A ladder that is 15 feet long is placed so that it reaches from level ground to the top of a vertical wall that is 13 feet high
A. Use the law of sines to find the angle that the ladder makes with the ground to the nearest hundredth
B. Is more than one position of the ladder possible? Explain your answer.
A rt triangle is formed:
hyp. = 15 Ft. = Length of ladder.
Ver. side = 13Ft.
A. sinA / 13 = sin90 / 15.
sinA = 13*sin90 / 15 = 60.07 Deg.
B. The bottom of the ladder can be moved closer to are farther from the
bottom of the wall. But the hor. and
ver. side should always be > 0 to keep
the rt triangle.
CORRECTION: sinA=13*sin90/15 = 0.8666.
A = 60.07 Deg.
To solve this problem, we can use the law of sines, which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all sides and angles in the triangle.
A. To find the angle that the ladder makes with the ground, we will label the ladder as side c, the vertical wall as side a, and the base of the triangle as side b.
By applying the law of sines, we have the following equation:
sin(A) / a = sin(B) / b = sin(C) / c
Since we want to find angle A, which is the angle that the ladder makes with the ground, we have:
sin(A) / 13 = sin(90° - A) / 15
Now, we can solve this equation to find angle A.
sin(A) / 13 = cos(A) / 15 (since sin(90° - A) = cos(A))
Cross multiplying the equation, we get:
15 * sin(A) = 13 * cos(A)
Dividing both sides by 13 * sin(A), we have:
15 / 13 = cos(A) / sin(A)
Using the trigonometric identity cot(A) = cos(A) / sin(A), we rearrange the equation as:
cot(A) = 15 / 13
Now, we can find the value of A by taking the inverse cotangent of both sides:
A = cot^(-1)(15 / 13)
Using a calculator, we find that A is approximately equal to 42.64 degrees. Therefore, the angle that the ladder makes with the ground is approximately 42.64 degrees to the nearest hundredth.
B. There is only one position of the ladder that is possible. Since the ladder is 15 feet long and the wall is 13 feet high, the ladder must be inclined at an angle such that its length is longer than the height of the wall. If the ladder was placed at a steeper angle, its length would not reach the top of the wall. Therefore, there is only one position of the ladder that allows it to reach the top of the wall.