Problem 37 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

A rock climber is using a rock climbing treadmill that is 10 feet long. The climber begins by lying horizontally on the treadmill, which is then rotated about its midpoint by \(110^{\circ}\) so that the rock climber is climbing toward the top. If the midpoint of the treadmill is 6 feet above the ground, how high above the ground is the top of the treadmill?

Step-by-Step Solution

Answer

The top of the treadmill is \(h\) feet above the ground where \(h\) can be calculated by using the law of cosines as described in the solution steps.

Step 1: Identify Given Information

The problem provides the following information: the initial height of the midpoint of the treadmill from the ground (6 feet), the length of the treadmill (10 feet), and the angle that the treadmill was rotated \(110^{\circ}\).

Step 2: Interpret the Problem in Terms of a Right Triangle

The rotation of the treadmill creates a right triangle. The hypotenuse of the triangle is the length of the treadmill. The angle opposite the hypotenuse is the angle by which the treadmill was rotated from horizontal. We can call the height we are trying to find as 'h'.

Step 3: Apply Trigonometric Principles

Since we have a right triangle, we can use the cosine of the angle to find the unknown side. Cosine of an angle in a right triangle is defined as the adjacent side divided by the hypotenuse. Thus, we can formulate the following equation based on the law of cosines: \( \cos (110^{\circ}) = 6 / 10 \). Solve this equation to calculate 'h'.

Step 4: Solve for h

To solve for 'h', first calculate the cosine of 110 degrees. Then multiply this value by the length of the treadmill (10 feet). Finally, add this result to the initial height of the midpoint above the ground (6 feet). The resulting value will be the height 'h'.

Key Concepts

Right Triangle

In the world of trigonometry, a right triangle is a triangle that features one angle measuring exactly 90 degrees. This type of triangle is fundamental because it leads to simple relationships among the angles and sides, which are defined by trigonometric functions. In our problem, the treadmill, after it rotates, forms a right triangle.

Here's what that looks like:

• Hypotenuse - This is the longest side of the right triangle, opposite the right angle. It refers to the length of the treadmill in our problem, which is 10 feet.
• Base - This typically corresponds to the horizontal side, initially represented by the ground in the problem.
• Height - This side is what connects the base at a 90-degree angle, analogous to the vertical side of the triangle, or the height 'h' above the initial midpoint in our problem.

The right triangle formation allows us to apply trigonometric principles to solve for the unknown height, effectively simplifying the complexity of rotational geometry.

Angle of Rotation

The angle of rotation is a measure of how much a figure is rotated around a central, fixed point. In our scenario, the treadmill is rotated by an angle of 110 degrees about its midpoint. Understanding this angle is crucial as it determines the configuration and orientation of the right triangle with respect to the horizontal or original position.

To visualize:

• The treadmill began parallel to the ground, essentially flat when the rotation starts.
• A rotation by 110 degrees moves it to a certain angle where the triangle is formed.
• This angle helps calculate how much vertical distance exists from the ground to the new highest point of the treadmill.

Understanding the angle of rotation helps break down the geometrical transformation, turning a straightforward horizontal layout into a triangular geometry that can be analyzed using trigonometry.

Cosine Function

One of the fundamental trigonometric functions is the cosine function, which relates the angle of a right triangle to the lengths of the sides. Specifically, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse.

In this context:

• The angle chosen is 110 degrees, due to the rotation of the treadmill.
• The length of the side adjacent to this angle can be linked to its projection on the vertical axis, given the problem's setup.
• The hypotenuse of interest is the 10-foot long treadmill.

By knowing these measurements, the cosine function can be used to compute the vertical height by rearranging the cosine formula: \[\cos(110^{\circ}) = \frac{\text{adjacent side}}{\text{hypotenuse}}\]Thus, this function allows us to isolate the unknown height while incorporating the given rotation and geometry of a right triangle.

Triangular Projection

The concept of a triangular projection has to do with relating a 3D orientation to a 2D frame through the right triangle's configuration. This is key in visualizing how the treadmill, once rotated, changes its effective height above the ground.

In our case:

• By observing the triangle formed by the rotation, the original length (10 feet) aligns along the hypotenuse.
• The cosine function is used to project this rotation back onto a vertical axis, giving us a height measurement.
• This projection translates the diagonal movement into a direct height increase, adjusting for both rotation and elevation.

Understanding these projections, we can calculate how high the treadmill's top reaches, solidifying a complex rotational concept with straightforward trigonometric methods.