a lader is leaning against a jouse and is forming a 60 degree angle with the ground.
if the top of the ladder is 24 feet off of the gound, how long is the ladder? leave your answer in exact form.
looks like routine trig
sin 60º = 24/h
h = 24/sin60
= 24/(√3/2) from the 30-60-90 triangle ratios
= 48/√3
= 16√3 after rationalizing the denominator
To find the length of the ladder, we can use trigonometric functions. In this case, since we know the angle and the opposite side length, we can use the trigonometric function sine.
The sine function relates the angle of a right triangle to the ratio of the length of the side opposite to the angle to the length of the hypotenuse of the triangle. In this case, the angle is 60 degrees, the opposite side is 24 feet, and we want to find the length of the hypotenuse (the ladder).
The formula for sine is:
sin(angle) = opposite/hypotenuse
In our case:
sin(60 degrees) = 24 feet / hypotenuse
To solve for the length of the ladder, we isolate the hypotenuse by multiplying both sides of the equation by the hypotenuse:
hypotenuse * sin(60 degrees) = 24 feet
Next, we can divide both sides of the equation by sin(60 degrees):
hypotenuse = 24 feet / sin(60 degrees)
Now, let's calculate the value of sin(60 degrees):
sin(60 degrees) = sqrt(3)/2
Substituting this value back into the equation:
hypotenuse = 24 feet / (sqrt(3)/2)
To simplify the expression, we multiply the numerator and denominator by 2:
hypotenuse = (24 feet * 2) / sqrt(3)
Finally, simplifying the expression:
hypotenuse = 48 feet / sqrt(3)
Thus, the length of the ladder is 48 feet divided by the square root of 3, and we leave the answer in exact form.