A 25-foot ladder stands against a vertical wall at an angle of n degrees with the ground. If sin n = 4 / 5, how far is the base of the ladder from the wall?
A. 12
B. 13
C. 14
D. 15
E. 16
Sin n = 4 / 5. Remember that the formula for sin would be Opposite / Hypotenuse. So:
Sin n = Opposite / Hypotenuse = 4 / 5
Sine n = Opposite / 25 = 4 / 5
5 Opposite = 100
5 / 5 Opposite = 100 / 5
Opposite = 20.
Now what we do is use Pythagorean Theorem to find the length of the adjacent side.
a^2 + b^2 = c^2
20^2 + b^2 = 25^2
400 + b^2 = 625
400 - 400 + b^2 = 625 - 400
b^2 = 225
sqrt(b^2) = sqrt(225)
b = 15.
Answer choice D
To find how far the base of the ladder is from the wall, we can use the equation:
sin(n) = opposite / hypotenuse
In this case, the opposite side is the distance between the base of the ladder and the wall, and the hypotenuse is the length of the ladder.
Given that sin(n) = 4/5, we can substitute the values into the equation:
4/5 = opposite / 25
To find the value of the opposite side, we can cross-multiply:
opposite = (4/5) * 25
opposite = 20
Therefore, the base of the ladder is 20 feet away from the wall.
The correct choice is A. 12.
To find the distance between the base of the ladder and the wall, we can use the trigonometric relationship between the sine, angle, and the sides of a right triangle.
In this problem, the ladder, the vertical wall, and the ground form a right triangle. The length of the ladder is the hypotenuse of the triangle, and the distance from the base of the ladder to the wall is one of the legs.
Given that sin n = 4/5, we can determine the length of the other leg of the triangle using the Pythagorean theorem. Recall that the ratio of the sine of an angle to the length of the hypotenuse is equal to the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the distance from the base of the ladder to the wall, and the hypotenuse is the length of the ladder.
Let's use x to represent the distance from the base of the ladder to the wall. According to the Pythagorean theorem, we have:
x^2 + (25^2) = (4/5)^2
x^2 + 625 = 16/25
x^2 = (16/25) - 625
Now, simplify and solve for x:
x^2 = (16 - 625(25^2)) / (25^2)
x^2 = (16 - 625(625)) / (25^2)
x^2 = (16 - 625(625)) / (625^2)
x^2 = (-625^2) / (625^2)
x^2 = -1
Since the square of a real number cannot be negative, this equation has no real solutions. Therefore, the base of the ladder is 0 feet away from the wall.
None of the given answer choices match the solution, so none of the options are correct.