diana works in a building that is 130 feet tall. she is outside, looking up at the building at an angle of…

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

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

Answer

Explanation:

Step1: Calculate the initial distance

We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let $x_1$ be the initial distance from Diana to the building. Given $\theta = 37^{\circ}$ and the height of the building (opposite side) $h = 130$ feet. Since $\tan37^{\circ}=\frac{130}{x_1}$, then $x_1=\frac{130}{\tan37^{\circ}}$. Using $\tan37^{\circ}\approx0.754$, we have $x_1=\frac{130}{0.754}\approx172.4$ feet.

Step2: Calculate the new distance

Let $x_2$ be the new distance from Diana to the building. Given $\theta = 40^{\circ}$ and $h = 130$ feet. Since $\tan40^{\circ}=\frac{130}{x_2}$, then $x_2=\frac{130}{\tan40^{\circ}}$. Using $\tan40^{\circ}\approx0.839$, we have $x_2=\frac{130}{0.839}\approx154.9$ feet.

Step3: Calculate the difference in distances

The difference $\Delta x=x_1 - x_2$. $\Delta x=172.4 - 154.9 = 17.5\approx17.6$ feet.

Answer:

17.6 ft