Measuring by ShadowsDate: 05/22/2001 at 07:55:02 From: Andreza Rio Branco Subject: Measuring by shadow How can I measure a tree using its and my shadows? Date: 05/23/2001 at 08:21:25 From: Doctor Rick Subject: Re: Measuring by shadow Hi, Andreza, thanks for writing to Ask Dr. Math! The secret is in the sun's rays: they fall on both you and the tree from the same direction. We say the rays are parallel. Also, the tree stands straight and so do you, so you and the tree are parallel. Finally, if the ground is flat, the tree's shadow and your shadow are parallel. We can draw the tree, its shadow, you, and your shadow as triangles. The top of the tree is joined to the shadow of the top of the tree by a line that points back up to the sun. The top of your head and the shadow of the top of your head are joined by another line pointing back up to the sun. TREE |\ | \ | \ | \ ? | \ | \ | \ YOU | \ |\ | \ 5'| \ __|_________\_____________|__\______ 50' 2.5' When each side of one triangle is parallel to a side of another triangle, the triangles are SIMILAR. When we use this word in everyday life, we just mean "they are sort of the same, but not quite." When we use it in math, it has a special meaning: "their shapes are exactly the same, though their sizes don't have to be the same." The "tree triangle" and the "you triangle" are similar. The sides of two similar triangles have equal proportions. If the horizontal side of one triangle is twice as long as the horizontal side of the other triangle, then the vertical side of the first triangle is twice as long as the vertical side of the other, and the diagonal sides follow the same pattern. These equal proportions let you figure out one length if you know three others. In the picture above, I marked "your height" as 5 feet, and your shadow as 2 1/2 feet. The tree's shadow is 50 feet. How long is the tree's shadow compared to your shadow? 50 100 --- = --- = 20 2.5 5 The ratio is 20 to 1; that is, the tree's shadow is 20 times as long as your shadow. Since the triangles are similar, the ratio of the tree's height to your height is also 20 to 1. If your height is 5 feet, and the tree is 20 times as tall, then the tree's height is 20 * 5 feet = 100 feet (The "*" is the multiplication sign.) This is how you can use shadow lengths and your own height to measure the height of a tree! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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