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

• MHB
• liang123993
In summary, the conversation discusses using the equations 5W+2L=550 and LW=A to determine the width and length of a desired area. It is suggested to solve for one variable in terms of the other to find the maximum area at the vertex of an inverted parabola.
liang123993
View attachment 9234

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

Attachments

• unknown.png
6 KB · Views: 67
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 said:
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.

1. What is the formula for finding the area of a rectangle?

The formula for finding the area of a rectangle is length x width.

2. How do I find the length and width that will maximize the area of a rectangle?

To find the length and width that will maximize the area of a rectangle, you can use the formula for area (length x width) and set it equal to a variable. Then, take the derivative of the variable and set it equal to 0 to find the critical points. The length and width at the critical points will maximize the area.

3. Can I use any units for length and width when finding the area of a rectangle?

Yes, you can use any units for length and width as long as they are consistent. For example, if you use meters for length, you must also use meters for width.

4. What is the difference between length and width?

Length refers to the longer side of a rectangle, while width refers to the shorter side. In the formula for area, length is represented by the longer side and width is represented by the shorter side.

5. How do I know if my calculated length and width are correct?

You can check if your calculated length and width are correct by plugging them back into the formula for area and seeing if it matches the given area. You can also graph the rectangle with the calculated length and width to visually confirm if it maximizes the area.

• General Math
Replies
21
Views
2K
• General Math
Replies
3
Views
1K
• General Math
Replies
1
Views
1K
• General Math
Replies
21
Views
3K
• General Math
Replies
10
Views
2K
• General Math
Replies
2
Views
1K
• General Math
Replies
10
Views
1K
• General Math
Replies
4
Views
1K
• General Math
Replies
6
Views
2K
• General Math
Replies
8
Views
5K