0.5 \times 166.67 \approx -83.335 \]
Now we add these values together:
\[ y \approx 41.67 - 83.335 + 50 \approx 8.335 \]
Thus, the lowest point of the bridge is approximately \( 8.3 \) meters.
Since the required clearance is \( 10 \) meters, and the calculated height at the lowest point of the bridge is approximately \( 8.3 \) meters, which is less than \( 10 \) meters, we conclude that:
No, the contractor should not use the model, because the bottom of the bridge will only be about 8.3 meters above the gorge, which will not allow for the proper clearance.
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