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Chapter 3: Problem 3

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

Short Answer

Expert verified
Answer: To maintain a constant area, the new width should be \frac{l 脳 w}{(l + x)}, where l is the initial length, w is the initial width, and x is the increase in the length.

Step by step solution

01

Introduce the variables

Let the length of the rectangle be l and the width be w. As given, two opposite sides increase in length, let's say that these are the length sides of the rectangle, so the new length will be l + x, where x is the increase in length. We need to find the change in width required to keep the area constant.
02

Write the initial area formula of the rectangle

The area of a rectangle is given as A = l 脳 w. The initial area of the rectangle will be A = l 脳 w.
03

Write the final area formula of the rectangle with the increased length

Now, we need to find the new width of the rectangle such that its area remains constant after the increase in length. Let's denote the new width as w_new. The area of the rectangle with the increased length will be A = (l + x) 脳 w_new.
04

Set up the equation to maintain constant area

Given that the area of the rectangle must remain constant, A = l 脳 w = (l + x) 脳 w_new. Now we will solve for w_new.
05

Solve for the new width of the rectangle

We have the equation, l 脳 w = (l + x) 脳 w_new. Divide both sides by (l + x): w_new = \frac{l 脳 w}{(l + x)} This equation represents the relationship between the new width (w_new) and the initial dimensions (l, w) and the increase in length (x) that keeps the area of the rectangle constant.
06

Conclusion

To keep the area of a rectangle constant, when the length is increased by x, the new width (w_new) should be \frac{l 脳 w}{(l + x)}, where l is the initial length, w is the initial width, and x is the increase in the length.

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