Did you know?
In a rectangle, the ratio of the width to the length always remains constant even as it expands or contracts. For instance, if the ratio of width to length is always 2 to 5, it means that for every 2 units increase in width, there will be a corresponding increase of 5 units in length.
Here's an interesting application: If the perimeter of the rectangle is increasing at a rate of 10 cm/s, and the perimeter is currently 35 cm, we can determine the rate of increase of the rectangle's area using the given ratio. By dividing the perimeter by 2, we find that the sum of the width and length is currently 17.5 cm.
To find the width and length separately, we multiply each by the ratio's constant factor: 2/7. This gives us a width of 5 cm and a length of 12.5 cm.
To calculate the rate of increase in the area, we use the formula A = w * l, where A is the area, w is the width, and l is the length. Therefore, at the current moment, the area of the rectangle is 62.5 cm².
Now, we can determine the rate of increase of the area. As the perimeter increases at a rate of 10 cm/s, the width and length both increase at half that rate, which is 5 cm/s. By finding the derivative of the area formula, we get:
dA/dt = (dw/dt) * l + w * (dl/dt)
Plugging in the known values, we have:
dA/dt = 5 cm/s * 12.5 cm + 5 cm * 0 cm/s
Simplifying this equation, we find that the rate of increase of the area is 62.5 cm²/s.
