It was observed that the shadow of a vertical pole of 6 meters longer when the angle of elevation of the sun was 30° than when it was 60°.By means of a sketched diagram,calculate,correct to two decimal places,the height of the pole.
tan 60deg = y/x y/x =1.7325
tan 30 deg = y/(x + 6) = .577
x = shadow and y = pole
y = 1.7325x
y = .577(x +6)
Since they both equal y, you can say..
1.7325x = .577x + 3.462
Solve for x to get the shadow for the 60 degrees.
Find y, by putting this answer into both equations. If you did everything correct, you should get the same value for y. I got around 5.2 meters. See if you get the same thing.
To solve this problem, let's assume the height of the pole is "h" meters.
When the angle of elevation of the sun is 30°, we can establish a right triangle between the pole, its shadow, and the sun. Let the length of the shadow be "x" meters.
Using trigonometry, we can write the equation: tan(30°) = h / x
Similarly, when the angle of elevation of the sun is 60°, we have another right triangle with length "x+6" as the shadow. The height of the pole (h) remains the same. So, the equation becomes: tan(60°) = h / (x+6)
Let's solve these two equations simultaneously:
tan(30°) = h / x
tan(60°) = h / (x+6)
Using the tangent values of 30° and 60° (0.577 and 1.732 respectively), we can rewrite the equations as:
0.577 = h / x
1.732 = h / (x+6)
By solving these two equations simultaneously, we can find the value of "h" (the height of the pole).
Divide the second equation by the first equation:
1.732 / 0.577 = (h / (x+6)) / (h / x)
2.999 = (x / (x+6))
Cross-multiply:
2.999(x+6) = x
2.999x + 17.994 = x
2.999x - x = -17.994
1.999x = -17.994
x = -17.994 / 1.999
x ≈ -9
Since negative lengths don't make sense in this context, we discard the negative solution.
Now, we have the length of the shadow "x" as approximately 9 meters.
Substitute this value back into the first equation to find the height of the pole:
0.577 = h / 9
h = 0.577 * 9
h ≈ 5.19
Therefore, the height of the pole is approximately 5.19 meters.
To solve this problem, we can use basic trigonometry and set up two equations based on the given information.
Let's sketch the diagram to visualize the situation:
A
/|
/ |
/ |
/ | h
/ |
/_____|
x
In the diagram, let A be the top of the pole, and let x be the length of the shadow when the angle of elevation of the sun is 60°. The length of the shadow is then x + 6 when the angle of elevation is 30°. The height of the pole is represented by h.
We can start by finding the relationship between the height of the pole and the length of the shadow using trigonometry.
When the angle of elevation is 60°:
tan(60°) = h/x
When the angle of elevation is 30°:
tan(30°) = h/(x+6)
Now we can set up the equations:
tan(60°) = h/x
tan(30°) = h/(x+6)
We need to solve these equations to find the value of h (the height of the pole).
Using a scientific calculator, we can find that the tangent of 60° is √3 and the tangent of 30° is 1/√3.
Equation 1: √3 = h/x
Equation 2: 1/√3 = h/(x+6)
To solve these equations, we can use algebraic manipulation:
From Equation 1, we can solve for x:
x = h/√3
Now substitute this value of x into Equation 2:
1/√3 = h/(h/√3 + 6)
Let's simplify the equation:
1/√3 = (√3 * h) / (h + 6√3)
√3/(3) = (√3 * h) / (h + 6√3) (rationalizing the denominator by multiplying top and bottom by √3)
Cancel out √3 on both sides:
1/3 = h / (h + 6√3)
Cross multiply to get rid of the denominator:
h + 6√3 = 3h
Rearrange the equation:
2h = 6√3
h = 3√3
So, the height of the pole is approximately 3√3 meters.
Note: To find the numerical approximation, replace √3 with its approximate value of 1.732.
Therefore, the height of the pole is approximately 5.20 meters (rounded to two decimal places).