A pine tree stands beside a road that has a 7 degree gradient. When the angle of elevation to the sun is 59 degrees, the tree casts a shadow of 31 ft down slope along the road. What is the height of the tree?

Draw a diagram.

If the tip of the shadow is x feet horizontally away from the tree of height x, we have

x/31 = cos 7°
(h + 31*sin7°)/x = tan 59°

I get h = 47.4 ft

To find the height of the tree, we can use trigonometry. We have the angle of elevation to the sun (59 degrees) and the length of the shadow (31 ft).

First, we need to find the length of the opposite side in the triangle formed by the tree, its shadow, and the road. This represents the height of the tree.

Given that the road has a gradient of 7 degrees, we can determine the length of the adjacent side by using the formula:

adjacent = shadow / tan(gradient)

In this case, the gradient is 7 degrees and the shadow length is 31 ft. Plugging in these values, we get:

adjacent = 31 / tan(7)

Calculate the value of adjacent by finding the tangent of 7 degrees and dividing the shadow length by this value.

Now, we can use the angle of elevation (59 degrees) and the adjacent side length to find the height of the tree. We can use the formula:

opposite = adjacent * tan(angle)

Substituting the values we have:

height = adjacent * tan(59)

To get the height of the tree, multiply the adjacent length by the tangent of 59 degrees.

Now, let's calculate it:

1. Calculate the adjacent side length (adjacent):
adjacent = 31 / tan(7)

2. Calculate the height of the tree (height):
height = adjacent * tan(59)

Note: Remember to use a scientific calculator that can compute trigonometric functions in degrees.

By following these steps and performing the calculations, you will find the height of the tree.