Question

find the first mistake in solving:
a ladder 20 ft long leans against a building making a 65° angle
with the ground.
find the height it reaches.

1. use \\( \sin ( 65 ^ { \circ } ) = \frac { h } { 20 } \\).
2. multiply: \\( h = 20 \times 0.9063 \\).
3. final answer: \\( h = 18.13 \\) feet.

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no mistake was made.
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Explanation:

Step1: Recall the sine function definition

In a right - triangle (ladder - building - ground forms a right - triangle), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 65^{\circ}\), the hypotenuse (length of the ladder) \(c = 20\) ft and the opposite side (height \(h\)) is the side we want to find. So, \(\sin(65^{\circ})=\frac{h}{20}\) is correct.

Step2: Calculate \(\sin(65^{\circ})\)

Using a calculator, \(\sin(65^{\circ})\approx0.9063\). Then \(h = 20\times\sin(65^{\circ})\approx20\times0.9063\) is correct.

Step3: Perform the multiplication

\(h=20\times0.9063 = 18.126\approx18.13\) (rounded to two decimal places).

Answer:

No mistake was made.