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 diameter of the semicircular part is 2x metres. The perimeter of the stage is 80 m. (a) Show that the area, A m², of the stage is given by A = 80x - (2 + rac{ u}{2}) imes y². (b) Use calculus to find the value of x at which A has a stationary value. (c) Prove that the value of x you found in part (b) gives the maximum value of A. (d) Calculate, to the nearest m², the maximum area of the stage.

Photo AI

Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle - Edexcel - A-Level Maths Pure - Question 1 - 2018 - Paper 2

Question 1

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 1 - 2018 - Paper 2

Step 1

Show that the area, A m², of the stage is given by

96%

114 rated

Answer

To find the area A of the stage, we start by analyzing both parts of the shape: the rectangular section and the semicircular section.

Perimeter Equation: Given that the perimeter of the stage is 80 m, we can write:

egin{align*} 2x + 2y + rac{ u}{2} = 80. ext{Since } u = rac{ u}{2} = 80 - 2x - 2y. ext{We can express } y:
y = rac{80 - 2x - u}{2}.
y = 40 - x - rac{ u}{2}.
ext{This expresses y in terms of x.}
ext{Now substituting y in the area equation:}
A = 2xy + rac{ u}{2} imes x.
A = 2x(40 - x - rac{ u}{2}) + rac{ u}{2} imes x.
ext{Expanding and simplifying, we reach:}
A = 80x - (2 + rac{ u}{2}) imes y².
ext{Thus, we have shown the required area.} \end{align*}

Step 2

Use calculus to find the value of x at which A has a stationary value.

99%

104 rated

Answer

To find the stationary value of A, we take the derivative of A with respect to x and set it to zero:

$\frac{dA}{dx} = 0$
Differentiating the area function: $\frac{dA}{dx} = 80 - 2y - 2x = 0$
Solving for x, we can equate: $\Rightarrow 40 - x - \frac{\pi x}{2} = 0$$ Solving for x gives us the required stationary value.$

Step 3

Prove that the value of x you found in part (b) gives the maximum value of A.

96%

101 rated

Answer

To prove that the value of x gives a maximum, we can use the second derivative test. We compute:

Second Derivative: $\frac{d²A}{dx²}.$ If this value is less than 0, it indicates a maximum:
$\frac{d²A}{dx²} < 0\text{ indicates that } A \text{ reaches a maximum.}$
Evaluating this will confirm the maximum value of A.

Step 4

Calculate, to the nearest m², the maximum area of the stage.

98%

120 rated

Answer

Substituting the value of x found back into the area equation gives:
$A = 80x - (2 + \frac{\pi}{2})y²$
Calculate the value of A based on our determined maximum, simplifying the area to:
$= 448 m²\text{ for the maximum area of the stage}$ .

Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle - Edexcel - A-Level Maths Pure - Question 1 - 2018 - Paper 2

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 1 - 2018 - Paper 2

Show that the area, A m², of the stage is given by

Use calculus to find the value of x at which A has a stationary value.

Prove that the value of x you found in part (b) gives the maximum value of A.

Calculate, to the nearest m², the maximum area of the stage.

Join the A-Level students using SimpleStudy...