Question

a 20 - foot - tall pine tree casts a 35 - foot - long shadow on the ground. which equation could be used to determine the distance from the top of the pine tree to the end of its shadow? 6 of 12 question $x^{2}=35^{2}-20^{2}+2(35)(20)\cos90^{circ}$ $20^{2}=35^{2}+x^{2}-2(35)(20)\cos90^{circ}$ $35^{2}=x^{2}+20^{2}-2(35)(20)\cos90^{circ}$ $x^{2}=35^{2}+20^{2}-2(35)(20)\cos90^{circ}$

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines is \(a^{2}=b^{2}+c^{2}-2bc\cos A\), where \(a\), \(b\), \(c\) are sides of a triangle and \(A\) is the angle opposite to side \(a\).

Step2: Identify the sides and angle of the triangle

Here, the height of the tree (\(20\) feet) and the length of the shadow (\(35\) feet) are two sides of a right - triangle (angle between them \(A = 90^{\circ}\)), and \(x\) is the hypotenuse (distance from the top of the tree to the end of the shadow).
Substitute \(a=x\), \(b = 35\), \(c = 20\), and \(A=90^{\circ}\) into the Law of Cosines formula.
We get \(x^{2}=35^{2}+20^{2}-2(35)(20)\cos90^{\circ}\)

Answer:

\(x^{2}=35^{2}+20^{2}-2(35)(20)\cos90^{\circ}\) (the fourth option)