Final answer:
To find the rate of change of the perimeter of the square, we need to differentiate the perimeter formula with respect to time and solve for the rate of change of the side length. Then, we can substitute the rate of change of the side length into the formula for the perimeter to find the rate of change of the perimeter. The rate of change of the perimeter of the square can be calculated using the expression ds/dt = 15 / (2s).
Step-by-step explanation:
To find the rate of change of the perimeter of the square, we need to differentiate the perimeter formula with respect to time. The perimeter of a square can be expressed as P = 4s, where s is the length of one side. Since the area of the square is increasing at a rate of 15 square inches per minute, we can express this as dA/dt = 15. We can relate the area and the side length using the formula A = s^2. Taking the derivative of both sides with respect to time, we have dA/dt = 2s(ds/dt). Substituting the given rate of change of the area and the equation for the area, we can solve for ds/dt, which will give us the rate of change of the side length. Finally, we can substitute the rate of change of the side length into the formula for the perimeter to find the rate of change of the perimeter.
Let's solve step by step:
- Given: dA/dt = 15 square inches per minute
- Given: A = 11 square inches
- Express the relationship between A and s: A = s^2
- Take the derivative of both sides with respect to time: dA/dt = 2s(ds/dt)
- Substitute the given values: 15 = 2s(ds/dt)
- Solve for ds/dt: ds/dt = 15 / (2s)
- Substitute the value of s into the formula for the perimeter: P = 4s
- Substitute the value of ds/dt into the formula for the perimeter to find the rate of change of the perimeter.
The rate of change of the perimeter of the square is given by the expression ds/dt = 15 / (2s). Substituting the value of s, we can calculate the rate of change of the perimeter.