a man 1.9 metres to observe the angles of elevation of the top and bottom of your mouth as 65° and 41 degrees respectively if the mast is mounted on a house 12m high find to the three significant figures a) the distance of the Man from the house and b) the length of the mast
Draw a diagram. You can see that if the mast (not your mouth!) has height h, and the distance to the house is x, then
(h+12-1.9)/x = tan65°
(12-1.9)/x = tan41°
eliminate x, and you have
(h+12-1.9) cot65° = (12-1.9)cot41°
Solve for h, and use that to find x.
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
To solve this problem, we can apply basic geometry principles and trigonometric functions. Let's break down the steps to find the solutions:
a) To find the distance of the man from the house:
Step 1: Set up a diagram based on the given information:
A (top of mast)
|\
h | \ d
| \
|_____\
B (bottom of mast)
C (height of house)
Step 2: Calculate the height of the mast:
We know that the difference in the angles of elevation for the top and bottom of the mast is due to the vertical difference between points A and B. We can use the tangent function to find this height:
tan(65°) = h / d
Rearranging the formula, we have:
h = d * tan(65°)
Step 3: Calculate the distance of the man from the house:
Now, we can use the tangent function again to find the distance between the man and the house:
tan(41°) = (d + x) / (h + C)
Rearranging the formula and substituting the value of h calculated before, we have:
(d + x) = (h + C) * tan(41°)
Step 4: Solve for x and round to three significant figures:
x = [(h + C) * tan(41°)] - d
Now, substitute the values and calculate:
x = [(12 + 1.9) * tan(41°)] - (12 * tan(65°))
Perform the calculations to find the value of x.
b) To find the length of the mast:
Step 1: Calculate the height of the mast (h) using the tangent function:
h = d * tan(65°)
Step 2: Calculate the length of the mast:
The length of the mast is the sum of the height of the mast (h) and the height of the house (C).
Length of the mast = h + C
Substitute the known values and calculate to find the length of the mast.
Remember to round the answers to three significant figures.
To solve this problem, we can use trigonometry and set up a right triangle.
Let's label the following points:
- A: Top of the mast
- B: Bottom of the mast
- C: Man's position
- D: Top of the house
Now, let's solve part a) first, which asks for the distance of the man from the house.
In the given information, we know that the man's height is 1.9 meters, the angle of elevation of the top of the mast (angle ACD) is 65 degrees, and the angle of elevation of the bottom of the mast (angle BCD) is 41 degrees.
Step 1: Calculate the height of the mast
We can use the tangent function:
tan(65 degrees) = height of the mast / distance of the man from the house
Let's represent the distance of the man from the house as "x."
tan(65 degrees) = 12 meters / x
Step 2: Solve for x
Rearrange the equation to solve for x:
x = (12 meters) / tan(65 degrees)
Using a calculator, evaluate the tangent of 65 degrees and divide 12 meters by that value. Round the answer to three significant figures to match the desired accuracy in the question.
Now, let's move on to part b), which asks for the length of the mast.
Step 1: Calculate the height difference between the top and bottom of the mast
To find the length of the mast, we need to calculate the height difference between points A and B. We can use the tangent function again:
tan(41 degrees) = height difference / distance of the man from the house
Using the value of distance (x) obtained from part a), we can solve for the height difference.
Step 2: Solve for the height difference
Rearrange the equation to solve for the height difference:
height difference = distance of the man from the house * tan(41 degrees)
Multiply the distance of the man from the house (x) by the tangent of 41 degrees to find the height difference.
Now, to find the length of the mast, we add the height difference to the height of the house.
Length of the mast = height difference + height of the house
Substitute the calculated values for the height difference and the height of the house into the equation to find the length of the mast. Round the answer to three significant figures.
By following these steps, you should be able to find the answers to both parts a) and b) of the question.