Question

Question

A bridge is being built between two 50-meter cliff edges. The contractor is contemplating whether to use the parabolic model y=0.0015x2−0.5x+50
when designing and constructing the bridge, where x is the distance between the two cliffs (in meters) and y is the height (in meters). At the bridge’s lowest point, there must be a 10-meter clearance from the gorge below and the bottom of the bridge. Decide whether the contractor should use this model and state why.(1 point)
Responses

Yes, the contractor should use the model, because the bottom of the bridge will be about 167 meters above the gorge, which will allow for the proper clearance.
Yes, the contractor should use the model, because the bottom of the bridge will be about 167 meters above the gorge, which will allow for the proper clearance.

Yes, the contractor should use the model, because the bottom of the bridge will be 200 meters above the gorge, which will allow for proper clearance.
Yes, the contractor should use the model, because the bottom of the bridge will be 200 meters above the gorge, which will allow for proper clearance.

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.
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.

Yes, the contractor should use the model, because the bottom of the bridge will be about 45 meters above the gorge, which will allow for the proper clearance.

Answer 1

To determine whether the contractor should use the given parabolic model for the bridge, we need to find the minimum height (y-value) of the parabola, as it will represent the lowest point of the bridge.

The given parabolic equation is:

\[ y = 0.0015x^2 - 0.5x + 50 \]

To find the x-coordinate of the vertex (which gives us the minimum value of y), we can use the vertex formula for a parabola \( y = ax^2 + bx + c \):

The x-coordinate of the vertex is given by:

\[ x = -\frac{b}{2a} \]

Here, \( a = 0.0015 \) and \( b = -0.5 \).

Now we can calculate this:

\[ x = -\frac{-0.5}{2 \cdot 0.0015} = \frac{0.5}{0.003} = 166.67 \text{ meters} \]

Next, we will use this x-value to find the corresponding y-value:

\[ y = 0.0015(166.67)^2 - 0.5(166.67) + 50 \]

Calculating \( (166.67)^2 \):

\[ (166.67)^2 \approx 27777.78 \]

Now substitute back:

\[ y = 0.0015 \times 27777.78 - 0.5 \times 166.67 + 50 \]

Calculating these:

\[ y \approx 41.67 - 83.335 + 50 \approx 8.335 \text{ meters} \]

Thus, at the lowest point of the bridge, the height would be approximately 8.335 meters above the gorge.

Now, since the required clearance from the gorge is 10 meters, the parabolic model does not provide sufficient clearance for safety.

Therefore, the correct decision is:

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.