Question

1. two office buildings are 51 meters apart. the height of the taller building is 207 meters. the angle of depression from the top of the taller building to the top of the shorter building is 15°. find the height of the shorter building to the nearest meter.9. an engineer is 980 feet from the base of a fountain at fountain hills, arizona. the angle of elevation to the top of the column of water is 29.7°. the surveyor’s angle measuring device is at the same level as the base of the fountain. find the height of the column of water to the nearest 10 feet.10. a firefighter on the ground sees fire break through a window near the top of the building. there is voice contact between the ground and firefighters on the roof. the angle of elevation to the top of the building is 42°. the firefighter is 75 feet from the building and her eyes are 5 feet above the ground. what roof - to - windowsill distance can she report to the firefights on the roof?

Explanation:

(Problem 8):

Step1: Define height difference

Let $h$ = height difference between buildings. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$
$\tan(15^\circ)=\frac{h}{51}$

Step2: Calculate height difference

Rearrange to solve for $h$:
$h=51\times\tan(15^\circ)$
$h\approx51\times0.2679=13.6629$ meters

Step3: Find shorter building height

Subtract $h$ from taller building height:
$\text{Shorter height}=207 - 13.6629$

(Problem 9):

Step1: Define water column height

Let $h$ = height of water column. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$
$\tan(29.7^\circ)=\frac{h}{980}$

Step2: Calculate water column height

Rearrange to solve for $h$:
$h=980\times\tan(29.7^\circ)$
$h\approx980\times0.5704=558.992$ feet

Step3: Round to nearest 10 feet

Round $558.992$ to nearest 10:
$h\approx560$ feet

(Problem 10):

Step1: Calculate height to roof

Let $H_r$ = height from ground to roof. Use $\tan(42^\circ)=\frac{H_r - 5}{75}$
$H_r=75\times\tan(42^\circ)+5$
$H_r\approx75\times0.9004+5=67.53+5=72.53$ feet

Step2: Calculate height to windowsill

Let $H_w$ = height from ground to windowsill. Use $\tan(28^\circ)=\frac{H_w - 5}{75}$
$H_w=75\times\tan(28^\circ)+5$
$H_w\approx75\times0.5317+5=39.8775+5=44.8775$ feet

Step3: Find roof-to-windowsill distance

Subtract $H_w$ from $H_r$:
$\text{Distance}=72.53 - 44.8775$

Answer:

1. Height of shorter building: $\boldsymbol{193}$ meters
2. Height of water column: $\boldsymbol{560}$ feet
3. Roof-to-windowsill distance: $\boldsymbol{28}$ feet