The length l of a rectangle is decreasing at the rate of 3cm/sec, while its width w is increasing at the rate of 3cm/sec. Find the rates of change of the perimeter, and the length of one diagonal at the instant when l=15 and w=6.

given : dl/dt = 3 cm/s, dw/dt 3 cm/s

P = 2l + 2w
dP/dt = 2 dl/dt + 2 dw/dt
plug in the values

d^2 = l^2 + w^2

I assume you want the rate of change of the perimeter, not just its length. Your wording is confusing.

2d dd/dt = 2l dl/dt + 2w dw/dt
you are given l and w, so you can find d.
plug in the values to find dd/dt

To find the rates of change of the perimeter and the length of one diagonal of the rectangle, we can use the formulas for these measurements in terms of the length and width.

The perimeter of a rectangle is given by P = 2(l + w), where l is the length and w is the width.

The length of the diagonal of a rectangle can be found using the Pythagorean theorem:
d^2 = l^2 + w^2, where d is the diagonal length.

Given that the length l is decreasing at a rate of 3 cm/sec and the width w is increasing at a rate of 3 cm/sec, we can differentiate the above formulas with respect to time (t) to find the rates of change.

1. Rate of change of the perimeter (dP/dt):
Differentiating P = 2(l + w) with respect to t, we get:
dP/dt = 2(dl/dt + dw/dt)

Given that dl/dt = -3 cm/sec (length is decreasing) and dw/dt = 3 cm/sec (width is increasing), we can substitute these values to find the rate of change of the perimeter.

dP/dt = 2(-3 + 3) = 0 cm/sec

Therefore, the rate of change of the perimeter is 0 cm/sec. This means that as the length decreases, the width increases in such a way that the change in their sum cancels out, resulting in no change in the perimeter.

2. Rate of change of the diagonal length (dd/dt):
Differentiating d^2 = l^2 + w^2 with respect to t, we get:
2d(dd/dt) = 2l(dl/dt) + 2w(dw/dt)

Given that dl/dt = -3 cm/sec, dw/dt = 3 cm/sec, l = 15 cm, and w = 6 cm, we can substitute these values to find the rate of change of the diagonal length.

2d(dd/dt) = 2(15)(-3) + 2(6)(3)
2d(dd/dt) = -90 + 36
2d(dd/dt) = -54
dd/dt = -27 cm/sec

Therefore, the rate of change of the diagonal length is -27 cm/sec. This means that at the instant when l = 15 cm and w = 6 cm, the length of one diagonal is decreasing at a rate of 27 cm/sec.

In summary:
- The rate of change of the perimeter is 0 cm/sec.
- The rate of change of the diagonal length is -27 cm/sec.