Ask a New Question
Menu

The angle of elevation from the ground is 68 degrees. Franklin has used 1890 feet of string. What is the approximate height of the kite? Round to the nearest whole number and make sure to show all of your work

5 answers

To find the height of the kite using the angle of elevation and the length of the string, we can model the situation as a right triangle. Here, the string acts as the hypotenuse, the height of the kite represents the opposite side, and the distance from the base of the triangle vertically beneath the kite to the point where the string meets the ground represents the adjacent side.

Here’s how we find the height:

  1. Identify the known quantities:

    • Length of string (hypotenuse, \( h \)): \( 1890 \) feet
    • Angle of elevation (\( \theta \)): \( 68^\circ \)

  2. Use the sine function: The height (\( H \)) of the kite can be found using the sine function: \[ H = h \cdot \sin(\theta) \]

  3. Substituting the known values: \[ H = 1890 \cdot \sin(68^\circ) \]

  4. Calculating \( \sin(68^\circ) \): Using a scientific calculator or sine values: \[ \sin(68^\circ) \approx 0.9272 \]

  5. Calculate the height: \[ H = 1890 \cdot 0.9272 \approx 1756.368 \]

  6. Rounding to the nearest whole number: \[ H \approx 1756 \text{ feet} \]

Therefore, the approximate height of the kite is 1756 feet.

Did you use SOH, CAH, or TOA

In solving for the height of the kite, we used the TOA aspect of the trigonometric mnemonic. Here's a breakdown:

  • TOA stands for "Tangent = Opposite / Adjacent". However, in our case, we actually used the Sine function, which is also part of this mnemonic when describing a right triangle:

  • SOH: Sine = Opposite / Hypotenuse

  • CAH: Cosine = Adjacent / Hypotenuse

  • TOA: Tangent = Opposite / Adjacent

Since we needed to find the height (opposite side) of the triangle, given the length of the string (hypotenuse), we specifically used SOH (Sine): \[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]

Thus, we rearranged it to calculate the height of the kite: \[ \text{Height} = \text{Hypotenuse} \cdot \sin(\theta) \]

In this analysis, we used SOH to find the height of the kite.

What did sin(68degrees) equal?

The value of \(\sin(68^\circ)\) is approximately \(0.9272\).

If you want a more precise value, using a calculator, \(\sin(68^\circ)\) is typically calculated as follows:

\[ \sin(68^\circ) \approx 0.9271838545667874 \]

For most practical purposes, \(\sin(68^\circ) \approx 0.9272\) is sufficiently accurate. If you need to compute values directly or round them, be sure to specify how many decimal places are necessary.