Zane and his favorite geometry student, , have just set sail for a day of fun on lake Erie. A spring squall quickly forms and catches you off guard, leaving you adrift on the lake. Zane suddenly discovers he has an irrational fear of water and is reduced to hugging the mast. You hear Sherman barking from the top of a lighthouse to help guide you to shore. In order to save Zane and yourself, you must calculate how far away you are from this lighthouse. Using a range finder, you expertly measure the distance from the sailboat to the top of the lighthouse at 350 yards. angle B is 40 degrees.

(a) Based on the above information, find the distance across the water from the sailboat to the island

where the lighthouse is standing. You can only use trigonometry to solve. You must show all work for

credit. Round your answer to the nearest whole number if needed.

(b)Calculate the height of the lighthouse. You can only use trigonometry or Pythagorean Theorem to

solve.

Water is the Adjacent of this right triangle. 350 yards is the length of the Hypotenuse. We have given angle B, 40 degrees.

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According to SOH CAH TOA, we use CAH (cosineΘ = adjacent/hypotenuse).

Θ = 40hyoptenuse = 350adjacent = x

cosine40 = x/350

[multiply both sides by 350]

350 · cosine40 = x

[put calculator in degree mode]

x ≈ 268 yards

So, the distance across the water from sailboat to the island is 268 yards.

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Let's use pythagorean theorem since we already know the adjacent and hypotenuse to find the opposite (opposite is also known as height).

a^2 + b^2 = c^2

a = ? (opposite or height)b = 268 (adjacent)c = 350 (hypotenuse)

a^2 + (268)^2 = (350)^2

a^2 + 71824 = 122500

a^2 = 122500 - 71824

a^2 = 50676

[square both sides]

=

a =

a ≈ 225

So, the height, opposite, of the lighthouse is 225 yards.

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ANSWER: a) 268 yards b) 225 yards

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.