Question

Mr. and Mrs. McClurrie are debating on whether or not to remove a tall tree in their front yard. Mr. McClurrie is concerned that the tree could fall during a storm and land on their home. Mrs. McClurrie disagrees. She thinks the tree poses no danger to their home and wants to keep the tree because of the wonderful shade it provides. They decide to estimate the height of the tree using special right triangles.
They stood 25 feet from the base of the tree to create a 30° angle from the ground to the top of the tree.

If their home is 20 feet from the base of the tree, could the McClurrie's wait a few more years before cutting down the tree or does it pose a risk now?

Answer 1

Check the picture.

Let A be the point where Mr. and Mrs. McClurrie are standing, let B be the base of the tree and C the top of the tree. Let H be the point representing the house.

ABC is a right triangle, with m(BAC)=30°.

so by special triangles, if the length of the hypotenuse is  2x, the length of BC is x, since it is the side opposite to the 30° angle.

by the pythagorean theorem :

$(AC)^{2}= (AB)^{2}+(BC)^{2}$
$(2x)^{2}= (25)^{2}+(x)^{2}$
$4x^{2}= (25)^{2}+x^{2}$
$3x^{2}= (25)^{2}$
$x= \sqrt{ \frac{(25)^{2} }{3}}= \frac{25}{ \sqrt{3} }= \frac{25 \sqrt{3} }{3}$
$\frac{25 \sqrt{3} }{3}=25* \frac{ \sqrt{3} }{3}=25*0.577= 14.43$ (feet)

so even if the tree falls, it falls more than 5 feet away from the house.

Answer: they can wait a few more years.