Question

Catching Rainwater A 1125 -ft \(^{3}\) open-top rectangular tank with a square base \(x\) ft on a side and \(y\) ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product \(x y\). (a) If the total cost is $$c=5\left(x^{2}+4 x y\right)+10 x y$$ what values of x and y will minimize it? (b) Writing to Learn Give a possible scenario for the cost function in (a).

Short answer

The values of x and y that will minimize the cost c are both 0.

Step-by-step solution

The Cost Function

The total cost function in terms of x and y is given by c=5(x^2+4xy)+10xy. The task now is to find the values of x and y that will minimize the cost. This means finding where the derivative(s) of the function equals to 0.

Differentiation with respect to x

Differentiate c wrt x: \(\frac{dc}{dx} = 5 * (2x + 4y) + 10y\). Now set \(\frac{dc}{dx} = 0\), which gives \(10x +20y + 10y = 0\). By simplifying, this equation results in \(x +3y = 0\). Let's call this Equation [1].

Differentiation with respect to y

Now, let's differentiate c wrt y: \(\frac{dc}{dy} = 5 * (4x) + 10x\). Setting \(\frac{dc}{dy} = 0\), leads to the equation \(20x + 10x = 0\). This simplifies to the equation: \(x = 0\).

Solving the system of equations

Substitute x = 0 into Equation [1], it gives y = 0. Using these values of x and y, the cost c = 5(0^2+4*0*0)+10*0*0 = 0.

Check for minimum

To check if these values actually represent a minimum of the function, we would consider the second derivatives of the cost function with respect to x and y. In this case as our second derivatives will equal to zero, as first derivatives were constants, this confirms that our solution is indeed a minimum.