Let's solve the problem step-by-step:
1. Understand the problem:
A 16-foot ladder is leaned against a wall, forming an angle of 78 degrees with the ground. We need to find out how high up the wall the top of the ladder reaches and round the answer to the nearest hundredth of a foot.
2. Visualize the scenario:
We have a right triangle formed by the ladder, the wall, and the ground. The length of the ladder is the hypotenuse of this right triangle. The height up the wall is the side opposite the given angle of 78 degrees.
3. Use the sine function:
In a right triangle, the sine of an angle is the ratio of the length of the side opposite to the length of the hypotenuse.
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
Here, [tex]\(\theta\)[/tex] is 78 degrees, the opposite side is the height up the wall we want to find, and the hypotenuse is 16 feet.
4. Set up the equation:
[tex]\[
\sin(78^\circ) = \frac{\text{height up the wall}}{16}
\][/tex]
5. Solve for the height:
[tex]\[
\text{height up the wall} = 16 \times \sin(78^\circ)
\][/tex]
6. Calculate the height (using trigonometric values):
- First, convert the angle from degrees to radians because trigonometric functions typically use radians.
[tex]\[
\text{height up the wall} = 16 \times \sin(78^\circ \text{in radians})
\][/tex]
- Now calculate the sine of 78 degrees.
7. Result:
- The height up the wall is approximately 15.65036161174089 feet when calculated.
8. Round the result:
- We need to round the height to the nearest hundredth of a foot.
[tex]\[
\text{height up the wall (rounded)} = 15.65 \text{ feet}
\][/tex]
So, the ladder reaches approximately 15.65 feet up the wall.
