Calculate to one decimal place, the height of triangle $\triangle$AIB in the diagram below.
Understand the Problem
The question asks to calculate the height of triangle $\triangle$AIB to one decimal place. We are given $\angle$ACD = $54^\circ$ and the length CD = 24m This appears to be a trigonometry problem.
Answer
$33.0m$
Answer for screen readers
$AD \approx 33.0m$
Steps to Solve
- Identify the relevant right triangle
The height of $\triangle AIB$ is AD. $\triangle ACD$ is a right triangle with a known angle $\angle ACD = 54^\circ$ and a known side $CD = 24m$.
- Apply the tangent function
We can use the tangent function to relate the angle $\angle ACD$ to the sides $AD$ (opposite) and $CD$ (adjacent).
$$ \tan(\angle ACD) = \frac{AD}{CD} $$
- Solve for AD
Rearrange to solve for $AD$: $$ AD = CD \cdot \tan(\angle ACD) $$ Substitute the given values: $$ AD = 24 \cdot \tan(54^\circ) $$
- Calculate AD using a calculator
$$ AD = 24 \cdot 1.37638 $$ (Since $\tan(54^\circ) \approx 1.37638$) $$ AD \approx 33.033 $$
- Round to one decimal place
Rounding $33.033$ to one decimal place gives $33.0$.
$AD \approx 33.0m$
More Information
The problem involves using the tangent function in trigonometry to find the length of the side opposite to a given angle in a right-angled triangle.
Tips
- Using the wrong trigonometric function: A common mistake is to use sine or cosine instead of tangent. Remember SOH CAH TOA to help you choose the correct function.
- Incorrect calculator mode: Make sure your calculator is in degree mode when calculating trigonometric functions with angles in degrees.
- Rounding errors: Rounding intermediate values can lead to inaccurate final answers. Try to keep as many decimal places as possible until the final step.
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