a hedgehog wishes to cross a road without being run over. he observes the angle of elevation of a lamp post on the other side of the road to be 27 degrees from the edge of the road and 15 degrees from a point 10m back from the raod. How wide is the road??
2 rt. triangles are formed. The ver
side(y) is common to both triangles.
x = hor side of smaller triangle = Width of the RD.
(x + 10) = hor side of larger triangle
tan27 = y/x,
Eq1. y = x*tan27.
tan15 = y / (x + 10),
Eq2. y = (10 + x)tan15.
Substitute Eq1 for y in Eq2:
x*tan27 = (x + 10)*tan15,
x*tan27 = x*tan15 + 10*tan15,
x*tan27 - x*tan15 = 10*tan15,
0.5095x - 0.2679x = 2.679,
0.2416x = 2.679,
X = 2.679 / 0.2416 = 11.1 m. = Width of RD.
i do not wish to cross a road
To find the width of the road, let's first draw a diagram to visualize the problem.
First, draw a straight line to represent the road. Then, draw a vertical line perpendicular to the road to represent the lamp post. Label the point where the lamp post meets the road as point A.
Next, draw a line from point A to the hedgehog's position 10 meters back from the road. Label this point as B.
Now, let's label the angles mentioned in the problem. The angle of elevation from the edge of the road to the lamp post is 27 degrees, so label this angle as ∠C. The angle of elevation from point B to the lamp post is 15 degrees, so label this angle as ∠D.
We need to find the width of the road, which we can label as x.
To solve the problem, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle to the opposite and adjacent sides of a right triangle.
In triangle ABC, we can apply the tangent function to ∠C to find the length of AB:
tan(∠C) = AB / x
Similarly, in triangle ABD, we can apply the tangent function to ∠D to find the length of AB:
tan(∠D) = AB / 10
Now we have two equations:
AB / x = tan(∠C)
AB / 10 = tan(∠D)
To determine x, we need to find AB in terms of x. Rearranging the first equation, we have:
AB = x * tan(∠C)
Substituting this expression for AB in the second equation, we have:
(x * tan(∠C)) / 10 = tan(∠D)
Now, we can solve for x by rearranging this equation:
x = (10 * tan(∠D)) / tan(∠C)
Plugging in the given values for ∠D (15 degrees) and ∠C (27 degrees), we can calculate x:
x = (10 * tan(15)) / tan(27)
By evaluating this expression, we find that the width of the road is approximately 20.53 meters.