# Rate/Relations Calc I Area of Rectangle

• QuarkCharmer
In summary, the problem involves finding the rate at which the area of a rectangle is increasing when its length and width are also increasing. The given equation is A = LW and the approach is to implicitly differentiate it to get A' = L'W + W'L. Plugging in the values of the rates of change for length and width gives the final answer of 140cm^2/s when the length is 20cm and the width is 10cm.
QuarkCharmer

## Homework Statement

Stewart Calculus 6E, 3.8 #4
4.) The length of a rectangle is increasing at a rate of 8cm/s and its width is increasing at a rate of 3cm/s. When the length is 20cm and the width is 10cm, how fast is the area of the rectangle increasing?

A = LW

## The Attempt at a Solution

I am not entirely sure how to set this problem up. I think I would start with A=LW, and then implicitly differentiate to:

$$A' = L'W + W'L$$

Then perhaps, plug in my values for the change in length and width as so:
$$A' = 8W + 3L$$

But that doesn't seem right?

I think that is correct.

So, in that equation the W and L non prime can be put, so that:
$$A' = 8(10)+3(20)$$

So the rate of the area when the sides are width:10 length 20 is 140cm^2/s ??

That should be correct or how I would do the question.

## 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. This can also be written as A = lw.

## 2. How do you calculate the area of a rectangle if you only know the perimeter?

If you only know the perimeter of a rectangle, you can use the formula P = 2(l + w) to find the sum of the length and width. Then, you can divide the perimeter by 2 and subtract the sum of length and width to find the length and width individually. Once you have the length and width, you can use the formula A = lw to calculate the area.

## 3. Can the area of a rectangle be negative?

No, the area of a rectangle cannot be negative. It is a measure of the space inside the rectangle and is always a positive value.

## 4. How does the area of a rectangle change when the length and width are increased or decreased?

The area of a rectangle is directly proportional to the length and width. This means that as the length and width are increased, the area will also increase. Similarly, if the length and width are decreased, the area will decrease as well.

## 5. How is the area of a rectangle related to other shapes?

The area of a rectangle is similar to the area of other shapes, such as squares, parallelograms, and triangles. The only difference is in the formulas used to calculate the area. For example, the area of a square is A = s^2, where s is the length of one side, while the area of a triangle is A = 1/2bh, where b is the base and h is the height.

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