Find the rate of change of the area of a square with respect to the length of a side, when the side is 5cm long
The correct answer and explanation is:
The rate of change of the area of a square with respect to the length of a side can be found by differentiating the area formula with respect to the side length.
The area AA of a square is given by: A=s2A = s^2
Where ss is the length of a side of the square. To find the rate of change of the area with respect to the side length, we differentiate AA with respect to ss: dAds=dds(s2)\frac{dA}{ds} = \frac{d}{ds}(s^2)
Using the power rule of differentiation: dAds=2s\frac{dA}{ds} = 2s
This means that the rate of change of the area of the square with respect to the length of a side is 2s2s, where ss is the side length.
Now, we are asked to find the rate of change when the side is 5 cm long. Substituting s=5s = 5 into the equation: dAds=2(5)=10 cm2/cm\frac{dA}{ds} = 2(5) = 10 \, \text{cm}^2/\text{cm}
Therefore, the rate of change of the area of the square with respect to the length of a side when the side is 5 cm long is 10 cm2/cm10 \, \text{cm}^2/\text{cm}.
In simpler terms, for every 1 cm increase in the side length, the area of the square increases by 10 square centimeters when the side is 5 cm long. This result shows how sensitive the area of a square is to changes in the side length, highlighting the quadratic nature of the area formula.