wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

A rectangular garden has a perimeter of 120ft.

How do you find an equation for the area of the rectangle as a function of the width, then determine the length and width of the rectangle which provide the maximum area?


Open in App
Solution

Finding the length and width of the given rectangular garden:

Step-1: Assumption:

Let the length and width of the rectangular garden be lft and wft respectively.

Step-2: Finding the relation between the length and the width of the rectangle using its perimeter

We know that the perimeter of a rectangle of length l unit and width w unit is S=2l+w unit.

So, the perimeter of the given rectangular garden is S=2l+wft.

Now, given that the perimeter is 120ft.

So, we must get:

S=1202l+w=120l+w=1202l=60-w

Step-3: Finding the area as a function of the width

We know that the area of a rectangle of length l unit and width w unit is A=lw square unit.

Here, the width is wft and the length is (60-w)ft.

So, the area will be: w60-w=60w-w2ft2.

Suppose, fw=60w-w2.

This is required function of w which calculates the area of the rectangle and it is the function to be maximized.

Step-4: Finding the critical points of fw

We know that the critical points of fw will be the solution of f'w=0.

Now, differentiating fw with respect to w, we get: f'w=60-2w.

Now,

f'w=060-2w=02w=60w=602w=30

Hence, w=30 is the critical point of fw.

Step-5: Checking if w=30 is a point of maximum of the function fw

We know that w=a is a point of maximum of the function fw if f''a<0.

Now, differentiating f'w with respect to w, we get: f''w=-2.

Then, at w=30, we have: f''30=-2<0.

Hence, w=30 is a point of maximum of the function fw.

Step-6: Finding the length and the width

Thus, the width of the rectangle for which the area will be maximum is w=30ft.

Now, we can find the length using the relation l=60-w. So, the length will be:

l=60-w=60-30=30ft

Therefore, the length and the width of the given rectangular garden are: l=30ft and w=30ft respectively i.e. in fact the garden will be square shaped.


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Ellipse and Terminologies
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon