Question

A repairman leans a 10 foot ladder against the wall of a building making a 77 degree angle with the ground. How high does the ladder reach on the building.

Answer 1

To find the height that the ladder reaches on the building, we can use trigonometry. Specifically, we can use the sine function, which relates the angle of the ladder, the height it reaches, and the length of the ladder.

The sine function is defined as:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this scenario:
- The angle \(\theta\) is 77 degrees.
- The opposite side (the height \(h\) that the ladder reaches on the wall) is what we want to find.
- The hypotenuse is the length of the ladder, which is 10 feet.

Rearranging the formula to solve for the height \(h\), we have:

\[
h = \text{hypotenuse} \times \sin(\theta)
\]

Substituting the known values:

\[
h = 10 \times \sin(77^\circ)
\]

Now we can calculate \(\sin(77^\circ)\):

Using a calculator, we find:

\[
\sin(77^\circ) \approx 0.9749
\]

Now substituting back into the equation for \(h\):

\[
h \approx 10 \times 0.9749 \approx 9.749 \text{ feet}
\]

Thus, the height that the ladder reaches on the building is approximately **9.75 feet**.