A 17 m ladder is placed against the side of a house and reaches a window that is 14 m above the ground.1)How far from the base of the house is the ladder?(Round your answer to one decimal place and put only numbers in the blank with no units.)2) What is the angle of elevation of the ladder?(Round your answer to one decimal place and put only numbers in the blank with no units.)
Question
A 17 m ladder is placed against the side of a house and reaches a window that is 14 m above the ground.1)How far from the base of the house is the ladder?(Round your answer to one decimal place and put only numbers in the blank with no units.)2) What is the angle of elevation of the ladder?(Round your answer to one decimal place and put only numbers in the blank with no units.)
Solution
- To find the distance from the base of the house to the ladder, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the ladder is the hypotenuse, the height of the window is one side, and the distance from the base of the house to the ladder is the other side.
So, we have:
(Length of the ladder)^2 = (Height of the window)^2 + (Distance from the base of the house to the ladder)^2
Substituting the given values:
(17 m)^2 = (14 m)^2 + (Distance from the base of the house to the ladder)^2
289 m^2 = 196 m^2 + (Distance from the base of the house to the ladder)^2
Rearranging the equation to solve for the distance:
(Distance from the base of the house to the ladder)^2 = 289 m^2 - 196 m^2
(Distance from the base of the house to the ladder)^2 = 93 m^2
Taking the square root of both sides:
Distance from the base of the house to the ladder = sqrt(93 m^2) = 9.6 m (rounded to one decimal place)
- To find the angle of elevation of the ladder, we can use the definition of the sine of an angle in a right triangle, which is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the angle of elevation is the angle at the base of the ladder, the height of the window is the side opposite the angle, and the ladder is the hypotenuse.
So, we have:
sin(Angle of elevation) = (Height of the window) / (Length of the ladder)
Substituting the given values:
sin(Angle of elevation) = (14 m) / (17 m)
Angle of elevation = arcsin[(14 m) / (17 m)] = 54.5 degrees (rounded to one decimal place)
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