A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company models the logo on an x-y plane as shown in the diagram. Use calculus to find the maximum area of the rectangle. Fully justify your answer.

A-Level AQA Maths Mechanics Quantities, Units & Modelling: A company is designing a logo. T

A company is designing a logo - AQA - A-Level Maths Mechanics - Question 13 - 2018 - Paper 1

Question 13

A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company models t... show full transcript

Worked Solution & Example Answer:A company is designing a logo - AQA - A-Level Maths Mechanics - Question 13 - 2018 - Paper 1

Step 1

Identify Variables

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Answer

Let the width of the rectangle be denoted as $2x$ and the height as $2y$ , where $x$ and $y$ are the distances from the center of the circle to the rectangle's sides. Therefore, the area $A$ of the rectangle can be expressed as:

$A = 2x imes 2y = 4xy$

The circle's equation is given by the relation:

$x^2 + y^2 = 16$

This is derived from the circle of radius 4 inches.

Step 2

Eliminate Variable

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Answer

From the circle's equation $x^2 + y^2 = 16$ , we can express $y$ in terms of $x$ :

$y = rac{ ext{sqrt}(16 - x^2)}{1}$

Substituting this expression into the area formula:

$A = 4x imes rac{ ext{sqrt}(16 - x^2)}{1} = 4x ext{sqrt}(16 - x^2)$

Step 3

Differentiate the Area Expression

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Answer

To find the maximum area, we need to differentiate $A$ with respect to $x$ :

$A' = 4 rac{d}{dx} ig( x ext{sqrt}(16 - x^2) \ ig)$

Using the product rule, we can differentiate to find critical points.

Step 4

Set Derivative to Zero

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Answer

Setting $A'$ to zero gives:

$rac{dA}{dx} = 4 ext{sqrt}(16 - x^2) - rac{4x^2}{ ext{sqrt}(16 - x^2)} = 0$

This leads to:

$4(16 - x^2) - 4x^2 = 0$

Simplifying reveals:

ightarrow x^2 = 8 ightarrow x = ext{sqrt}(2)$$

Step 5

Calculate Maximum Area

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Answer

To find the height at this value of $x$ :

$y = ext{sqrt}(16 - x^2) = ext{sqrt}(16 - 8) = ext{sqrt}(8)$

Thus, substituting into the area formula yields:

$A = 4xy = 4(2)(2) = 32 ext{ square inches}$

A company is designing a logo - AQA - A-Level Maths Mechanics - Question 13 - 2018 - Paper 1

Worked Solution & Example Answer:A company is designing a logo - AQA - A-Level Maths Mechanics - Question 13 - 2018 - Paper 1

Identify Variables

Eliminate Variable

Differentiate the Area Expression

Set Derivative to Zero

Calculate Maximum Area

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