can you just show me how to set this up? or a formula that could help? im confused.
The length of a rectangle is increasing at 3 ft/min and the width is decreasing at 2 ft/min. When the length is 50 ft and the width is 20 ft, at what rate is the rectangle decreasing?
Let the width of the rectangle be x ft
let the length of the rectangle be y ft
Area = xy
d(Area)/dt = x dy/dt + y dx/dt
given: dx/dt = -2 and dy/dt = 3
so when x = 20 and y = 50
d(Area)/dt = 20(3) + 50(-2)
= -40
At that moment, the area is decreasing at a rate of 40 ft^2/min
thank you so much!!!!
To find the rate at which the rectangle is decreasing, we need to use the chain rule in calculus. The chain rule states that if a variable depends on another variable that is changing, we can find the rate of change of the dependent variable by taking the derivative of the dependent variable with respect to the independent variable, and multiplying it by the rate of change of the independent variable.
In this case, the length of the rectangle is the independent variable, and the width of the rectangle is the dependent variable. We can let L represent the length of the rectangle and W represent the width of the rectangle.
Given that dL/dt = 3 ft/min (rate of change of length) and dW/dt = -2 ft/min (rate of change of width), we want to find dW/dt when L = 50 ft and W = 20 ft.
To find dW/dt, we need to find dW/dL (the rate of change of width with respect to length), and then multiply it by dL/dt.
We can find dW/dL by differentiating the equation that relates length and width, which is W = f(L) (since the width depends on the length).
Differentiating both sides of the equation with respect to L gives:
dW/dL = df(L)/dL
To solve for df(L)/dL, we need to know the specific equation that relates the length and width of the rectangle. If we have the equation, we can differentiate it and find df(L)/dL.
Once we have df(L)/dL, we can then multiply it by dL/dt to find dW/dt.
Without knowing the specific equation that relates the length and width of the rectangle, we cannot provide an exact formula to find the rate at which the rectangle is decreasing.