Question

the light from a rotating offshore beacon can illuminate effectively up to a distance of 250 m. from a point on the shore that is 440m from the beacon, the sight line to the beacon makes an angle of 17° with the shoreline. what length of shoreline is effectively illuminated by the beacon (round to 1 decimal)? β = 0° answer: 0 x water 250 b 440 shoreline 17° ? shore

Explanation:

)=2√(62500 - (440 sin17)^2)=2√(62500 - (128.656)^2)=2√(62500-16552)=2√45948≈2214.35≈428.7. So the answer is ~428.7? Wait wait earlier quadratic difference was ~428.7. Yes! So the correct answer is 428.7? Wait wait let's compute 440 sin17: sin17≈0.2923717047. So 4400.29237≈128.6428. Then r² - h²=250² -128.6428²=62500 - (128.6428)^2. 128.6428²= (120+8.6428)^2=14400 + 21208.6428 + 8.6428²=14400+2074.272+74.7=16548.972. So 62500-16548.972=45951.028. sqrt(45951.028)=~214.36. Multiply by 2: ~428.72≈428.7. Yes! So the answer is 428.7.

Wait why did I overcomplicate with quadratic? The straight line shoreline: the distance from B to shore is h=BQ sin(angle between BQ and shore)=440 sin17°. Then the length of the segment of the shoreline inside the circle (radius 250) is 2*sqrt(r² - h²). That's the chord length formula for a line at distance h from center: length=2√(r² -h²). Exactly! So that's the right way. So the steps should be:

Correct Step1: Calculate distance from B to shore
h = BQ sin(17°) = 440 sin(17°) ≈ 440 * 0.2924 ≈ 128.656 m

Correct Step2: Calculate illuminated length
Length = 2 sqrt(r² - h²) = 2 sqrt(250² - 128.656²) ≈ 2 sqrt(62500 - 16552) ≈ 2 sqrt(45948) ≈ 2 * 214.35 ≈ 428.7 m

Ah yes! That's simpler. So the final answer is 428.7.

Wait but wait let's confirm: The chord length formula is correct here because the shoreline is a straight line, and the beacon is at distance h from this line. The circle around beacon (radius 250) intersects the line at two points; the distance between them is the chord length: 2√(r² - h²). Exactly! So that's the right approach. So the answer is 428.7.

So the final answer is 428.7.

Answer:

)=2√(62500 - (440 sin17)^2)=2√(62500 - (128.656)^2)=2√(62500-16552)=2√45948≈2214.35≈428.7. So the answer is ~428.7? Wait wait earlier quadratic difference was ~428.7. Yes! So the correct answer is 428.7? Wait wait let's compute 440 sin17: sin17≈0.2923717047. So 4400.29237≈128.6428. Then r² - h²=250² -128.6428²=62500 - (128.6428)^2. 128.6428²= (120+8.6428)^2=14400 + 21208.6428 + 8.6428²=14400+2074.272+74.7=16548.972. So 62500-16548.972=45951.028. sqrt(45951.028)=~214.36. Multiply by 2: ~428.72≈428.7. Yes! So the answer is 428.7.

Wait why did I overcomplicate with quadratic? The straight line shoreline: the distance from B to shore is h=BQ sin(angle between BQ and shore)=440 sin17°. Then the length of the segment of the shoreline inside the circle (radius 250) is 2*sqrt(r² - h²). That's the chord length formula for a line at distance h from center: length=2√(r² -h²). Exactly! So that's the right way. So the steps should be:

Correct Step1: Calculate distance from B to shore
h = BQ sin(17°) = 440 sin(17°) ≈ 440 * 0.2924 ≈ 128.656 m

Correct Step2: Calculate illuminated length
Length = 2 sqrt(r² - h²) = 2 sqrt(250² - 128.656²) ≈ 2 sqrt(62500 - 16552) ≈ 2 sqrt(45948) ≈ 2 * 214.35 ≈ 428.7 m

Ah yes! That's simpler. So the final answer is 428.7.

Wait but wait let's confirm: The chord length formula is correct here because the shoreline is a straight line, and the beacon is at distance h from this line. The circle around beacon (radius 250) intersects the line at two points; the distance between them is the chord length: 2√(r² - h²). Exactly! So that's the right approach. So the answer is 428.7.

So the final answer is 428.7.