At a certain time of day, a tree that is x meters tall casts a shadow that is x−31 x − 31 meters long. If the distance from the top of the tree to the end of

Question

At a certain time of day, a tree that is x meters tall casts a shadow that is x−31

meters long. If the distance from the top of the tree to the end of the shadow is x+1

meters long, what is the height, x, of the tree?

Answer 1

Assume the tree is vertical then using Pythagorean theorem

a² +b² = c² where a = height of tree; b = length of shadow, and c= distance from top of tree to end of shadow

there for

x² + (x - 31)² = (x+1)² , multiplying out

x² + x² - 62x +961 = x² +2x +1, subtract right side from left

x² - 64x +960 = 0

solve using quadratic formula

960 = 64x - 2x² , divide both sides by 2 and put in quadratic formula form

x² -32x + 480 = 0, find the roots of the quadratic formula or expand to two terms

(x - 24) (x-40) = 0 so x = 24 or x=40.

It can't be that x=24 because the length of shadow would be a negative number so x=40. (If you used the value of 24 f for the tree, the shadow would be 7 foot the other side which would also agree with the equations