The width across the bottom of the tunnel opening is 60 feet.
To find the width across the bottom of the tunnel opening, we need to identify the distance at which the height function, h(x)=− 1/30x^2 +2x, is maximized. The width corresponds to the horizontal distance between the two points where the height function intersects the x-axis.
To determine the maximum height, we can find the vertex of the quadratic function. The vertex of a quadratic function in the form ax^2+bx+c is given by the coordinates (−b/2a,f(−b/2a)). In this case, a=−1/30, b=2, and c=0. Plugging these values into the formula, we get the x-coordinate of the vertex as −(−2)/(2×(−1/30))=30.
So, the maximum height occurs at x=30. To find the width, we need to determine the distance between the two points where h(x) intersects the x-axis. These points are when h(x)=0. Setting h(x)=0, we get 1/30x^2+2x=0. Factoring out x, we get -1/30x(x−60)=0. This equation is satisfied when x=0 and x=60.
Therefore, the width across the bottom of the tunnel opening is the horizontal distance between these two points, which is 60−0=60 feet.