A square is increasing in area at a rate of 20 mm² each second. Calculate the rate of change of each side when it's 1,000 mm long. O 0.02 mm/s O.50 mm/s O 0.01 mm/s O 100 mm/s
We know that the formula for the area of a square is A = s², where A is the area and s is the length of a side.
We can take the derivative of both sides with respect to time:
dA/dt = 2s(ds/dt)
Where dA/dt is the rate of change of area, ds/dt is the rate of change of the length of a side, and we want to solve for ds/dt.
Given that the area is increasing at a rate of 20 mm²/s, we have:
dA/dt = 20 mm²/s
When the length of a side is 1000 mm, the area is:
A = s² = (1000 mm)² = 1,000,000 mm²
Substituting into the equation above, we get:
20 mm²/s = 2(1000 mm)(ds/dt)
Simplifying:
ds/dt = 0.01 mm/s
Therefore, the rate of change of each side is 0.01 mm/s when the square is 1000 mm long.
The answer is: O 0.01 mm/s.
To find the rate of change of each side of the square when it has a length of 1,000 mm, first, we need to determine the relationship between the area and the side length of a square.
Let's denote the side length as "s" and the area as "A". The formula for the area of a square is given by A = s^2.
We are given that the area is increasing at a rate of 20 mm²/s. Therefore, dA/dt (the rate of change of the area with respect to time) is 20 mm²/s.
To find the rate of change of each side length, ds/dt (the rate of change of the side length with respect to time), we need to differentiate the area formula with respect to time.
Differentiating both sides of the equation A = s^2 with respect to t, we get:
dA/dt = 2s * ds/dt
Substituting the given values, we have:
20 mm²/s = 2(1000 mm) * ds/dt
Now we can solve for ds/dt:
ds/dt = 20 mm²/s / 2000 mm
ds/dt = 0.01 mm/s
Therefore, the rate of change of each side length when it is 1,000 mm long is 0.01 mm/s. Hence, the correct option is O 0.01 mm/s.