Question

1. you have a 20-foot ladder that you are using as you paint your house. you place the ladder so that it forms an angle of 30° at the point of contact between the ladder and the ground. how high will the top of the ladder be above the ground? (6 points)

Explanation:

Step1: Identify the triangle type

This is a right - triangle problem where the ladder is the hypotenuse ($c = 20$ feet), the angle between the ladder and the ground $\theta=30^{\circ}$, and we want to find the height $h$ (opposite side to the angle $\theta$). We use the sine function, which is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Apply the sine formula

We know that $\sin\theta=\frac{h}{c}$. Given $\theta = 30^{\circ}$ and $c = 20$ feet, and $\sin(30^{\circ})=\frac{1}{2}$. Substituting the values into the formula: $h=c\times\sin\theta$.
So $h = 20\times\sin(30^{\circ})$.
Since $\sin(30^{\circ})=\frac{1}{2}$, we have $h = 20\times\frac{1}{2}$.

Step3: Calculate the height

$20\times\frac{1}{2}=10$.

Answer:

The top of the ladder will be 10 feet above the ground.