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.
To find the tangent of the angles, we need to use the information given about the shadow and the height of the yardstick.
First, let's convert the shadow length into a single unit. Since the shadow is given in feet and inches, we need to convert the inches into feet. We can do this by dividing the number of inches by 12.
1 foot = 12 inches
So, 8 inches = 8/12 = 2/3 feet
Now, we can calculate the total length of the yardstick. The shadow represents the opposite side, and the height of the yardstick represents the adjacent side of a right triangle formed by the rays of the sun, where the angle formed between the horizon and the rays is the angle we are looking for.
Using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides, we can find the total length of the yardstick.
Let's denote the length of the yardstick as 'x,' so we have:
x^2 = (1.67)^2 + x^2
Simplifying the equation, we get:
0 = (1.67)^2 - x^2
To solve for x, we take the square root of both sides of the equation:
x = square root((1.67)^2)
Calculating the square root, we have:
x = 1.67 feet
Now, let's draw a diagram to represent the situation:
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In the diagram, the horizontal line represents the surface, and the vertical line represents the yardstick. The angle we are looking for is the angle between the yardstick and the horizon.
Lastly, to find the tangent of the angle, we use the formula:
tangent(angle) = height (opposite) / base (adjacent)
tangent(angle) = x / 1.67
Calculating this, we find:
tangent(angle) ≈ (1.67 feet) / 1.67
tangent(angle) ≈ 1
Therefore, the tangent of the angle that the rays of the sun make with the horizontal is approximately 1.