Download Calculus 12 Unit V: Trigonometry – More Word Problems Two sides

Calculus 12
Unit V: Trigonometry – More Word Problems
1. Two sides of a triangle have lengths 8m and 12m. The angle between them is increasing at a rate of

rad min . How fast is the length of the third side increasing when the angle between the fixed
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
c 2  a 2  b 2  2ab cos C
sides is ?
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2. A ladder 6m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a
speed of 1.5m/s, how fast is the angle between the top of the ladder and the wall changing when the

angle is radians?
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3. A lighthouse is on a small island 6 km away from the nearest point P on a straight shoreline and its light
makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it 2
km from P.
4. An observer is 2 miles from a spot where a rocket is launched vertically. When the rocket is 1mile high,
it is gaining altitude at the rate of 70mph. How fast is the angle of elevation from the observer
increasing at that instant?
5. A man meanders along a straight path at a speed of 2m/s. A searchlight is located on the ground 12m
from the path, and is kept focused on the man. At what rate is the searchlight rotating when the man is
10m from the point on the path closest to the searchlight?
6. The length of the hypotenuse of a right triangle is 8cm. One of the acute angles is increasing at a rate
of 10 / s . How fast is the area increasing when this angle is 20 ?

rad hr . When it is 45 , what is the rate
7. The angles of elevation of the sun is decreasing at a rate of
15
of increase of the length of the shadow cast by a 8m pole?
8. A steel pipe is being carried down a hallway 7ft wide. At the end of the hall there is a right-angled turn
into a narrower hallway 5ft wide. What is the length of the longest pipe that can be carried horizontally
around the corner?
9. Two sides of a triangle are 10cm and 17cm. The angle between these two sides is increasing at
15 min . When the angle is 30 how fast is the length of the third side increasing?
10. Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of
0.09rad s . Find the rate at which the area of the triangle is increasing when the angle between the

sides of fixed length is .
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11. Determine the maximum area of a triangle inscribed in a semicircle of radius 30cm.
12. An aircraft spotter observes a plane flying toward him at an altitude of 2.4km. He notes that when the


rad sec . What is the ground speed of the
angle of elevation is , it is increasing at a rate of
30
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aircraft?
13. A wall of height 6m stands parallel to and 7m from a tall building. What angle will the shortest
ladder make with the ground if it is to pass over the wall and lean on the building?
Solutions
dc
 2.8m / min
1.
dt
d
 28rad / hr
4.
dt
dx
 3.351m / hr
7.
dt
dA
 0.45m 2 / s
10.
dt
13.   0.76rad
d
 0.5rad / s
dt
d
 0.098rad / s
5.
dt
2.
8. length = 16.89m
11. A = 900m2
dx
 167.3km / min
dt
dA
 4.278cm 2 / s
6.
dt
dc
 2.289cm / min
9.
dt
dx
 1.005km / s
12.
dt
3.