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makvit [3.9K]
3 years ago
14

Find the first five terms of each sequence an = -2(n-7)

Mathematics
1 answer:
Scrat [10]3 years ago
8 0

Answer:

when n=1

= -2(1-7) = 12

when n=2

= -2(2-7) = 10

when n=3

= -2(3-7) = 8

when n=4

= -2(4-7) = 6

when n=5

= -2(5-7) = 4

Step-by-step explanation:

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The angles of depression of two ships from the top of the light house are 45° and 30° towards east. If the ships are 200 m apart
azamat

Answer:

273 meters

Step-by-step explanation:

See image attached for the diagram I used to represent this scenario.

The distance between the ships, at angles 30 and 45, is 200 meters. The distance between the left ship and the lighthouse is x meters.

We can use trigonometric ratios to solve this problem. We can use the tangent ratio \big{(} \frac{\text{opposite}}{\text{adjacent}} \big{)} to create an equation with the two angles.

  • \displaystyle \text{tan(45)} = \frac{h}{x}
  • \displaystyle \text{tan(30)} = \frac{h}{x+200} }

Let's take these two equations and solve for x in both of them.

<h2>\textbf{Equation I}</h2>
  • \displaystyle \text{tan(45)} = \frac{h}{x}  

tan(45) = 1, so we can rewrite this equation.

  • \displaystyle 1=\frac{h}{x}

Multiply x to both sides of the equation.

  • \displaystyle x = h
<h2>\textbf{Equation II}</h2>
  • \displaystyle \text{tan(30)} = \frac{h}{x+200} }

Multiply x + 200 to both sides and divide h by tan(30).

  • \displaystyle \text{x + 200} = \frac{h}{\text{tan (30)}}  

Subtract 200 from both sides of the equation.

  • \displaystyle \text{x} = \frac{h}{\text{tan (30)}} - 200

Simplify h/tan(30).

  • x=\sqrt{3}h - 200  
<h2>\textbf{Equation I = Equation II}</h2>

Take Equation I and Equation II and set them equal to each other.

  • h=\sqrt{3}h-200

Subtract √3 h from both sides of the equation.

  • h-\sqrt{3}h=-200

Factor h from the left side of the equation.

  • h(1-\sqrt{3}) =-200

Divide both sides of the equation by 1 - √3.

  • \displaystyle h=\frac{-200}{1-\sqrt{3} }

Rationalize the denominator by multiplying the numerator and denominator by the conjugate.

  • \displaystyle h=\frac{-200}{1-\sqrt{3} } \big{(} \frac{1+\sqrt{3} }{1+\sqrt{3}} \big{)}
  • \displaystyle h=\frac{-200+200\sqrt{3} }{1-3}
  • \displaystyle h =\frac{-200+200\sqrt{3} }{-2}

Simplify this equation.

  • \displaystyle h=100+100\sqrt{3}
  • h=273.20508075

The height of the lighthouse is about 273 meters.

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