Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 5
Question 2
Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is 2x metres and the width is y metres. The... show full transcript
Worked Solution & Example Answer:Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 5
Step 1
Show that the area, A m², of the stage is given by
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Answer
To show that the area of the stage is given by the expression provided, we start from the perimeter equation:
ho}{2} = 80$$
Solving for y:
$$y = rac{80 - 2x - rac{
ho}{2}}{2}$$
Now substituting y in the area formula:
$$A = 2xy + rac{
ho}{2} x$$
Replacing y:
$$A = 2x igg( rac{80 - 2x - rac{
ho}{2}}{2}igg) + rac{
ho}{2} x$$
On simplifying this, we derive the expression for A as:
$$A = 80x - (2 + rac{
ho}{2})x^2$$
Step 2
Use calculus to find the value of x at which A has a stationary value.
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Answer
To find the stationary value, differentiate A with respect to x:
ho}{2}$$
Setting this equal to zero for stationary points:
$$80 - 4x - rac{
ho}{2} = 0$$
Solving for x gives:
$$x = rac{80 - rac{
ho}{2}}{4}$$
Step 3
Prove that the value of x you found in part (b) gives the maximum value of A.
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Answer
To confirm that this x value provides a maximum, check the second derivative:
rac{d^2A}{dx^2} = -4
Since the second derivative is negative, this indicates a maximum value at the previously found x.
Step 4
Calculate, to the nearest m², the maximum area of the stage.
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Answer
Substituting the x value into the area expression:
ho}{2}}{4}igg) - igg(2 + rac{
ho}{2}igg) igg( rac{80 - rac{
ho}{2}}{4}igg)^2$$
Calculating this, we find the maximum area A to be approximately 448 m².
