Question

A flagpole is 23 feet tall and it casts a shadow that is 5 feet in length from the base of the flagpole. Imagine you were to draw an imaginary line from the top of the flag pole to the end of the shadow. Use the inverse of tangent to determine the approximate angle formed at the top of the flagpole.

Answers

GPT-4o mini
To determine the angle formed at the top of the flagpole with respect to the ground, we can use the inverse tangent function (also known as arctan). The angle can be calculated using the ratio of the height of the flagpole to the length of the shadow.

Let:
- \( h \) = height of the flagpole = 23 feet
- \( s \) = length of the shadow = 5 feet

The tangent of the angle \( \theta \) at the top of the flagpole can be expressed as:

\[
\tan(\theta) = \frac{h}{s}
\]

Substituting the values:

\[
\tan(\theta) = \frac{23}{5}
\]

Now we calculate \( \tan(\theta) \):

\[
\tan(\theta) = 4.6
\]

Next, we use the inverse tangent function to find the angle \( \theta \):

\[
\theta = \tan^{-1}(4.6)
\]

Using a calculator to find the angle:

\[
\theta \approx 77.63^\circ
\]

Therefore, the approximate angle formed at the top of the flagpole is:

\[
\boxed{77.63^\circ}
\]