To find the vertical distance that Jamal has climbed when driving up a 45-degree incline for 82 miles, we can use trigonometry.
In a right triangle where the angle of elevation is 45 degrees, the opposite side (the vertical distance climbed) is equal to the adjacent side (the horizontal distance traveled). The hypotenuse is the distance traveled along the slope.
We can use the sine function: \[ \text{Vertical Distance} = \text{Hypotenuse} \times \sin(\text{angle}) \]
Given:
- Hypotenuse = 82 miles
- Angle = 45 degrees
Since \(\sin(45°) = \frac{\sqrt{2}}{2}\):
\[ \text{Vertical Distance} = 82 \times \frac{\sqrt{2}}{2} = 41\sqrt{2} \text{ miles} \]
So, the vertical distance that Jamal has climbed is \(41\sqrt{2}\) miles.