College Algebra Question

Question

At a certain time of day, a tree that is meters tall casts a shadow that is x-49 meters long. If the distance from the top of the tree to the end of the shadow is x+1 meters, what is the height,x , of the tree?

Answer 1

If we assume that the tree stands straight up and the ground is level, then they form a right angle. So the tree, the ground, and the distance from the top of the tree to the tip of the shadow on the ground form a right triangle. To find the lengths of the sides of a right triangle, use Pythagorean Theorem, a² + b² = c²

Let’s make the height of the tree a, the length of the shadow b, and the distance from the top of the tree to the tip of the shadow c.

x² + (x-49)² = (x+1)²

Be sure to FOIL the binomials. (x-49)² means (x-49)(x-49).

x² + x² -98x +2401 = x² + 2x +1

Combine like terms.

2x² -98x +2401 = x² +2x +1

Move everything over to one side of the equal sign so that the equation is set equal to zero so it can be solved.

x² - 100x +2400 =0

Find two numbers that multiply to 2400 and add to -100 to factor (or use the Quadratic Formula).

(x-60)(x-40) =0

The two solutions are x=40 and x=60. However, if we plug 40 back into the side lengths of the original problem, the tree would be 40 meters high, but the shadow would be x-49 or 40-49= -9 meters long. The length of a shadow can’t be negative, so we can reject that answer.

So the only solution is x=60 meters.

Answer 2

Pythagorean:

x² + (x-49)²=(x+1)²

x² + x² -98x + 49²= x² + 2x +1

Simplify and move all stuffs to one side:

x² -100x +2400=0

Solve quadratic function:

x= 60 (applicable)

x=40 (not applicable)

So, the answer is 60