Question

cost, Revenue, and Profit A company can manufacture \(x\) items at a cost of \(c(x)\) dollars, a sales revenue of \(r(x)\) dollars and a profit of \(p(x)=r(x)-c(x)\) dollars (all amounts in thousands). Find \(d c / d t, d r / d t,\) and \(d p / d t\) for the following values of \(x\) and \(d x / d t\) (a) \(r(x)=9 x, \quad c(x)=x^{3}-6 x^{2}+15 x\) and \(d x / d t=0.1\) when \(x=2 .\) (b) \(r(x)=70 x, \quad c(x)=x^{3}-6 x^{2}+45 / x\) and \(d x / d t=0.05\) when \(x=1.5\)

Short answer

The solutions to part (a) are \(dc/dt = 3\), \(dr/dt = 0.9\) and \(dp/dt = 0.6\) while for part (b) the solutions are \(dc/dt = -7.5\), \(dr/dt = 3.5\) and \(dp/dt = 11\).

Step-by-step solution

Calculate the Derivatives for Part (a)

First, calculate the derivatives of the cost and revenue functions with respect to \(x\): \nThe derivative of \(c(x)\) is \(dc/dx = 3x^2 - 12x + 15\). The derivative of \(r(x)\) is \(dr/dx = 9\). Now, the chain rule can be applied to find \(dc/dt\) and \(dr/dt\) by multiplying \(dc/dx\) and \(dr/dx\) respectively by \(dx/dt\): \(dc/dt = dc/dx * dx/dt = (3x^2 - 12x + 15)(0.1)\) \(dr/dt = dr/dx * dx/dt = 9(0.1)\). For \(x = 2\), these become:\(dc/dt = 3*2^2 - 12*2 + 15 = 3\) and \(dr/dt = 9*0.1 = 0.9\). Then, using the given formula for profit, \(dp/dx = dr/dx - dc/dx\), So \(dp/dt = dp/dx * dx/dt = (9 - (3*2^2 - 12*2 + 15))(0.1) = 0.6\).

Calculate the Derivatives for Part (b)

Now, calculate the derivatives of the cost and revenue functions with respect to \(x\) for part (b).The derivative of \(c(x)\) is \(dc/dx = 3x^2 - 12x - 45/x^2\). The derivative of \(r(x)\) is \(dr/dx = 70\). Apply the chain rule to find \(dc/dt\) and \(dr/dt\) by multiplying \(dc/dx\) and \(dr/dx\) respectively by \(dx/dt\): \(dc/dt = dc/dx * dx/dt = (3x^2 - 12x - 45/x^2)(0.05)\) \(dr/dt = dr/dx * dx/dt = 70(0.05)\). For \(x = 1.5\), these become:\(dc/dt = 3*1.5^2 - 12*1.5 - 45/1.5^2 = -7.5\) and \(dr/dt = 70*0.05 = 3.5\). Using the given formula for profit, \(dp/dx = dr/dx - dc/dx\), So \(dp/dt = dp/dx * dx/dt = (70 - (3*1.5^2 - 12*1.5 - 45/1.5^2))(0.05) = 11\).