Question

MODELING WITH MATHEMATICS You are standing 120 feet from the base of a 260-foot building. You watch your friend go down the side of the building in a glass elevator. a. Write an equation that gives the distance \(d\) (in feet) your friend is from the top of the building as a function of the angle of elevation \(\theta\). b. Graph the function found in part (a). Explain how the graph relates to this situation.

Short answer

Equation for the distance your friend is from the top of the building as a function of the angle of elevation \(\theta\) is \(d = 260 - \frac{120}{cos(\theta)}\). The graph of this function gives a representation of how far your friend is from the top of the building as the elevator descends, with respect to the angle of elevation from your vantage point.

Step-by-step solution

Developing Equation for Distance

The scenario presented is a right-angled triangle situation. Triangulating the scenario, we use the building as the adjacent side, the distance from the base of the building as the base, and the distance your friend is from you while in the elevator as the hypotenuse. Considering the standard expression for cosine function in a right-angled triangle, cos(θ) = adjacent/hypotenuse, the hypotenuse could be found by rearranging the expression as follows: hypotenuse = adjacent/cos(θ). However, the distance \(d\) from the top of the building to your friend is the difference between the total height of the building and the adjacent side. Hence, the equation that gives the distance \(d\) from your friend to the top of the building when is \(d = 260 - \frac{120}{cos(\theta)}\) which is gotten by substituting 260 for the hypotenuse and 120 for the adjacent side.

Graphing the function

In this step, we graph the function \(d = 260 - \frac{120}{cos(\theta)}\). 'd' is the dependent variable while '\(\theta\)' is the independent variable. The graph will begin at a point (0,260) when \(\theta\) is 0 degrees, representing your friend at the top of the building before the elevator starts to move downwards. The graph will then curve downwards as \(\theta\) increases until \(\theta\) is 90 degrees representing your friend at the bottom of the building.

Relation of graph to situation

The graphical representation provides a visual view of the how the distance of the friend from the top of the building changes as the angle of elevation \(\theta\) increases from 0 degrees at the top of the building to 90 degrees at the bottom. Therefore, this gives an understanding of the relationship between the elevation angle and the height above the ground.

Key concepts

Modeling with Mathematics

In this problem, you have to think about real-life situations in mathematical ways. You are standing a certain distance from a tall building, and you want to know how far your friend is from the top of the building as the elevator moves. This involves creating a mathematical model, which is an equation, to represent this scenario. You can think of it as translating a real-world situation into a math problem! This equation uses trigonometry because you're dealing with angles and distances in a right-angled triangle. By modeling this situation with an equation, you're able to predict and calculate the distance based on different angles.

Right Triangle Trigonometry

The core of this problem involves understanding right triangle trigonometry. Imagine standing on the ground, looking up at a building; the base of this imaginary triangle is the distance from you to the base of the building. The height of the building forms one of the sides, known as the adjacent side in terms of the angle of elevation. However, the real challenge is understanding how your friend's distance from the top of the building can be measured through trigonometric functions.

If you recall, cosine is a trigonometric function that relates the angles of a triangle to the lengths of the sides. Here, cosine is used because you have the adjacent side and you want to find out how it relates to the hypotenuse as the elevator descends. The formula we derive, using the cosine function, is \(\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\). Solving for the hypotenuse gives us \(\text{hypotenuse} = \frac{\text{adjacent}}{\text{cos}(\theta)}\). This makes it possible to then identify the distance from the top of the building using the formula \(d = 260 - \frac{120}{\cos(\theta)}\).

It's fascinating how trigonometry lets us tackle such problems by simply knowing one length and one angle!

Graphing Functions

Graphing is a great way to visualize how changes in one variable affect another. In this exercise, graphing the function \(d = 260 - \frac{120}{\cos(\theta)}\) helps you see how your friend's distance from the top changes with the angle of elevation. This function features \(d\) as the dependent variable and \(\theta\) as the independent variable, demonstrating their relationship visually.

In the graph, you start at point \((0, 260)\), which is when your friend is at the very top of the building. As \(\theta\) increases from 0 to 90 degrees, \(d\) decreases, showing your friend moving further down the building. The graphical curve helps students understand that as the angle of elevation rises, the distance from the top decreases, providing a clearer perspective on the impact of changing one variable on the other. It's not just about plotting points; it's about connecting the graph's visual with the real-world scenario to enhance understanding.