4. A tree stands vertically on top of a hill whose angle of inclination is 30°. The top and base of the tree are 4√3 and 4 meters, respectively, from a point P at the foot of the hill. What is the angle of subtended by the tree at point P?
Did you make a sketch?
On mine I let the top of the tree be A and its bottom be B
look at triangle APB.
angle ABP = 120° , AP = 4√3, BP = 4
by the sine law:
sinA/4 = sin120/4√3
sinA = 4(√3/2)/(4√3) = .5
A = 30°
so angle APB = 180-120-30 = 30°
Well, isn't this just a tree-mendously mathematical question! Let's branch out and solve it together.
Alright, so we have a hill with an angle of inclination of 30°, the top and base of the tree are 4√3 and 4 meters from point P. Now, what we need to find is the angle subtended by the tree at point P.
To do that, let's create a right triangle. We'll call the height of the tree "h" and the distance from point P to the base of the tree "x". We can use the tangent of the angle of inclination to relate the height and distance.
The tangent of 30° is the opposite side (h) divided by the adjacent side (x), so we have:
tan(30°) = h / x
Since the opposite side is h = 4√3 and the adjacent side is x = 4, we can substitute those values in:
tan(30°) = 4√3 / 4
Simplifying this, we get:
√3 = √3
No surprises there! The square root of 3 is equal to the square root of 3. So, we've come to a conclusion - the angle subtended by the tree at point P is... drumroll, please... 30°!
Now, isn't that oak-ay? Leaf it to me to bring some laughs to geometry!
To find the angle subtended by the tree at point P, we can use trigonometry. Let's call the angle subtended by the tree at point P as x.
Since the tree stands vertically on top of the hill, the angle between the ground and the tree is 90°.
Now, we can observe that we have a right-angled triangle formed by the hill, the tree, and the ground. The opposite side to the angle x is the height of the tree, which is 4√3 meters. The adjacent side to the angle x is the distance from point P to the base of the tree, which is 4 meters.
Using the tangent function, we can write:
tan(x) = opposite/adjacent
tan(x) = (4√3)/4
tan(x) = √3
To find the value of x, we can take the inverse tangent (arctan) of both sides:
x = arctan(√3)
x ≈ 60°
Therefore, the angle subtended by the tree at point P is approximately 60°.
To find the angle subtended by the tree at point P, we can use trigonometry.
First, let's label the triangle formed by the tree, the hill, and point P. Let's call the top of the tree T, the base of the tree B, and the angle of subtended by the tree at point P as θ.
From the given information, we know that the distance from point P to the base of the tree (BP) is 4 meters and the distance from point P to the top of the tree (TP) is 4√3 meters.
Next, let's draw a perpendicular line from point P to the base of the hill, which will intersect the base of the tree at point Q. This forms a right triangle BPQ.
Since we know the lengths of BP and TP, we can use the tangent function to find the angle θ. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side of a right triangle.
In this case, the opposite side is TP and the adjacent side is BP. Therefore, we can write the equation:
tan(θ) = TP / BP
Substituting the given values, we have:
tan(θ) = 4√3 / 4
Simplifying the equation, we get:
tan(θ) = √3
To find the angle θ, we can take the inverse tangent (or arctan) of both sides:
θ = arctan(√3)
Using a calculator, we can evaluate this to:
θ ≈ 60°
Therefore, the angle subtended by the tree at point P is approximately 60°.