Question

A 10ft long stem of bamboo is broken in such a way that its tip touches the ground 3ft from the base of the stem, as shown in the figure. What is the height of the break?

Short answer

The height of break is $4.55\text{ft}$.

Step-by-step solution

Step-1. Formula used.

Pythagorean Theorem: $H^2=P^2+B^2$

Step-2. Setup the model.

Let the height of the break$=x\mathrm{feet}.$

The hypotenuse of the given triangle $=10-x.$

The base of the stem to the broken tip $=3\mathrm{feet}$

We need to find the height of the break

Step-3. Solve for height h.

Substitute the respective values into the Pythagorean theorem.

$\begin{array}{l}{H}^2=P^2+B^2\\ (10-x{)}^2=(x{)}^2+(3{)}^2\\ (10{)}^2-2\times 10\times x+(x{)}^2=x^2+9\\ 100-20x+x^2=x^2+9\\ 100-20x=9\\ -20x=9-100\\ -20x=-91\\ x=\frac{-91}{-20}\\ x\approx 4.55\end{array}$

The height of the break is $4.55\mathrm{ft}$.