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

liang123993

New member
Aug 12, 2019
1
unknown.png

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

Greg

Perseverance
Staff member
Feb 5, 2013
1,382
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
Feb 5, 2013
1,382
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.