Question

At noon, a tree casts a shadow that is 40 feet long. The distance from the top of the tree to the furthest tip of the shadow is 60 feet.

A tree with height h forms a triangle with its shadow. A side has length of 40 feet and the hypotenuse is 60 feet.
[Not drawn to scale]

What is the height of the tree? Round to the nearest hundredth.
10.00 feet
20.00 feet
44.72 feet
72.11 feet

Answer 1

To find the height of the tree, we can use the Pythagorean theorem. In this case, we have a right triangle formed by the height of the tree, the length of the shadow, and the hypotenuse (which is the distance from the top of the tree to the tip of the shadow).

Let:
- \( h \) = height of the tree
- \( a \) = length of the shadow = 40 feet
- \( c \) = length of the hypotenuse = 60 feet

According to the Pythagorean theorem, we have:

\[
h^2 + a^2 = c^2
\]

Substituting in the known values:

\[
h^2 + 40^2 = 60^2
\]

Calculating \( 40^2 \) and \( 60^2 \):

\[
h^2 + 1600 = 3600
\]

Now, subtract 1600 from both sides:

\[
h^2 = 3600 - 1600
\]
\[
h^2 = 2000
\]

Next, take the square root of both sides to find \( h \):

\[
h = \sqrt{2000}
\]

Calculating \( \sqrt{2000} \):

\[
h \approx 44.72 \text{ feet}
\]

So the height of the tree rounded to the nearest hundredth is:

\[
\boxed{44.72 \text{ feet}}
\]