Question

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 $y=2x-0.1x^2$ where $y$ is the height of the arch in meters. The maximum possible height of the arch is

• A. $8$ meters
• B. $10$ meters
• C. $12$ meters
• D. $14$ meters

Answer 1

The correct option is B $10$ meters

$y=2x-\left(0.1\right)x^2,\frac{dy}{dx}=2-0.2x,\frac{d^{2}y}{d{x}^2}=-0.2$
$\frac{dy}{dx}=0,x=10$ is a critical point
and ${\left(\frac{d^{2}y}{d{x}^2}\right)}_{x=10}=-0.2<0$
$\Rightarrow y$ is maximum at $x=10$
$\&$ Maximum height $=y=2\left(10\right)-\left(0.1\right)\left(100\right)=10m$