A bird at the top of a tree looks down at a field mouse with an angle of depression of 65 degrees. If the field mouse is 30 meters from the base of the tree, find the distance from which the field mouse to the bird's eyes. Round the answer to the nearest tenth.

I feel the question required not the ht of tree but distance between bird's eye and mouse which will be=hypoteneuse =30/cos65. May please check.

To find the distance from the field mouse to the bird's eyes, we can use trigonometry.

Let's assume that the distance from the bird's eyes to the base of the tree is x meters.

We can set up a right triangle with the bird's eyes at the top of the tree, the field mouse at the base of the tree, and the distance from the bird's eyes to the mouse as the hypotenuse.

Using the angle of depression, we can find the length of the adjacent side of the triangle.

In this case, the adjacent side would be x - 30 meters (the distance from the base of the tree to the field mouse).

Now we can use the cosine function to find the length of the hypotenuse:

cos(65 degrees) = (x - 30) / x

To solve for x, we can rearrange the equation:

x - 30 = x * cos(65 degrees)

x - x * cos(65 degrees) = 30

x(1 - cos(65 degrees)) = 30

x = 30 / (1 - cos(65 degrees))

Calculating this, we find:

x ≈ 53.6 meters

Therefore, the distance from the field mouse to the bird's eyes is approximately 53.6 meters.

To find the distance from the bird's eyes to the field mouse, we can use trigonometry and the given angle of depression.

Let's call the distance from the bird's eyes to the field mouse "x" (in meters).

Since we have an angle of depression, we can use the tangent function, which is defined as the opposite side divided by the adjacent side.

In this case, the opposite side is the height of the tree, and the adjacent side is the distance from the base of the tree to the field mouse.

We can set up the equation as follows:

tan(65°) = height of the tree / distance from the base of the tree to the field mouse

Now, substitute the known values into the equation:

tan(65°) = height of the tree / 30 meters

To solve for the height of the tree, we need to isolate it. We can do this by multiplying both sides of the equation by 30 meters:

30 meters * tan(65°) = height of the tree

Using a scientific calculator, we can find the value of tan(65°) to be approximately 2.14451.

Therefore:

30 meters * 2.14451 = height of the tree

Now, let's calculate the height of the tree:

30 meters * 2.14451 = 64.3353 meters

So, the height of the tree is approximately 64.3 meters.

Now that we know the height of the tree, we can find the distance from the bird's eyes to the field mouse (x). We can use the tangent function again:

tan(65°) = height of the tree / x

Substitute the known values:

tan(65°) = 64.3353 meters / x

To solve for x, we need to isolate it. We can do this by dividing both sides of the equation by tan(65°):

x = 64.3353 meters / tan(65°)

Using a scientific calculator, we can find the value of tan(65°) to be approximately 2.14451.

Therefore:

x = 64.3353 meters / 2.14451

Calculating the value:

x ≈ 29.99 meters

Rounding to the nearest tenth:

x ≈ 30.0 meters

So, the distance from the bird's eyes to the field mouse is approximately 30.0 meters.

did you make a sketch of the right-angled triangle?

Remember the line of sight is parallel to the ground, and alternate angles are equal.

I have
tan 65° = h/30 , where h is the height of the tree
h = 30tan65 = 64.3 metres