Question

at a certain time of day, a tree that is x meters tall casts a shadow that is x - 34 meters long. if the distance from the top of the tree to the end of the shadow is x + 2 meters, what is the height, x, of the tree? answer how to enter your answer (opens in new window) 6 points keyboard shortcuts keypad

Explanation:

Step1: Apply Pythagorean theorem

We know that in a right - triangle formed by the tree, its shadow and the line from the top of the tree to the end of the shadow. According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = x\) (height of the tree), \(b=x - 34\) (length of the shadow) and \(c=x + 2\) (distance from the top of the tree to the end of the shadow). So we have the equation \(x^{2}+(x - 34)^{2}=(x + 2)^{2}\).

Step2: Expand the equations

Expand \((x - 34)^{2}=x^{2}-68x + 1156\) and \((x + 2)^{2}=x^{2}+4x+4\). The equation becomes \(x^{2}+x^{2}-68x + 1156=x^{2}+4x + 4\).

Step3: Simplify the equation

Rearrange the terms: \(x^{2}+x^{2}-x^{2}-68x-4x+1156 - 4 = 0\), which simplifies to \(x^{2}-72x+1152 = 0\).

Step4: Solve the quadratic equation

We can solve the quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b=-72\), \(c = 1152\)) using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or by factoring. Factoring: \(x^{2}-72x + 1152=(x - 12)(x - 96)=0\). So \(x=12\) or \(x = 96\). But if \(x = 12\), then the length of the shadow \(x-34=12 - 34=-22\) which is not possible. So we reject \(x = 12\).

Answer:

96 meters