To determine the measure of the angle that the ramp forms with the freeway, we analyze the given information. We have a right triangle in which:
- The rise (opposite side to the angle of interest) is 29 feet.
- The run (adjacent side to the angle of interest) is 200 feet.
To solve for the angle, we use the trigonometric function that relates the opposite side and the adjacent side of a right triangle. This function is the tangent (tan), and we use the formula:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{29}{200} \][/tex]
Now, we need to find the angle [tex]\(\theta\)[/tex] given the tangent value. This requires the inverse tangent (also known as arctangent) function:
[tex]\[ \theta = \tan^{-1}\left(\frac{29}{200}\right) \][/tex]
Using a calculator or appropriate mathematical tools, we find:
[tex]\[ \theta \approx 0.14399642170889201 \text{ radians} \][/tex]
To convert the angle from radians to degrees:
[tex]\[ \theta \approx 8.250387228905497 \text{ degrees} \][/tex]
Thus, the measure of the angle formed between the ramp and the freeway is approximately [tex]\(8.25\)[/tex] degrees.
The correct answer from the provided options is:
d) [tex]\(\tan^{-1}\left(\frac{29}{200}\right)\)[/tex]
