the length of a rectangle is increasing at a rate of and its width is increasing at a rate of . when the length is and the width is how fast is the area of the rectangle increasing?

the length of a rectangle is increasing at a rate of and its width is increasing at a rate of . when the length is and the width is how fast is the area of the rectangle increasing?

A rectangle is a two-dimensional shape, a quadrilateral with all four right angles. It follows from this that the rectangle has two pairs of opposite and equal sides; That is, a rectangle is a special case of a parallelogram whose angles are all right. A square is also a special case of a rectangle in which the lengths of all four sides are equal.

the length of a rectangle is increasing at a rate of and its width is increasing at a rate of . when the length is and the width is how fast is the area of the rectangle increasing?

When is a quadrilateral a rectangle?
We say of a simple quadrilateral that it is a rectangle if and only if one of the conditions is met:

All angles are equal.
All angles are straight.
As the length of the diagonal are equal.
The rectangle ABCD and the two triangles produced when we set the diameter: ABD and CDA are identical.

The length of a rectangle is increasing at a rate of 9 cm per s and its width is increasing at a rate of 7 cm per s. When the length is 15 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

$140 {cm}^{2} / s$

Explanation:

We are told that:

$\frac{d l}{d t} = 8$ cm/s (const), and, $\frac{d w}{d t} = 3$ cm/s (const)

The Area of the rectangle is:

$A = l w$

Differentiating wrt $t$ (using the product rule) we get;

$\frac{d A}{d t} = (l) (\frac{d w}{d t}) + (\frac{d l}{d t}) (w)$
$∴ \frac{d A}{d t} = 3 l + 8 w$

So when $l = 20$ and $w = 10 \Rightarrow$
$\frac{d A}{d t} = 3 \cdot 20 + 8 \cdot 10$
$∴ \frac{d A}{d t} = 60 + 80$
$∴ \frac{d A}{d t} = 140 {cm}^{2} / s$