Question

question 5 of 12, step 1 of 1. at a certain time of day, a tree that is x meters tall casts a shadow that is x - 35 meters long. if the distance from the top of the tree to the end of the shadow is x + 5 meters, what is the height, x, of the tree?

Explanation:

Step1: Apply Pythagorean theorem

We know that if the height of the tree is $x$ meters, the length of the shadow is $x - 35$ meters and the distance from the top of the tree to the end of the shadow is $x + 5$ meters. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a=x$, $b = x - 35$ and $c=x + 5$. So we have the equation $x^{2}+(x - 35)^{2}=(x + 5)^{2}$.

Step2: Expand the equation

Expand the squares: $x^{2}+x^{2}-70x + 1225=x^{2}+10x + 25$.

Step3: Simplify the equation

Combine like - terms: $x^{2}+x^{2}-70x + 1225-(x^{2}+10x + 25)=0$. Which simplifies to $x^{2}-80x+1200 = 0$.

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 1$, $b=-80$, $c = 1200$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-80)^{2}-4\times1\times1200=6400 - 4800 = 1600$. Then $x=\frac{80\pm\sqrt{1600}}{2}=\frac{80\pm40}{2}$. We get two solutions: $x_1=\frac{80 + 40}{2}=60$ and $x_2=\frac{80-40}{2}=20$. But if $x = 20$, then the length of the shadow $x-35=20 - 35=-15$ which is not possible. So we take $x = 60$.

Answer:

60