Exercise 11 Page 558

We are asked what measures we must know to calculate the height of a tall object using shadow reckoning. First, let's recall how shadow reckoning works. Let's say we want to find the height $h$ of a transmitting tower.

To find this height using shadow reckoning, we use the fact that the tower and its shadow form two sides of a right triangle.

Another object in the same location and its shadow would form a similar right triangle. For example, a person standing near the tower and their shadow form two sides of a right triangle that is similar to the triangle formed by the tower and its shadow.

We know that in similar figures, the ratios of lengths of corresponding sides are proportional. Therefore, the ratio of the height $t$ of the person to the height $h$ of the tower is equal to the ratio of the length $s$ of the shadow cast by the person to the length $\ell$ of the shadow cast by the tower.

This lets us write a proportion. If we know how tall the person is, the length of the shadow the person casts, and the length of the shadow the tower casts, we can use this equation to find the height of the tower. In general, to calculate the height of a tall object using shadow reckoning, we have to know the height of another object, the length of the shadow cast by the other object, and the length of the shadow cast by the object we want to measure.