Question

the area of a rectangle, whose length is twice its width, is increasing at the rate of 8cm^2/sec. Find the rate at which the length is increasing when the width is 5cm.

Answer 1

The rate at which the length is increasing when the width is 5cm is 0.8 cm/sec. This result was found by using the related rates method in calculus, considering that the area of the rectangle is increasing at a rate of 8 cm^2/sec and the length is twice the width.

Step-by-step explanation:

To find the rate at which the length of a rectangle is increasing when the width is 5cm and the area is increasing at a rate of 8 cm2/sec, we use related rates in calculus. Since the length is twice the width, let the width be w and the length be 2w. The area A of the rectangle can then be expressed as A = w × 2w = 2w2.

Given that the area is increasing at a rate of 8 cm2/sec, we can write dA/dt = 8. Differentiating both sides of the area equation with respect to t gives us 2 × 2w(dw/dt), or 4w(dw/dt) = 8. When w is 5 cm, we can solve for dw/dt by substituting w into the equation:

1. 4 × 5(dw/dt) = 8
2. 20(dw/dt) = 8
3. dw/dt = 8 / 20
4. dw/dt = 0.4 cm/sec

Since l = 2w, the rate of change of the length dl/dt will be twice the rate of change of the width, dl/dt = 2 × dw/dt. Therefore, dl/dt = 2 × 0.4 cm/sec = 0.8 cm/sec.

The length of the rectangle is increasing at a rate of 0.8 cm/sec when the width is 5cm.