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When my friend was showing me around the Southern Island of New Zealand, he took me to Baldwin Street, which had been billed as the "World's Steepest Street." We walked up it, and while it certainly felt steep, I thought that there must be steeper streets in San Francisco. So we turned to the internet to do some research.
First, it’s important to discuss the different ways to measure the steepness of a road. We could measure a road’s steepness by using its slope, which we know to be the rise over run. And we could also measure a road’s steepness by taking its angle of inclination, the angle the road makes with a horizontal surface. But scientists often express the steepness of a road as a percentage, which is called the grade. This is effectively the same ratio as slope but expressed as a percentage.
Let’s examine a few right triangles and look at the slope and grade of the hypotenuse with respect to the angle of inclination:
Since the steepness of roads is usually measured in grades, we need to be able to convert between slopes, angles, and grades. Converting between slope and grade is relatively easy because they’re two representations of the same number, just one using a ratio and the other using percentage. But to convert between angles and grades, we must use the following:
grade = tan(angle of inclination)
So if we measure an angle of elevation of 37 degrees, then the grade is given by tan(37) ≈ 0.75 = 75%. Pretty easy! So let’s get back to the streets.
According to the Guinness Book of World Records, Baldwin Street was actually the second steepest street in the world, but the winner wasn't in San Francisco: it was actually a street in Wales called Ffordd Pen Llech. Ffordd Pen Llech had a grade of 37.45%, whereas Baldwin Street only had a grade of 35%. But what about San Francisco?
The SF Bureau of Engineering listed the steepest streets as Filbert between Leavenworth and Hyde and 22nd between Church and Viksburg with a 31.5% grade. However, this list appears to be incomplete and outdated, according to a 7x7 article that claims that the steepest street in San Francisco is Bradford Street, which has a whopping 41% grade above Tompkins!
Sample Problems
Sample Problems
1. Go to Bradford Street above Tompkins and carefully measure the angle of elevation of the street, making sure to avoid traffic. Then calculate the grade. Is the grade actually 41%? Are there parts of the street that are even steeper? (You can measure the angle of elevation with a level tool app on a smartphone.)
2. Carefully measure the angles of elevation of some of the other steepest streets in San Francisco, calculate their grades, and compare them to the list in the 7x7 article, which can be found online. How close are your measurements to the ones in the article? If they're different, why might that be?
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