Answer to Question #251527 in Real Analysis for peter

Question

From a piece of wire 30 cm long, it is required to bend a rectangle of the largest area. What are the dimensions of this rectangle

Answer 1

Let $x=$ the width of the rectangle.

Then the length of the rectangle will be $\dfrac{30-2x}{2}=15-x.$

The area of the rectangle is

$$A=A(x)=x(15-x), 0<x<15$$

$$A(x)=15x-x^2=-(x^2-15x+56.25)+56.25$$

$$=-(x-7.5)^2 +56.25$$

Vertex $(7.5, 56.25).$

The quadratic function $A(x)$ has the absolute maximum with value of $56.25$ at $x=7.5.$

width $7.5$ cm

length $7.5$ cm