Question

Pranav and Rowen went hiking on different trails near the same mountain. They are both standing at the same angle of elevation from the mountain, on different sides. Rowen is 1,200 feet from the mountain and Pranav is 900 feet from the mountain. They see a goat climbing up the mountain. The goat is 200 feet, vertically, above Rowen's head. Use similar triangle ratios to determine the vertical distance in feet the goat is above Pranav's head. (1 point) Responses The goat is 300 feet above Pranav's head. The goat is 300 feet above Pranav's head. The goat is 200 feet above Pranav's head. The goat is 200 feet above Pranav's head. The goat is 150 feet above Pranav's head. The goat is 150 feet above Pranav's head. The goat is 100 feet above Pranav's head.

Answer 1

To solve the problem using similar triangles, we can set up the proportions based on the heights and distances of Rowen and Pranav from the mountain.

Let \( h_R \) be the height above Rowen's head (which is \( 200 \) feet) and \( h_P \) be the height above Pranav's head that we need to find. The distances from the mountain are \( d_R = 1200 \) feet for Rowen and \( d_P = 900 \) feet for Pranav.

Since the angles of elevation from both positions are the same, we can use the ratio of their distances to set up the following proportion based on similar triangles:

\[
\frac{h_R}{d_R} = \frac{h_P}{d_P}
\]

Now substituting the known values into the proportion:

\[
\frac{200}{1200} = \frac{h_P}{900}
\]

To solve for \( h_P \), we can cross-multiply:

\[
200 \times 900 = 1200 \times h_P
\]

Calculating the left side:

\[
180000 = 1200 \times h_P
\]

Now, divide both sides by \( 1200 \):

\[
h_P = \frac{180000}{1200} = 150
\]

Thus, the vertical distance that the goat is above Pranav's head is \( 150 \) feet.

So, the answer is: **The goat is 150 feet above Pranav's head.**