Question

diana works in a building that is 130 feet tall. she is outside, looking up at the building at an angle of 37° from her feet to the top of the building. if diana walks forward and her angle looking to the top of the building changes to 40°, how much closer is she to the building? round the answer to the nearest tenth of a foot. 10.3 ft 17.6 ft 30.2 ft 97.2 ft

Explanation:

Step1: Calculate initial distance

We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the initial distance be $x_1$. Given $\theta = 37^{\circ}$ and opposite side (height of building) $h = 130$ feet.
Since $\tan37^{\circ}=\frac{130}{x_1}$, then $x_1=\frac{130}{\tan37^{\circ}}$.
We know that $\tan37^{\circ}\approx0.7536$, so $x_1=\frac{130}{0.7536}\approx172.5$ feet.

Step2: Calculate new distance

Let the new distance be $x_2$. Given $\theta = 40^{\circ}$ and $h = 130$ feet.
Since $\tan40^{\circ}=\frac{130}{x_2}$, then $x_2=\frac{130}{\tan40^{\circ}}$.
We know that $\tan40^{\circ}\approx0.8391$, so $x_2=\frac{130}{0.8391}\approx154.9$ feet.

Step3: Calculate the difference

The difference in distance $\Delta x=x_1 - x_2$.
$\Delta x=172.5-154.9 = 17.6$ feet.

Answer:

17.6 ft