Discussion
A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equation y = 2x - 0.1x2 where y is the height of the arch in meters. The maximum possible height of the arch is
Answer: B
The maximum height is 10 meters.
Differentiate the profile and set the slope to zero:
\[ \begin{aligned} \frac{dy}{dx} &= 2-0.2x = 0 \Rightarrow x = 10 \\ y_{max} &= 2(10)-0.1(10)^{2} = 10\ \text{m} \end{aligned} \]
The second derivative, -0.2, is negative, confirming a maximum. The arch meets the ground at x = 0 and x = 20 m, and the top is midway.
12 and 14 m exceed the largest value the parabola can reach. 8 m is reached at x = 5.53 m and again at x = 14.47 m, on the way up and down, not at the peak.
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