Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.
The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]
(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")
d. Prove this property holds for any a ≥ 0 and c > 0:
The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass
(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

Accepted Answer

First, write down the equations of the tangent lines to the curves $$y = e$$^{-c\cdot x}$$ and y$$ = $$-e^{-c\cdot x} at $$the point $$x = a. To do $$this, find the derivatives of both functions at $$x = a$$: Calculate the derivative of $$y = e$$^{-c\cdot x}$$, which is y$$' = -c $$e^{-c\cdot x}. $$Evaluate this at $$x = a to $$get the slope of the tangent line: $$m = -c e$$^{-c a}$$. Similarly, the derivative of y$$ = $$-e^{-c\cdot x} is$$ $$y' = c e$$^{-c\cdot x}$$, and at x = a$ the slope is m = c$$ $$e^{-c a}. $$Write the equations of the tangent lines using point-slope form. For the upper curve, the tangent line at $$(a, e$$^{-c a}$$) is$$:

$$
 y = e^{-c a} + (-c e^{-c a})(x - a) = e^{-c a} - c e^{-c a} (x - a)
$$

For the lower curve, the tangent line at $$(a, -e$$^{-c a}$$) is$$:

$$
 y = -e^{-c a} + (c e^{-c a})(x - a) = -e^{-c a} + c e^{-c a} (x - a)
$$ Find the intersection point $$(x_I, y_I) of $$these two tangent lines by setting their $$y$$ values equal:

$$
 e^{-c a} - c e^{-c a} (x_I - a) = -e^{-c a} + c e^{-c a} (x_I - a)
$$

Simplify this equation to solve for $$x_I. $$After solving for $$x_I$$, observe that the $$y-$$coordinate of the intersection $$y_I is $$zero due to symmetry. Then, compare $$x_I$$ with the expression for the center of mass $$\bar{x} = \frac{\int_a$$^{\infty}$$ x e$$^{-c x}$$ dx}{\int_a$$^{\infty}$$ e$$^{-c x}$$ dx}. $$Show that $$x_I = \bar{x}$$, thus proving that the tangent lines intersect at the center of mass for any $$a \geq 0$$ and $$c \u003e 0.$$

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Key Concepts

Center of Mass for a Planar Region

Tangent Lines to Exponential Curves

Improper Integrals and Convergence on [a, ∞)