Related Rate Problems: Airplane Distance and Water Level Changes

[Zerkor]

Homework Statement

This is not my homework I'm a self-learner and I faced a problem solving these two problems and I found nobody to help me to solve them, I hope somebody here will help. The two problems are :

1- An airplane is flying at a constant speed of 360 mile\hr and climbing at an angle of 45. At the moment the plane's altitude is 10560 ft. , it passes directly over an air traffic control tower on the ground. Find the rate at which the airplane's distance from the tower is changing 1 minute later (neglect the height of the tower)

2- A paper cup containing water has the shape of a frustum of a right circular cone of altitude 6 in. and lower and upper base radii 1 in. and 2 in. respectively. If water is leaking out of the cup at a rate of 3 in^3\hr. , at what rate is the water level decreasing when the depth of the water is 4 in.? (Note: the volume V of a frustum of a cone of a right circular cone of altitude h and base radii a and b is given by V=1\3*pi*h*(a^2+b^2+ab))

Homework Equations

The Attempt at a Solution

In problem (1) : I drew a triangle with radius r and an opposite side to the 45° angle of length 10560 ft (2 miles) but I don't know should I find dr\dt or should I find the rate of change of the side drawn from the point the tower into intersect the extent of the hypotenuse (r)? and if so how can I get it?
In problem (2) : I don't know whether if a is changing or not .. I think that a is changing and b is constant but whenever I treat them as constants or as variables my solution is always different from the provided answers to this problem .. (the answer is -27\25pi)

 

[Science4ver]

try posting your drawing of the situation.

 

[HallsofIvy]

Homework Statement

This is not my homework I'm a self-learner and I faced a problem solving these two problems and I found nobody to help me to solve them, I hope somebody here will help. The two problems are :

1- An airplane is flying at a constant speed of 360 mile\hr and climbing at an angle of 45. At the moment the plane's altitude is 10560 ft. , it passes directly over an air traffic control tower on the ground. Find the rate at which the airplane's distance from the tower is changing 1 minute later (neglect the height of the tower)

2- A paper cup containing water has the shape of a frustum of a right circular cone of altitude 6 in. and lower and upper base radii 1 in. and 2 in. respectively. If water is leaking out of the cup at a rate of 3 in^3\hr. , at what rate is the water level decreasing when the depth of the water is 4 in.? (Note: the volume V of a frustum of a cone of a right circular cone of altitude h and base radii a and b is given by V=1\3*pi*h*(a^2+b^2+ab))

Homework Equations

The Attempt at a Solution

In problem (1) : I drew a triangle with radius r and an opposite side to the 45° angle of length 10560 ft (2 miles) but I don't know should I find dr\dt or should I find the rate of change of the side drawn from the point the tower into intersect the extent of the hypotenuse (r)? and if so how can I get it?

Do you understand that you have a triangle with one side of length 10560 ft and angle 135 degrees? The length of the other side of that angle is changing at 360 mph so 31680 feet per minute and can be represented as 31680t feet. Use the cosine law to find the other side of the triangle.

In problem (2) : I don't know whether if a is changing or not .. I think that a is changing and b is constant but whenever I treat them as constants or as variables my solution is always different from the provided answers to this problem .. (the answer is -27\25pi)

What do you mean by "a"? Your problem says "base radii a and b" but does not specify which is which. The base of the cup stays the same while the top of the water changes. If you are using "b" to represent the radius of the base of the cup that stays the same: 1 inch. If you are using a to represent the radius of the top of the water, it is changing. It is impossible to tell you what you are doing wrong if you don't show what you did!

 

[Zerkor]

Do you understand that you have a triangle with one side of length 10560 ft and angle 135 degrees? The length of the other side of that angle is changing at 360 mph so 31680 feet per minute and can be represented as 31680t feet. Use the cosine law to find the other side of the triangle.


What do you mean by "a"? Your problem says "base radii a and b" but does not specify which is which. The base of the cup stays the same while the top of the water changes. If you are using "b" to represent the radius of the base of the cup that stays the same: 1 inch. If you are using a to represent the radius of the top of the water, it is changing. It is impossible to tell you what you are doing wrong if you don't show what you did!

The cosine law .. that's it :) .. I used it and got the right answer .. in the second problem the provided answer is as you said (-27\25pi) but every time I try to solve it I get -9\7pi not -25\24pi , could you please tell me how did you get it in detail. And thanks for your help :)