# Year 10 Maths Find the length and width that will maximize the area of rectangle

#### liang123993

##### New member

The question is in the image. Working out with every step would be much appreciated.

#### Greg

##### Perseverance
Staff member
Here's a start:

Let $W$ be width, $L$ be length an $A$ be the desired area. Then,

$$\displaystyle 5W+2L=550$$

$$\displaystyle LW=A$$

Can you make any progress from there?

#### Greg

##### Perseverance
Staff member
Here's a start:

Let $W$ be width, $L$ be length an $A$ be the desired area. Then,

$$\displaystyle 5W+2L=550$$

$$\displaystyle LW=A$$

Can you make any progress from there?
$$\displaystyle W=\frac AL$$

$$\displaystyle \frac{5A}{L}+2L=550$$

$$\displaystyle 5A+2L^2=550L$$

$$\displaystyle A=110L-\frac{2L^2}{5}$$

$A$ has a maximum at the vertex of this inverted parabola, so $L=\frac{275}{2}$. Finding $A$ and $W$ from here should be straightforward.