In traveling across flat land, you notice a mountain in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain.
make a sketch
draw the height of the mountain as AB
C is the point of first observation (3.5°)
D is the point of second observation (9°)
angle B = 90° , CD=13, angle C=3.5, and angle ADB=9°
In triangle ACD, angle CDA = 171° and angle CAD=5.5°
by the Sine Law:
AD/sin 3.5 = 13/sin 5.5
AD = 13sin3.5/sin5.5 = 8.28029
now in the right-angled triangle,
sin 9° = AB/AD
AB =8.28029sin9 = 1.295 miles
check my arithmetic
Thank you so much! I was stuck on that one
To approximate the height of the mountain, we can use trigonometry and the concept of similar triangles.
Step 1: Draw a diagram
Draw a diagram to visualize the situation. Let the horizontal distance from your starting point to the base of the mountain be x miles. The height of the mountain will be h miles.
Step 2: Identify relevant angles and sides
In the original position, the angle of elevation is 3.5 degrees, and after driving 13 miles closer, the angle of elevation becomes 9 degrees.
Step 3: Set up the trigonometric relationship
In the first position, we can set up the following relationship:
tan(3.5) = h / x
In the second position, the relationship becomes:
tan(9) = h / (x - 13)
Step 4: Solve the equations
We have two equations with two unknowns (h and x). We can solve these equations simultaneously to find the values.
tan(3.5) = h / x -- eq. (1)
tan(9) = h / (x - 13) -- eq. (2)
To solve the equations, we can use algebraic methods or a graphing calculator. Solving the equations will give us the values of h and x.
Step 5: Calculate the height of the mountain
After finding the values of h and x, we can substitute those values into either eq. (1) or eq. (2) to find the height of the mountain (h).
Once you have determined the value of h, that will be the approximate height of the mountain.