Question

Hot-air Balloon While taking a ride in a hot-air balloon in Napa Valley, Francisco wonders how high he is. To find out, he chooses a landmark that is to the east of the balloon and measures the angle of depression to be \(54^{\circ} .\) A few minutes later, after traveling 100 feet east, the angle of depression to the same landmark is determined to be \(61^{\circ}\). Use this information to determine the height of the balloon.

Short answer

248.8 feet

Step-by-step solution

Draw a Diagram

Draw a diagram to visually represent the situation. The initial position of the balloon creates an angle of depression of \(54^{\circ}\) to a landmark. After moving 100 feet east, the angle of depression changes to \(61^{\circ}\). Let the height of the balloon be \( h \) and the horizontal distance from the initial position to the landmark be \( d \).

Apply Trigonometric Relationships

In the initial position, use the tangent function to relate the height of the balloon to the distance \( d \): \[ \tan(54^{\circ}) = \frac{h}{d} \] After moving 100 feet east, the new distance to the landmark is \( d + 100 \). Use the tangent function again for the new angle: \[ \tan(61^{\circ}) = \frac{h}{d + 100} \].

Set Up Equations

From Step 2, we have two equations: \[ \tan(54^{\circ}) = \frac{h}{d} \] \[ \tan(61^{\circ}) = \frac{h}{d + 100} \] Simplify these to: \[ d = \frac{h}{\tan(54^{\circ})} \] and \[ d + 100 = \frac{h}{\tan(61^{\circ})} \].

Solve for \( d \) and \( h \)

Set the two equations for \(h\) equal to each other: \[ \frac{h}{\tan(54^{\circ})} + 100 = \frac{h}{\tan(61^{\circ})} \]. Use algebra to solve for \(d\) and \(h\): \[ \frac{h}{\tan(61^{\circ})} - \frac{h}{\tan(54^{\circ})} = 100 \] Factor out \(h\): \[ h \left( \frac{1}{\tan(61^{\circ})} - \frac{1}{\tan(54^{\circ})} \right) = 100 \] Solve for the height \(h\): \[ h = \frac{100}{\frac{1}{\tan(54^{\circ})} - \frac{1}{\tan(61^{\circ})}} \].

Calculate the Height

Use a calculator to find the values of \( \tan(54^{\circ}) \) and \( \tan(61^{\circ}) \). Compute the expression for \( h \): \[ \tan(54^{\circ}) \approx 1.3764 \] \[ \tan(61^{\circ}) \approx 1.804 \] Substitute these values: \[ h = \frac{100}{\frac{1}{1.3764} - \frac{1}{1.804}} \approx 248.8 \text{ feet} \].

Key concepts

angle of depression

The angle of depression is the angle formed between the horizontal line of sight and the line of sight down to an object. Here, Francisco measures the angle of depression from the hot-air balloon to a landmark. Initially, this angle is \(54^{\circ}\) and later it changes to \(61^{\circ}\).
To visualize this, picture yourself looking straight out from the balloon. The angle of depression is the angle you would look down to spot the landmark.
Understanding this concept helps us relate the balloon’s height to how far it moved east.

tangent function

The tangent function (\text{tan}) in trigonometry links an angle of a right triangle to the ratio of its opposite side to its adjacent side. In our problem:

• For the initial position, \(\text{tan}(54^{\circ}) = \frac{h}{d}\), where \(h\) is the balloon's height, and \(d\) is the distance to the landmark.

• For the second position, \(\text{tan}(61^{\circ}) = \frac{h}{d + 100}\). The distance changed to \(d + 100\) feet after moving east.

Using the tangent function allows us to set up equations to solve for \(h\), linking the balloon’s height with the distances.

algebraic equations

Algebraic equations are used here to solve for the balloon’s height. From the tangent function, we derived two equations:

• \(\text{tan}(54^{\circ}) = \frac{h}{d}\)
• \(\text{tan}(61^{\circ}) = \frac{h}{d + 100}\)

We rearranged these to solve for \(d\) and then set the two expressions for \(h\) equal:

• \(d = \frac{h}{\text{tan}(54^{\circ})}\)
• \(d + 100 = \frac{h}{\text{tan}(61^{\circ})}\)

Equating the two expressions involving \(h\), we get:
\(\frac{h}{\text{tan}(61^{\circ})} - \frac{h}{\text{tan}(54^{\circ})} = 100\)
Finally, solving this algebraic equation, we determine \(h\), the height of the balloon, which is roughly 248.8 feet.