Question

select the correct answer from the drop - down menu.
a boat is 200 feet from shore. the angle its line of sight makes with pier a relative to the shoreline is 35°. the angle its line of sight makes with pier b relative to the shoreline is 75°. how far is pier a from pier b rounded to the nearest foot?
(there is an image showing a right - angled triangle - like figure with the boat at the top, the perpendicular distance from the boat to the shore is 200 ft, the angle between the line from the boat to pier a and the shore is 35°, and the angle between the line from the boat to pier b and the shore is 75°. there is a drop - down menu for choosing the distance between pier a and pier b with options 54 ft, 286 ft, 232 ft, 339 ft, and buttons for reset and next.)

Explanation:

Step1: Find distance from boat to shore projection

Let the projection of the boat on shore be point C. The boat to C is 200 ft (perpendicular).

Step2: Calculate AC using tan(35°)

$\tan(35^\circ) = \frac{200}{AC} \implies AC = \frac{200}{\tan(35^\circ)} \approx \frac{200}{0.7002} \approx 285.6$ ft

Step3: Calculate BC using tan(75°)

$\tan(75^\circ) = \frac{200}{BC} \implies BC = \frac{200}{\tan(75^\circ)} \approx \frac{200}{3.7321} \approx 53.6$ ft

Step4: Compute AB = AC + BC

$AB \approx 285.6 + 53.6 = 339.2 \approx 339$ ft (Wait, correction: angles are from pier A and B to boat's projection. If angles are on opposite sides of C, AB = AC + BC. But wait, original angles: 35° from A to boat, 75° from B to boat relative to shoreline. So AC = 200/tan35 ≈285.6, BC=200/tan75≈53.6, so AB=285.6+53.6≈339. But wait, the dropdown has 339 ft. Wait, earlier miscalculation: tan35≈0.7002, so 200/0.7002≈285.6; tan75≈3.732, so 200/3.732≈53.6. Sum≈339.2→339. So correct answer is 339 ft.

Step1: Define projection point C

Boat's perpendicular to shore is point C.

Step2: Calculate AC

$AC = \frac{200}{\tan(35^\circ)} \approx 286$ ft

Step3: Calculate BC

$BC = \frac{200}{\tan(75^\circ)} \approx 54$ ft

Step4: Sum AC and BC for AB

$AB = 286 + 54 = 340$? No, precise values: 285.6+53.6=339.2→339. So final answer 339 ft.

Step1: Find AC length

$AC = \frac{200}{\tan35^\circ} \approx 285.6$ ft

Step2: Find BC length

$BC = \frac{200}{\tan75^\circ} \approx 53.6$ ft

Step3: Add AC and BC

$AB = 285.6 + 53.6 = 339.2 \approx 339$ ft

Answer:

232 ft