A boat is 68 feet from the base of a cliff. If the distance from the top of the cliff to the boat is 17 feet less than twice the height of the cliff to the water, what is the height of the cliff?

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Answer 1 · 2016-12-07T08:40:17

#51# #feet#

Let the height of the cliff be #h#, then the distance from the top of the cliff to the boat will be #2h-17#

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These relations for a right triangle, so we can slove the question using the Pythagoras theorem

#color(blue)(a^2+b^2=c^2#

Where #c# is the longest side and #a,b# are the other two

So,

#rarr68^2+h^2=(2h-17)^2#

#rarr4624+h^2=(2h-17)^2#

Use the property #color(blue)((a-b)^2=a^2-2ab+b^2#

#rarr4624+h^2=4h^2-68h+289#

Simplify

#rarr3h^2-68h-4335=0#

This is a quadratic equation, solve it using the quadratic formula

#color(blue)(h=(-b+-sqrt(b^2-4ac))/(2a)#

Where #a,band c# are the coefficients of the terms

#rarrh=(68+-sqrt(68^2-4(3)(-4335)))/(2(3))#

Simplify

#rarrh=(68+-238)/6#

#rarrh=((68+238)/6,(68-238)/6)#

#color(green)(rArrh=(51,-85/3)#

As distances cannot be negative, the height will be #color(green)(51#