1. ## Trig Problem

Problem that is annoying me:
One of the steeper streets in the US is the 500 block highland drive on queen anne hill in Seattle. To measure the slope of the street, I held a builder's level so that one end touched pavement. The pavement was 14.4 inches below the level at the other end. The level itself was 71 inches long.

a) What angle does the pavement make with the level?

b) A map of Seattle show sthe horizontal length of this block of highland rive is 365 ft. How much longer than 365 feet is the slant distance up this hill.

c) How high does the street vertically rise up in this block?

If anyone could break down the steps of how do these that would be greatly appreciated.

2. First, let us recognize that what the problem is describing is a triangle. Let's call the length of the street the hypotenuse (c), then we'll call the length of the level a, and then we'll call the distance from the level to the pavement b. Recall the Pythagorean Theorem:

$a^2 + b^2 = c^2$

Where a and b must be the legs (straight) of the triangle and c must be the hypotenuse (diagonal). We know that:

$a = 71 in$

$b = 14.4 in$

So, we plug those in and solve for c:

$(71)^2 + (14.4)^2 = c^2$

$c^2 = 5248.36$

$c = \sqrt{5248.36} = 72.45 in$

That is the length of the street in the triangle.

Now, (a)

The angle we are looking for is between a (the level) and c (the street), so we can say that the level is adjacent (next to) the angle and c is the hypotenuse, so we can recall that:

$cos(\theta) = \frac{a}{c}$

Which means:

$\theta = \cos^{-1}\left(\frac{a}{c}\right)$

Just plug in a and c and solve.

Now (b),

You now have the angle from the street to some horizontal surface, now imagine that the street itself, with length 365 ft. (this is a), is replacing the level. The angle remains the same, but the lengths b (distance from horizontal to street) and c (length of the street) change. So you know the angle theta, now we use the formula we used previously:

$cos(\theta) = \frac{a}{c}$

Which means that:

$c = \frac{a}{cos(\theta)}$

You know a = 365 ft and $\theta$. Just solve for c and compare your answer to 365.

Now (c),

This is simple, you know side a (length of the horizontal street) and side c (length of diagonal street), all you need is side b (length of vertical street), recall the Pythagorean Theorem:

$a^2 + b^2 = c^2$

Plug in a and plug in c and solve for b, and you'll have your answer.

Hope this helps.