a yardstick, held vertically on a level surface, cast a shadow 1 foot 8 inch long. Find the tangent of the angles that the rays of the sun make with the horizontal
draw a diagram
review the tan function
see that
tanθ = 36/20
To find the tangent of the angles that the rays of the sun make with the horizontal, we can use the concept of similar triangles.
Let's assume that the height of the yardstick is h and the length of the shadow is s.
From the given information, the length of the shadow (s) is 1 foot 8 inches, which is equivalent to 1.67 feet. So, s = 1.67 feet.
Since the yardstick is held vertically, the height of the yardstick and the length of the shadow form the legs of a right triangle.
Using the Pythagorean theorem, we have:
h^2 + s^2 = L^2
where L is the hypotenuse. As the yardstick is held vertically, the hypotenuse is equal to the height of the yardstick.
Therefore, h^2 + s^2 = h^2
Simplifying the equation, we get:
s^2 = 0
This implies that the length of the shadow is zero, which is not possible. Thus, the given information is not consistent.
Please double-check the information provided and make sure it is accurate.
To find the tangent of the angles that the rays of the sun make with the horizontal, we can use the concept of similar triangles.
Let's assume that the height of the yardstick is "h" and the length of the shadow it casts is "s".
From the given information, we know that the shadow length is 1 foot 8 inches, which can be converted to 20 inches.
Using the concept of similar triangles, we can set up the following proportion:
(height of the yardstick) / (length of the shadow) = (distance from the top of the yardstick to the top of the shadow) / (distance from the bottom of the yardstick to the bottom of the shadow)
Let's denote the angles that the rays of the sun make with the horizontal as angle A and angle B, where A is the acute angle.
From the vertical yardstick, we can determine that angle A is the angle between the yardstick and the shadow, and angle B is the angle between the yardstick and the ground.
Now, let's solve for angle A:
tan(A) = (distance from the top of the yardstick to the top of the shadow) / (distance from the bottom of the yardstick to the bottom of the shadow)
Since the yardstick is held vertically on a level surface, the distance from the top of the yardstick to the top of the shadow is equal to the height of the yardstick (h), and the distance from the bottom of the yardstick to the bottom of the shadow is equal to the height of the yardstick plus the length of the shadow (h + s).
Therefore:
tan(A) = h / (h + s)
Substituting the known values:
tan(A) = h / (h + 20)
Similarly, we can solve for angle B:
tan(B) = (shadow length) / (height of the yardstick)
Since the shadow length is 20 inches and the yardstick height is h, the equation becomes:
tan(B) = 20 / h
Thus, you have the equations to find the tangent of angles A and B.
Note: To find the actual values of the angles, you would need to know the precise measurement of the yardstick's height.