Question

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

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 30$ feet), the height up the wall is the opposite side ($a$) to the given angle ($\theta=72^{\circ}$). We use the sine function: $\sin\theta=\frac{a}{c}$.

Step2: Solve for the height $a$

Substitute $c = 30$ and $\theta = 72^{\circ}$ into the formula $a=c\sin\theta$. So, $a = 30\times\sin(72^{\circ})$.
Since $\sin(72^{\circ})\approx0.951057$, then $a=30\times0.951057 = 28.53171$.

Answer:

$28.53$