From a point 150 ft from the base of a building the top and the base of a flagpole on a roof edge have angles of elevation of 27degrees49' and 21degrees42' respectively. Find the height of the pole.
height of building from ground:
tan 21 42/60 = h/150
height of flagpole top from ground:
tan 27 49/60 = H/150
then height of pole = H-h
22.115
To find the height of the flagpole, we can use trigonometry. Let's denote the height of the flagpole as h.
Step 1: Calculate the distance from the top of the building to the flagpole's base.
Since we know the angles of elevation, we can use tangent (tan) function.
tan(27 degrees 49') = h / x, where x is the distance from the top of the building to the flagpole's base.
Step 2: Calculate the distance from the point 150 ft from the building to the flagpole's base.
We can use tangent (tan) again.
tan(21 degrees 42') = h / (x + 150), where x is the distance from the top of the building to the flagpole's base.
Step 3: Solve the equations for x and h.
From Step 1, we have x = h / tan(27 degrees 49').
From Step 2, we have x = (h / tan(21 degrees 42')) - 150.
Step 4: Equate the expressions for x and solve for h.
h / tan(27 degrees 49') = (h / tan(21 degrees 42')) - 150
To solve for h, we can clear the fractions:
h * tan(21 degrees 42') = h * tan(27 degrees 49') - 150 * tan(21 degrees 42')
h * [tan(21 degrees 42') - tan(27 degrees 49')] = -150 * tan(21 degrees 42')
h = (-150 * tan(21 degrees 42')) / [tan(21 degrees 42') - tan(27 degrees 49')]
By plugging in the values and performing the calculation, you will obtain the value of h, which represents the height of the flagpole.
To find the height of the flagpole, we can use trigonometry and create a right triangle with the flagpole as the height, the horizontal distance from the base of the building to the flagpole as the base, and the distance from the top of the flagpole to the observer as the hypotenuse.
Let's break down the information given:
1. The angle of elevation from the observer to the top of the flagpole is 27 degrees 49'. This means that if we draw a line from the observer's eye to the top of the flagpole, it will make an angle of 27 degrees 49' with the horizontal ground.
2. The angle of elevation from the observer to the base of the flagpole is 21 degrees 42'. This means that if we draw a line from the observer's eye to the base of the flagpole, it will make an angle of 21 degrees 42' with the horizontal ground.
3. The horizontal distance from the base of the building to the flagpole is 150 ft.
To solve for the height of the flagpole, we need to find the vertical distance from the base of the flagpole to the observer's eye level.
1. First, let's find the length of the hypotenuse of the right triangle by using the sine function for the angle of elevation to the top of the flagpole:
sin(angle of elevation) = opposite / hypotenuse
sin(27 degrees 49') = height of flagpole / hypotenuse
We want to find the height of the flagpole, so let's solve for the hypotenuse:
hypotenuse = height of flagpole / sin(27 degrees 49')
2. Next, let's find the length of the horizontal base of the right triangle, which is the distance from the base of the building to the flagpole. We already know that this distance is 150 ft.
3. Now, we can use the tangent function to find the vertical distance from the base of the flagpole to the observer's eye level. We'll use the angle of elevation to the base of the flagpole:
tan(angle of elevation) = opposite / adjacent
tan(21 degrees 42') = height of flagpole / 150 ft
We want to find the height of the flagpole, so let's solve for the height:
height of flagpole = tan(21 degrees 42') * 150 ft
4. Finally, we can substitute the value of the height of the flagpole from step 3 into the equation from step 1 to find the hypotenuse:
hypotenuse = height of flagpole / sin(27 degrees 49')
This will give us the horizontal distance from the observer to the top of the flagpole.
By solving these equations, you will find the height of the flagpole.