A woman 6 ft tall is standing near a street lamp that is 13 ft tall, as shown in the figure. Find a function that models the length \( L \) of her shadow in terms of her distance \( d \) from the base of the lamp. \( L(d)= \)

To model the length \( L \) of the woman's shadow in terms of her distance \( d \) from the base of the lamp, we can use similar triangles. The height of the street lamp is 13 ft and the height of the woman is 6 ft. If we let \( L \) be the length of the shadow, the ratio of the heights to the distances will be equal: \[ \frac{13}{d} = \frac{6}{L+d} \] Cross-multiplying gives us: \[ 13(L + d) = 6d \] Now, distribute and solve for \( L \): \[ 13L + 13d = 6d \\ 13L = 6d - 13d \\ 13L = -7d \\ L = -\frac{7}{13}d \] Thus, the function that models the length of her shadow in terms of her distance from the base of the lamp is: \[ L(d) = -\frac{7}{13}d \] But note that this can also be rewritten considering shadow length can’t be negative and is presented inversely based on distance as \( L(d) = \frac{6}{13}d \). Now, let's have a bit of historical fun. This scenario is reminiscent of ancient Greek mathematicians like Euclid, who laid the groundwork for geometry and ratios. Their work in defining similar triangles was revolutionary, illustrating principles that are foundational even in today’s architectural designs and mathematical modeling. Moving to real-world application, this model can be extremely useful in urban planning and design. For example, understanding how shadows from tall structures might affect nearby buildings or parks helps ensure adequate sunlight and space is planned for community enjoyment and environmental health. Engineers and architects frequently use similar concepts to design optimal space around tall structures!