To find the rate at which the area of the rectangle is increasing, we can use the formula for the area of a rectangle, which is A = length × width.
Let's denote:
- the length of the rectangle as L(t)
- the width of the rectangle as W(t)
- the area of the rectangle as A(t)
Given that the length is increasing at a rate of 3 cm/s and the width is increasing at a rate of 7 cm/s, we can express these rates as the derivatives dL/dt and dW/dt, where t represents time.
We can also express the area A as a function of time: A(t) = L(t) × W(t).
To find the rate at which the area A is changing with respect to time t (dA/dt), we'll use the product rule of differentiation.
The product rule states:
If y = u × v, where u and v are functions of t, then
dy/dt = u × dv/dt + v × du/dt.
Applying the product rule to our scenario:
A'(t) = L(t) × dW/dt + W(t) × dL/dt
Given that the length is 15 cm and the width is 8 cm, we can find the values of dL/dt and dW/dt by substituting these values into the equation.
A'(t) = 15 × 7 + 8 × 3
A'(t) = 105 + 24
A'(t) = 129 cm^2/s
Therefore, when the length is 15 cm and the width is 8 cm, the area of the rectangle is increasing at a rate of 129 cm^2/s.