Problem 3 - Calculus Early Transcendentals

Question

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?

Step-by-Step Solution

Answer

Answer: When two opposite sides increase in length from 'a' to 'x', the other two sides must change from 'b' to 'y', where y = \frac{a * b}{x}. This means that as 'x' increases, the value of 'y' must decrease by the factor of \frac{a * b}{x} to keep the area constant.

Step 1: Define variables for the sides and area of the rectangle

Let's call the original length of the two opposite sides that are changing 'a', and the original length of the other two sides 'b'. Then let's say the length of the increased sides is 'x'. We'll call the new length of the other two sides 'y'. The area of the rectangle will be represented by 'A'.

Step 2: Write the formula for the area of a rectangle

The formula for the area of a rectangle is given by the product of the lengths of its two adjacent sides. Since we want the area to remain constant, we can write the following equation to represent the situation: A = a * b = x * y

Step 3: Solve for y in terms of a, b, and x

We need to find how the other two opposite sides (y) must change. To do that, we can solve our equation from Step 2 for y, which will give us the relationship between a, b, x, and y. Let's solve for y: y = \frac{a * b}{x}

Step 4: Interpret the results

The equation from Step 3, y = \frac{a * b}{x}, tells us the length of the other two sides (y) that will make the area of the rectangle constant while the first two sides (a) increase in length to x. This means that as x increases, the value of y must decrease by the factor of \frac{a * b}{x} to keep the area constant.