a rectangle is 10 inches by 6 inches whose sides are changing. write a formula for both the perimeter and how fast each is changing in terms of L and W
how do i do this, an explanation instead of an answer would be most appreciated, thanks in advance.
express P and A as a function of L and W
Then just use the chain rule to find the derivative (d/dt)
To write the formula for the perimeter and how fast each side is changing in terms of the length (L) and width (W) of a rectangle, we need to understand that the perimeter of a rectangle is the sum of all its sides.
The perimeter of a rectangle can be calculated by adding twice the length (L) to twice the width (W), which can be represented as:
Perimeter = 2L + 2W
Now, to find out how fast each side is changing, we need to take the derivatives of the formula with respect to time (t). Let's assume that both L and W are changing with respect to time. Hence, we need to apply the chain rule of differentiation:
dP/dt = d(2L)/dt + d(2W)/dt
To find dL/dt and dW/dt, we can consider that the lengths of the sides L and W are both functions with respect to time (t). Therefore, we can write:
L = L(t)
W = W(t)
Now, differentiating both L and W with respect to t:
dL/dt = d(L(t))/dt
dW/dt = d(W(t))/dt
So, to summarize:
Perimeter formula: Perimeter = 2L + 2W
Rate of change of perimeter: dP/dt = 2(dL/dt) + 2(dW/dt)
Rate of change of length: dL/dt = d(L(t))/dt
Rate of change of width: dW/dt = d(W(t))/dt
By applying these formulas, you can calculate the perimeter of the rectangle and determine how fast each side is changing with respect to time.