Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semircle - Edexcel - A-Level Maths Pure - Question 3 - 2005 - Paper 2

To find the stationary points, we will differentiate the area function A with respect to x and set the derivative to zero:

$\frac{dA}{dx} = 80 - 2(2 + \frac{ ext{π} }{ 2 })x$

Setting the derivative equal to zero:

$80 - 2(2 + \frac{ ext{π} }{ 2 })x = 0$

Solving for x gives:

$x = \frac{80}{2(2 + \frac{ ext{π} }{ 2 })}$

This value indicates where the area A has a stationary value.

To determine if the stationary point is a maximum, we analyze the second derivative:

Calculating the second derivative:

$\frac{d^2A}{dx^2} = -2(2 + \frac{ ext{π} }{ 2 })$

Since this is negative, it indicates that the area function A is concave down at this stationary point, confirming that it is indeed a maximum.

Substituting the value of x back into the area function:

$A = 80 \left( \frac{80}{2(2 + \frac{ ext{π} }{ 2 })} \right) - (2 + \frac{ ext{π} }{ 2 }) \left( \frac{80}{2(2 + \frac{ ext{π} }{ 2 })} \right)²$

After computing this expression, the maximum area A evaluates to approximately 448 m². Thus, the maximum area of the stage, rounded to the nearest m², is:

448 m².