Hello, Kayden,
I believe you are being asked to use the horizontal distance from the road's start to a point under the mountain peak, but at the same elevation as the start. I've drawn a diagram:
Since we have a right triangle, we can calculate the third side, the base distance, along the bottom. The percent grade change can be calculated using this measure of distance, not the 7.6 miles. I will say that using the 7.6 miles, instead, for the grade change measurement makes sense if want to know the rate of elevation change as one is driving up the road. But using the base distance provides an absolute percent grade change.
Use the Pythagorean x2 + y2 = Hypot.2, where the hypotenuse is the 7.6 miles. We need to convert miles into feet, or vice versa. I'll use 1 mile/5280 feet.
x = 1600feet * (1 mile/5280 feet) = 0.303 miles
We can then use the theorem: X2 + (0.303)2 = (7.6)2
where x is the base distance from the start to under the peak, at the same elevation as the start.
By the time you square 0.303 and subtract it from 7.62, the resulting number isn't much different than the 7.6, after you take the square root.. I get 7.59 miles. Seems like a whole lotta trouble for nothing, especially when the 7.6 miles given us only has 2 sig figs. If I use 2 sig figs here, I get, hold on, drum roll, 7.6 miles. Maybe I've made an error in the calculation. Please check carefully.
Assuming 7.6 miles is correct, the percent grade would be (0.303 miles/7.6 miles) *(100), or 3.99% --- 4.0% with 2 sig figs.
I hope this helps,
Bob
