David the monkey is looking down at a 33° angle from the top of a tree. There are bananas on the ground 42.7 ft from the base of the tree. If David flies down from the tree in a diagonal path, as shown in the diagram, what distance would he travel? Round to the nearest tenth.
We form a rt. triang:
X = 42.7 Ft = Hor. side.
Y = Ht of tree = Ver. side.
Z = Diag. = Hyp.
A = 33 Deg. = Angle bet. hyp and gnd.
Z = X/cosA = 42.7 / cos33 = 50.9 Ft. =
Dist .traveled.
To find the distance David would travel, we can use trigonometry. We need to find the length of the diagonal path, which is the hypotenuse of a right triangle.
In this case, the height of the tree forms one side of the triangle, the distance from the tree to the bananas forms the other side, and the diagonal path represents the hypotenuse.
First, we can find the height of the tree by using the tangent function. Tangent is defined as the opposite side length divided by the adjacent side length in a right triangle.
Let's call the height of the tree h and the distance from the tree to the bananas d.
The tangent of the angle (33°) is equal to the height of the tree divided by the distance from the tree to the bananas:
tangent(33°) = h / d
To find h, rearrange the equation:
h = d * tangent(33°)
Now, we can calculate the height of the tree using this equation:
h = 42.7 ft * tan(33°)
Using a calculator, we find h ≈ 26.6 ft.
Next, we can find the length of the diagonal path (hypotenuse) by using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's call the length of the diagonal path L.
L^2 = h^2 + d^2
Plugging in the values we've already found:
L^2 = 26.6^2 + 42.7^2
Calculating this with a calculator, we find L^2 ≈ 2561.45.
Taking the square root of both sides:
L ≈ √2561.45 ≈ 50.6 ft
Therefore, David would travel approximately 50.6 ft along the diagonal path to reach the bananas on the ground.