A piece of molding 153 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area?

We have been given that a piece of molding 153 cm long is to be cut to form a rectangular picture frame. We are asked to find the dimensions of frame that will enclose the largest area.

We can represent our given information in two equation as:

[tex]2w+2l=153...(1)[/tex] This is our constraint equation.

[tex]\text{Area}=l\cdot w...(2)[/tex] This is our objective equation.

Since our objective is to maximize the area, so we will convert area equation in terms of one variable as:

[tex]2(w+l)=153[/tex]

[tex]\frac{2(w+l)}{2}=\frac{153}{2}[/tex]

[tex]w+l=76.5[/tex]

[tex]w-w+l=76.5-w[/tex]

[tex]l=76.5-w[/tex]

Upon substituting this value in area equation, we will get:

[tex]A=(76.5-w) w[/tex]

[tex]A=76.5w-w^2[/tex]

Now, we will find the 1st derivative of area equation as:

[tex]A'=76.5-2w[/tex]

Let us set our derivative equal to 0 to solve for w.

[tex]76.5-2w=0[/tex]

[tex]76.5-2w+2w=0+2w[/tex]

[tex]76.5=2w[/tex]

[tex]\frac{76.5}{2}=\frac{2w}{2}[/tex]

[tex]38.25=w[/tex]

Upon substituting [tex]w=38.25[/tex] in [tex]l=76.5-w[/tex], we will get:

[tex]l=76.5-38.25[/tex]

[tex]l=38.25[/tex]

Therefore, the length and width of the frame would be 38.25 cm each and these dimensions will enclose the largest area.