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

[liang123993]

View attachment 9234

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

 

[Greg]

Here's a start:

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

$$5W+2L=550$$

$$LW=A$$

Can you make any progress from there?

 

[Greg]

Here's a start:

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

$$5W+2L=550$$

$$LW=A$$

Can you make any progress from there?

$$W=\frac AL$$

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

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

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

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