Question

katherine leans a 30-foot ladder against a wall so that it forms an angle of \\(72^\circ\\) with the ground. how high up the wall does the ladder reach? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Identify given values and target

We are given:

• Length of the ladder (hypotenuse, \(h\)) = \(30\text{ ft}\)
• Angle with the ground (\(\theta\)) = \(72^\circ\)

We need to find:

• Height up the wall (opposite side, \(y\))

Set up the trigonometric ratio

Using Right Triangle Trigonometry, we relate the opposite side and the hypotenuse:

$$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $$

Using Trigonometric Applications, we substitute our known values:

$$ \sin(72^\circ) = \frac{y}{30} $$

Solve for the height

Isolate \(y\) by multiplying both sides by \(30\):

$$ y = 30 \cdot \sin(72^\circ) $$

Calculate the numerical value

Using a calculator to find \(\sin(72^\circ)\):

$$ \sin(72^\circ) \approx 0.9510565 $$

Multiply by \(30\):

$$ y \approx 30 \cdot 0.9510565 \approx 28.531695\text{ ft} $$

Round to the nearest hundredth

Rounding \(28.531695\) to two decimal places:

$$ y \approx 28.53\text{ ft} $$

Answer:

\(28.53\text{ ft}\)