To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding :

Remember that this is not the full height of the larger building. To find that, we need to add  feet. Therefore, the taller building is 635.3 feet tall.

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground.  We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high.  We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle.  Our base of the triangle is 3 feet and the leg is 10 feet.

And so we need a support of 10.44 feet long.

We begin by drawing a picture. Let  stand for where the plane reaches a maximum altitude of 31,000 ft. We can also assume that if we draw a line straight down from the plane that it will be perpendicular to the ground. This forms a right triangle.

Now we will use our knowledge of right triangles.  We know the trigonometric identity, .  We can plug 55 in for our  and 31,000 in for the opposite side. Solving for the hypotenuse is solving for the distance from the point of takeoff to the plane when it reaches maximum altitude.

And so the distance from point  to the plane at maximum altitude is 37,844 ft.

We begin by drawing a picture.  Let point be where your eyes are looking on the billboard and point be where you are standing on the ground.

We must solve for the height of the billboard.  We have information of the adjacent side of the right triangle formed and we wish to know the height, which will be the opposite side from you.  Using trigonometric identities, we will use .  We can plug in 500 ft for our adjacent side and 67 degrees for our .

And the height of the section of the billboard you are looking at is 1,178 ft above the ground.

The important thing to note for this problem is that we are working with similar triangles.  Your shadow forms a right triangle with its three points being your feet, the top of your head, and the top of your shadow’s head.  We have enough information to find the angle of your head’s elevation to your shadow on the ground.  Just like your shadow forms a right triangle to your body, a similar right triangle will also be formed to the sun’s altitude.  We know that this triangle is similar because it shares the angle opposite your body and it shares a right angle, so it’s third angle must also be equal.  Since it shares the angle opposite your body, we must only solve for the angle of your head’s elevation and it will be the same as the sun’s elevation.

We do this by using .  We can plug in 60cm for the opposite since that is your height, and 66cm for the adjacent since that is the length of the shadow.  We are solving for .

And so the sun is at an elevation of 48 degrees.