Question

tamika leans a 30 - foot ladder against a wall so that it forms an angle of 65° 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:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 30$ feet), the angle between the ladder and the ground is $\theta=65^{\circ}$, and we want to find the height $h$ up the wall (opposite side to the given angle). We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
$\sin\theta=\frac{h}{c}$

Step2: Substitute the values

Substitute $\theta = 65^{\circ}$ and $c = 30$ into the formula $\sin\theta=\frac{h}{c}$. So, $\sin(65^{\circ})=\frac{h}{30}$.
$h = 30\times\sin(65^{\circ})$

Step3: Calculate the value

We know that $\sin(65^{\circ})\approx0.9063$. Then $h = 30\times0.9063=27.189$.
Rounding to the nearest hundredth, $h\approx27.19$ feet.

Answer:

$27.19$ feet