algebra problem

Question

A railroad bridge spans a gorge 40 feet wide and connects two cliffs at heights of 98 and 158

feet above the bottom of the gorge. A train is crossing this gorge from the higher cliff to the

lower. When the front of the train has traveled three-fourths of the bridge’s length, how many

feet is it above the bottom of the gorge?

Answer 1

This is really hard to explain in text because it involves a fairly complex diagram.

Start by drawing the high cliff on the left and the low cliff on the right. Then, the edge of the high cliff is P, and the edge of the low cliff is Q. Draw also a line across the gorge from Q to the wall on the other side, and call that point R.

Now: the length of QR is specified in the problem as the distance across the gorge: 40. The length of PR is also specified in the problem as the difference in elevation: 60.

The bridge is then represented as the segment PQ. This is the hypotenuse of triangle PQR, and since we specified point R to be directly across from Q, PQR is a right triangle. Therefore, Pythagoras' theorem gives its length as ~72.

Three-fourths of the way across the bridge means to move down along the hypotenuse PQ a distance of ~54 to a point we will call S. That is, S divides PQ into two segments: PS of length ~54, and SQ of length ~18. Drop a vertical line from S to a point on line QR; we will call that point T.

Call the angle formed between PQ and QR angle q. Since PR and QR are measured in the same units, we have tan q = length PR/length QR. Arctan of both sides gives q ~56.3.

Now, notice that baby triangle QST exists. By the rules for similarity it is also a right triangle, so we have sin q = length of ST/ length of QS. Rearrangement and arcsin of both sides gives the length of segment ST as being ~15.

Lastly, notice that what we are asked to find is the vertical distance between the bottom of the gorge and point S; this is simply the height of the shorter cliff plus the length of segment ST, for a final answer of 113.

Answer 2

Solution 1:

The width of the gorge = 40 feet

The height of the higher cliff = 158 feet

The height of the lower cliff = 98 feet

The length of the bridge = √((158-98)² + 40²) = 72.11 feet

The slope of the bridge = (158-98)/40 = 1.5

The length of 1/4 of the bridge from the lower cliff =72.11 - 3/4×72.11 = 18.03 feet

The angle of inclination of the bridge = tan⁻¹(1.5) = 56.31°

The height above the bottom at 3/4 from the higher cliff = The height above the bottom at 1/4 from the lower cliff = 98+ 18.03×sin(56.31 ) = 113 feet

Solution 2:

Which can also be found directly from the heights of the two cliffs knowing that 3/4 from the higher cliff = 1/4 from the lower cliff giving;

Height above the bottom gorge = 98 + 1/4×(158 - 98) = 113 feet.