Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 10 - Geometry - 10.2 Triangles - Exercise Set 10.2 - Page 628: 37

Answer

The height of tree is\[71.7\text{ feet}\].

Work Step by Step

The figure shows that the large triangle with the tree on the left-hand side and the small triangle with the man on the left-hand side both contain \[{{90}^{\circ }}\]angles. They also share a common angle. Thus, two angles of the large triangle have an equal measurement with that of the small triangle. Therefore, the triangles are identical and their corresponding sides are proportional. The figure shows that the height of a man is 5 feet and length of the shadow of a man is 6 feet. It also shows that the length of the shadow of the tree is 86 feet. Let the height of tree be\[x\]feet and\[x\] is the height of the tree and is opposite to the common angle in the large triangle. 5 ft. is the height of a man and is opposite to the common angle in the small triangle.The shadow of the tree is 86 feet in a large triangle and the shadow of a man is 6 feet in the small triangle. opposite Thus, \[\frac{x}{5}=\frac{86}{6}\] Compute \[x\]by applying the cross-multiplication principle for proportion which states that the following: \[\begin{align} & \frac{a}{b}=\frac{c}{d} \\ & ad=bc \end{align}\] Now, apply the cross-product principle as shown below: \[6x=\left( 86 \right)5\] Multiply 5 and 86 which is equal to 430 as shown below: \[6x=430\] Divide both sides by 6 to compute value of\[x\]as shown below: \[\begin{align} & \frac{6x}{6}=\frac{430}{6} \\ & x=71.66\approx 71.7 \end{align}\]
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