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The quadratic function describing the tunnel is \( y = -\frac{5}{16}x^2 + 5 \). The truck will not fit through the tunnel as the height at the edges is 3.75 m, which is less than the truck's height of 5 m.
Solución
To solve this problem, we need to find the quadratic function that describes the shape of the parabolic tunnel and then check if the truck can fit through it.
### a) Finding the Quadratic Function
Assuming the tunnel is symmetric and opens downwards, we can express the shape of the tunnel with a quadratic function in the form:
\[
y = ax^2 + bx + c
\]
To find the specific function, we need some key points about the tunnel. Typically, we would need the coordinates of the vertex and the width of the tunnel at a certain height. However, since we don't have specific points provided, let's assume the following:
1. The vertex of the parabola (the highest point of the tunnel) is at the origin (0, h), where h is the maximum height of the tunnel.
2. The width of the tunnel at the base (where it meets the ground) is given.
Let's assume the tunnel has a maximum height of \( h \) at \( x = 0 \) and opens downwards. If the width of the tunnel at the base is \( w \), then the points where the tunnel meets the ground can be represented as \( (-w/2, 0) \) and \( (w/2, 0) \).
For example, if the width of the tunnel at the base is 8 m (which is a common width for a tunnel), then the points would be \( (-4, 0) \) and \( (4, 0) \).
Using these points, we can set up the equations:
1. The vertex form of the parabola is given by:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex. In our case, the vertex is at \( (0, h) \).
2. The points where the tunnel meets the ground give us:
\[
0 = a(-4)^2 + h \quad \text{(for } x = -4\text{)}
\]
\[
0 = a(4)^2 + h \quad \text{(for } x = 4\text{)}
\]
From these equations, we can solve for \( a \) if we know \( h \).
Assuming \( h = 5 \) m (the height of the truck), we can set up the equation:
\[
0 = 16a + 5 \implies 16a = -5 \implies a = -\frac{5}{16}
\]
Thus, the quadratic function describing the shape of the tunnel is:
\[
y = -\frac{5}{16}x^2 + 5
\]
### b) Determining Whether the Truck Will Fit
To determine if the truck fits, we need to check the height of the tunnel at the edges of the truck. The truck is 4 m wide, so we need to evaluate the height of the tunnel at \( x = -2 \) m and \( x = 2 \) m (the edges of the truck).
Calculating the height at these points:
1. For \( x = -2 \):
\[
y = -\frac{5}{16}(-2)^2 + 5 = -\frac{5}{16}(4) + 5 = -\frac{20}{16} + 5 = -\frac{20}{16} + \frac{80}{16} = \frac{60}{16} = 3.75 \text{ m}
\]
2. For \( x = 2 \):
\[
y = -\frac{5}{16}(2)^2 + 5 = -\frac{5}{16}(4) + 5 = -\frac{20}{16} + 5 = -\frac{20}{16} + \frac{80}{16} = \frac{60}{16} = 3.75 \text{ m}
\]
The height of the tunnel at both edges of the truck is 3.75 m, which is less than the height of the truck (5 m).
### Conclusion
The truck will **not fit** through the tunnel, as the height of the tunnel at the edges of the truck is only 3.75 m, which is less than the truck's height of 5 m.
Respondido por UpStudy AI y revisado por un tutor profesional
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