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? Round to the nearest tenth.

Answer 1

To find out how high the ladder reaches on the building, we can use the sine of the angle formed between the ladder and the ground. The height reached by the ladder on the building is the opposite side of the triangle formed, while the ladder serves as the hypotenuse.

We can use the following relationship:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
where:
- \(\theta\) is the angle with the ground (77 degrees),
- the opposite side is the height (h) the ladder reaches on the building,
- the hypotenuse is the length of the ladder (10 feet).

Rearranging the sine formula to find the height (h):
\[
h = \text{hypotenuse} \times \sin(\theta)
\]
Now we can substitute the known values:
\[
h = 10 \times \sin(77^\circ)
\]

Next, we calculate \(\sin(77^\circ)\). Using a calculator:
\[
\sin(77^\circ) \approx 0.9749
\]

Now substituting back into the equation for height:
\[
h = 10 \times 0.9749 \approx 9.749
\]

Rounding to the nearest tenth, the height the ladder reaches on the building is:
\[
\boxed{9.7} \text{ feet}
\]