A retail sporting goods store estimates that weekly sales S and weekly advertising costs x are related by the equation S = 60,000 - 40,000e^(-0.0005x) . The current weekly advertising costs are $2,000, and these costs are increasing at a rate of $300 per week. Find the current rate of change of weekly sales.
Jeff Aaron
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Let u = -0.0005x, so du/dx = -0.0005
Since S = 60,000 - 40,000e^u, dS/du = -40,000e^u = -40,000e^(-0.0005x)
By the Chain Rule:
dS/dx = dS/du * du/dx
dS/dx = (-40,000e^(-0.0005x)) * (-0.0005)
dx/dt = 300
By the Chain Rule:
dS/dt = dS/dx * dx/dt
dS/dt = (-40,000e^(-0.0005x)) * (-0.0005) * 300
dS/dt = (-40,000e^(-0.0005x)) * (-0.0005) * 300
dS/dt = (-40,000e^(-0.0005x)) * (-0.15)
If x = 2,000, we have:
dS/dt = (-40,000e^(-0.0005 * 2,000)) * (-0.15)
dS/dt = (-40,000e^(-1)) * (-0.15)
dS/dt = 6,000/e
dS/dt =~ 2,207