Applications of Derivatives: Optimization

Example 3: Find the largest rectangle which will fit within a circle with a 4 inch

radius.

What are we trying to optimize?

Maximize the area of the rectangle, A.

What are we solving for?

The dimensions of the rectangle, x and y.

NEED SKETCH.

What is the equation of the area of the rectangle?

A = xy

Is there another equation we can use to get y in terms of x? (Hint: use the sketch.)

NEED SKETCH.

x^{2} + y^{2} = 8^{2}

So y = (64 – x^{2})^{1/2}

A = x(64 – x^{2})^{1/2}

Take the derivative using the Product Rule and the Chain Rule:

dA/dx = x * 1/2 (64 – x^{2})^{-1/2}(-2x) + (1) * (64 – x^{2})^{1/2}

^{ }dA/dx = -x^{2} /(64 – x^{2})^{1/2} + (64 – x^{2})^{1/2}

0 = -x^{2} /(64 – x^{2})^{1/2} + (64 – x^{2})^{1/2}

This can be simplified by multiplying the equation by (64 – x^{2})^{1/2}

Then: 0 = -x^{2} + (64 – x^{2})

0 = -2x^{2} + 64

x = 32^{1/2} = 5.66inches

y = 32^{1/2} = 5.66inches

Another way to do this problem is with implicit differentiation:

Instead of solving for y, leave it as x^{2} + y^{2} = 64.

Now take the derivative of A= xy with respect to x:

dA/dx = x * dy/dx + (1) * y

dA/dx = x dy/dx + y

Now take the derivative of x^{2} + y^{2} = 64 with respect to x and solve for dy/dx:

2x + 2y * dy/dx = 0

dy/dx = -2x/2y = -x/y

Plug this into the equation for dA/dx and set it equal to zero:

dA/dx = x (-x/y) + y

0 = -x^{2} + y^{2}

x = y

Plug this into x^{2} + y^{2} = 64 to get

x = 32^{1/2} = 5.66inches

y = 32^{1/2} = 5.66inches