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A spotlight is mounted on the eaves of a house 24 feet above the ground. A flower bed runs between the house and the​ sidewalk, so the closest the ladder can be placed to the house is 18 feet. How long a ladder is needed so that an electrician can reach the place where the light is​ mounted?

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Part 1
The length of the ladder needs to be
  
enter your response here ft.

1 answer

To determine the length of the ladder needed for the electrician to reach the spotlight mounted 24 feet above the ground, with the base of the ladder positioned 18 feet away from the house, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse \(c\) (the ladder, in this case) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\):

\[ c^2 = a^2 + b^2 \]

Here:

  • \(a\) is the height from the ground to the spotlight (24 feet),
  • \(b\) is the distance from the base of the ladder to the house (18 feet),
  • \(c\) is the length of the ladder.

Now you can plug in the numbers:

\[ c^2 = 24^2 + 18^2 \] \[ c^2 = 576 + 324 \] \[ c^2 = 900 \] \[ c = \sqrt{900} \] \[ c = 30 \]

Therefore, the length of the ladder needed is \(30\) feet.