A man 1.9m tall observes the angle of elevation of the top and bottom of a mast as 65° and 41° respectively.If the mast is mounted on a house 12m tall,find correct to 3 significant figures
I.distance of the man from the house
ii.length of the mast
I need solution
I. Well, let's call the distance of the man from the house "x". Now, we can form a right triangle with the man at one corner, the top of the mast at another corner, and the bottom of the mast at the third corner. The height of the mast is the sum of the height of the man (1.9m) and the height of the house (12m). So, the total height of the mast is 1.9m + 12m = 13.9m.
Now, using trigonometry, we can set up the following equations:
tan(65°) = (13.9m)/(x)
tan(41°) = (12m)/(x)
To solve these equations, we can rearrange to express x in terms of the known values:
x = (13.9m) / tan(65°)
x = (12m) / tan(41°)
x ≈ 12.29m
So, the distance of the man from the house is approximately 12.29m.
II. To find the length of the mast, we can use the right triangle formed earlier. The length of the mast is the hypotenuse of this triangle. We can use the Pythagorean theorem to solve for it:
Length of mast = √((13.9m)^2 + (x)^2)
Length of mast ≈ √((13.9m)^2 + (12.29m)^2)
Length of mast ≈ √(193.21m^2 + 151.0841m^2)
Length of mast ≈ √(344.2941m^2)
Length of mast ≈ 18.54m
So, the length of the mast is approximately 18.54m.
To find the distance of the man from the house and the length of the mast, we can use the tangent function.
Let's calculate the distance of the man from the house first:
Step 1: Draw a diagram to represent the situation.
/ |
/ |
12m |
/ |
/ | 41° <- Angle of elevation from man to the bottom of the mast
/ |
/ |
/ |
/ |
/ |
Man /_65° |
| |
| | <- Top of the mast
Step 2: We can create a right triangle with the height of the house (12m) as one side and the distance from the man to the house (let's call it x) as another side. The angle of elevation from the man to the top of the mast (65°) will be the angle of the triangle.
Step 3: Apply the tangent function to the triangle:
tan(65°) = height of the house / distance from the man to the house
tan(65°) = 12 / x
x = 12 / tan(65°)
Using a calculator, we find that x is approximately 6.436 meters.
Therefore, the distance of the man from the house is approximately 6.436 meters (correct to 3 significant figures).
Now let's calculate the length of the mast:
Step 1: We can create another right triangle, this time with the height of the mast as one side (let's call it y) and the distance from the man to the house (6.436m) plus the distance from the man to the bottom of the mast (let's call it z) as another side. The angle of elevation from the man to the bottom of the mast (41°) will be the angle of the triangle.
Step 2: Apply the tangent function to the triangle:
tan(41°) = height of the mast / (distance from the man to the house + distance from the man to the bottom of the mast)
tan(41°) = y / (6.436 + z)
Step 3: We can substitute the value of x we found earlier into the equation:
tan(41°) = y / (6.436 + z)
Step 4: Rewrite the equation to solve for z:
z = y / tan(41°) - 6.436
Step 5: Substitute the value of y (12m) into the equation:
z = 12 / tan(41°) - 6.436
Using a calculator, we find that z is approximately 6.676 meters.
Therefore, the length of the mast is approximately 6.676 meters (correct to 3 significant figures).
To solve this problem, let's break it down into two parts:
i. Distance of the man from the house:
We can use trigonometry to find the distance of the man from the house. Let's consider the triangle formed by the man, the house, and the top of the mast. The angle of elevation from the man to the top of the mast is 65°.
Using the tangent function, we can set up the following equation:
tan(65°) = (12m + x) / x
Simplifying the equation, we get:
tan(65°) = 12m / x + 1
To isolate x (distance of the man from the house), we rearrange the equation:
x * tan(65°) = 12m + x
Next, we can substitute the value of tan(65°) from a calculator:
1.880726465 = 12m / x + 1
Now, we solve for x:
x = 12m / (1.880726465 - 1)
Calculating the value, we get:
x ≈ 16.417 meters
Hence, the distance of the man from the house is approximately 16.417 meters.
ii. Length of the mast:
Again, using trigonometry, we can consider the triangle formed by the man, the house, and the bottom of the mast. The angle of elevation from the man to the bottom of the mast is 41°.
Similarly, we can set up an equation using the tangent function:
tan(41°) = 12m / (x + h)
Where h is the height of the mast.
Simplifying the equation, we get:
tan(41°) = 12m / (x + h)
To isolate h (length of the mast), we rearrange the equation:
h = 12m / tan(41°) - x
Next, we can substitute the value of tan(41°) from a calculator:
h = 12m / 0.869286737 - 16.417m
Now, we solve for h:
h ≈ 12.448 meters
Hence, the length of the mast is approximately 12.448 meters.
To summarize, the solution to the problem is:
i. Distance of the man from the house: approximately 16.417 meters
ii. Length of the mast: approximately 12.448 meters