10.2 Related Rates
Chapter
Chapter 10
Section
10.2
Solutions 26 Videos

Express the following statements in symbols:

• The area, A, of a circle is increasing at a rate of 4 m^2/s.
0.44mins
Q1a

Express the following statements in symbols:

• The surface area, S, of a sphere is decreasing at a rate of 3 m^22/min.
0.28mins
Q1b

Express the following statements in symbols:

• After travelling for 15 min, the speed of a car is 70 km/h.
0.43mins
Q1c

Express the following statements in symbols:

• The x- and y-coordinates of a point are changing at equal rates.
0.15mins
Q1d

Express the following statements in symbols:

• The head of a short-distance radar dish is revolving at three revolutions per minute.
0.22mins
Q1e

The function T(x) = \dfrac{200}{1 +x^2} represents the temperature, in degrees Celsius, perceived by a person standing x metres from a fire.

• If the person moves away from the fire at 2m/s, how fast is the perceived temperature changing when the person is 5 m away.
2.20mins
Q2

The side of a square is increasing at a rate of 5 cm/s. At what rate is the area changing when the side is 10 cm long? At what rate is the perimeter changing when the side is 10 cm long?

0.54mins
Q3

Each edge of a cube is expanding at a rate of 4 cm/s.

How fast is the volume changing when each edge is 5 cm?

0.49mins
Q4a

Each edge of a cube is expanding at a rate of 4 cm/s.

At what rate is the surface area changing when each edge is 7 cm?

0.49mins
Q4b

The width of a rectangle increases at 2 cm/s, while the length decreases at 3 cm/s. How fast is the area of the rectangle changing when the width equals 20 cm and the length equals 50 cm?

1.05mins
Q5

The area of a circle is decreasing at the rate of 5 m^2/s when its radius is 3 m.

a) At what rate is the radius decreasing at that moment?

b) At what rate is the diameter decreasing at that moment?

2.14mins
Q6

Oil that is spilled from a ruptured tanker spreads in a circle. The area of the circle increases at a constant rate of 6 km^2/h. How fast is the radius of the spill increasing when the area is 9\pi km^2 ?

1.55mins
Q7

The top of a 5 m wheeled ladder rests against a vertical wall. If the bottom of the ladder rolls away from the base of the wall at a rate of \frac{1}{3} m/s, how fast is the top of the ladder sliding down the wall when it is 3 m above the base of the wall?

2.59mins
Q8

How fast must someone let out line if a kite is 30 m high, 40 m away horizontally, and continuing to move away horizontally at a rate of 10 m/min?

2.15mins
Q9

If the rocket shown below is rising vertically at 268 m/s when it is 1220 m up, how fast is the camera-to-rocket distance changing at that instant? 2.56mins
Q10

Two cyclists depart at the same time from a starting point along routes that make an angle of \frac{\pi}{3} radians with each other. The first cyclist is travelling at 15 kmh. How fast are the two cyclists moving apart after 2h?

6.11mins
Q11

A spherical balloon is being filled with helium at a rate of 8 cm^3/s. At what rate is its radius increasing at the following moments.

a. when the radius is 12 cm.

b. when the volume is 1435 cm^3 (Your answer should be correct to the nearest hundredth.)

6.39mins
Q12

A cylindrical tank, with a height 15 m and diameter 2 m, is being filled with gasoline at a rate of 500 L/min. At what rate is the fluid level in the tank rising? (1L = 1000 cm^3).

About how long will it take to fill the tank?

Coming Soon
Q13

If V = \pi r^2 h, determine \frac{dV}{dt} if r and h are

both variables that depend on t. In your journal, write three problems that involve the rate of change of the volume of a cylinder such that

a. r is a variable and h is a constant

b. r is a constant and h is a variable

c. r and h are both variables

Coming Soon
Q14

The trunk of a tree is approximately cylindrical in shape and has a diameter of 1 m when the height of the tree is 15 m. If the radius is increasing at 0.003 m per year and the height is increasing at 0.4 m per year, determine the rate of increase of the volume of the trunk at this moment.

Coming Soon
Q15

A conical paper cup, with radius 5 cm and height 15 cm, is leaking water at a rate of 2 cm3/min. At what rate is the water level decreasing when the water is 3 cm deep?

3.43mins
Q16

The cross-section of a water trough is an equilateral triangle with a horizontal top edge. If the trough is 5 m long and 25 cm deep, and water is flowing in at a rate of 0.25 m^3/min,

how fast is the water level rising when the water is 10 cm deep at the deepest point?

Coming Soon
Q17

The shadow cast by a man standing 1 m from a lamppost is 1.2 m long. If the man is 1.8 m tall and walks away from the lamppost at a speed of 120 m/min, at what rate is the shadow lengthening after 5 s?

Coming Soon
Q18

A railroad bridge is 20 m above, and at right angles to, a river. A person in a train travelling at 60 km/h passes over the centre of the bridge at the same instant that a person in a motorboat travelling at 20 km/h passes under the centre of the bridge. How fast are the two people separating 10 s later?

Coming Soon
Q19

Liquid is being poured into the top of a funnel at a steady rate of 200 $cm^3/s$. The funnel is in the shape of an inverted right circular cone, with a radius equal to its height. It has a small hole in the bottom, where the liquid is flowing out at a rate of 20$cm^3/s$.

a. How fast is the height of the liquid changing when the liquid in the funnel is 15 cm deep?

b. At the instant when the height of the liquid is 25 cm, the funnel becomes clogged at the bottom and no more liquid flows out. How fast does the height of the liquid change just after this occurs?

Coming Soon
Q20

A ladder of length l, standing on level ground, is leaning against a vertical wall. The base of the ladder begins to slide away from the wall. Introduce a coordinate system so that the wall lies along the positive y-axis, the ground is on the positive x-axis, and the base of the wall is the origin.

a. What is the equation of the path followed by the midpoint of the ladder?

b. What is the equation of the path followed by any point on the ladder? (Hint: Let k` be the distance from the top of the ladder to the point on the ladder.)

Coming Soon
Q21
Lectures 8 Videos

1 Introduction to Related Rates  6.15mins
1 Introduction to Related Rates

2 Related Rates ex1 Radius and Sphere 3.37mins
2 Related Rates ex1 Radius and Sphere

## Related Rates ex2 Ladder sliding down 4.22mins
3 Related Rates ex2 Ladder sliding down

## Related Rates ex3 Water poured into Cone tank 4.38mins
4 Related Rates ex3 Water poured into Cone tank

## Related Rates ex4 Two Cars approaching a corner 4.58mins
5 Related Rates ex4 Two Cars approaching a corner 5.44mins
6 Related Rates ex5 rate of change of shadow

## Related Rates ex6 Area of Equilateral Triangle 