a yardstick, held vertically on a level surface, cast a shadow 1 foot
5 inch long. Find the tangent of the angles that the rays of the sun make with the horizontal.
1'5" = 17"
so,
tanθ = 36/17
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 consider the vertical yardstick and its shadow. We have a right triangle formed by the yardstick, its shadow, and the sun's rays.
Let's label the length of the shadow as 1 foot 5 inches, which is equal to 1.42 feet (there are 12 inches in a foot). We'll also assume that the height of the yardstick is h feet.
In a similar triangle, the ratio of corresponding sides is the same. So, the ratio of the height of the yardstick to the length of the shadow is equal to the ratio of the distance from the top of the yardstick to the sun's rays to the length of the shadow.
Let's represent the angle that the rays of the sun make with the horizontal as θ.
The tangent of θ is the ratio of the height of the yardstick (h) to the length of the shadow.
Therefore, tan(θ) = h / 1.42.
To find the value of tan(θ), we need to know the height of the yardstick (h). Without that information, we cannot determine the exact tangent of the angles.
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. Here's how you can do it step by step:
Step 1: Convert the length of the shadow into a single unit. In this case, we will convert it to inches. Since there are 12 inches in a foot, the shadow length becomes:
1 foot 5 inches = (12 inches x 1 foot) + 5 inches = 17 inches.
Step 2: Since the yardstick is held vertically, it forms a right triangle with the horizontal surface (ground) and its shadow. Let's denote the length of the yardstick as y and the length of the shadow as s. Based on the problem, we have:
s = 17 inches.
Step 3: Now, we can use the tangent function to find the angle θ that the sun rays make with the horizontal. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the length of the shadow (s) and the adjacent side is the length of the yardstick (y). Therefore:
tan(θ) = s / y.
Step 4: Rearrange the equation to solve for tan(θ):
tan(θ) = s / y.
Step 5: Substitute the values for s and y:
tan(θ) = 17 inches / y.
Since the length of the yardstick (y) is not provided in the problem, we cannot determine the exact value of tan(θ) without that information. However, if you have the length of the yardstick, you can plug it into the equation to find the tangent of the angles that the rays of the sun make with the horizontal.