1. A ladder 25 ft long is leaning against the side of a house. The base of the l
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Question
1. A ladder 25 ft long is leaning against the side of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top moving down the wall when the base of the ladder is 7 feet from the wall?
2. A 16 foot plank is leaning on the side of a wall and is being pulled up the wall at a rate of 1/2 ft/sec by a worker on top of the wall. How fast is the end of the plank sliding along the ground when it is 8 feet from the wall?
3. An airplane is flying at an altitude of 6 miles and passes over a radar antenna. The rate of change of the distance between the plane and the antenna is 240 miles per hour when the distance between the plane and the antenna is 10 miles. What is the speed of the plane?
4. A kite is flying at a height of 40 feet. A child is flying it so that it is moving horizontally at a rate of 3 ft/sec. If the string is taut, at what rate is the string being let out when the amount of string released is 50 feet?
5. Two cars, one going due east at 25 m/sec and the second going due south at 50/3 m/sec are traveling toward the intersection of the two roads they are driving on. At what rate are the two cars approaching each other at the instant when the first car is 200 m and the second car is 150 m from the intersection?
6. A balloon is being inflated from a helium tank at a constant rate of 50 cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 5 inches? Assume that the balloon is a perfect sphere.
Explanation / Answer
1. If a ladder is leaning on a wall it is making a right angles ladder with the ladder as its hypotenuse.So
Base^2 + Height^2 = 625 (where height is the distance o the top of the ladder to the ground)
at base = 7 feet
height = sqrt(625-49)=24
Now
b^2+h^2=625
Differentiating both sides
2b db/dt + 2h dh/dt = 0
7*2= -24*dh/dt
dh/dt= -7/12 feet /sec
Top moving down the wall at the rate of 7/12 ft/sec
2. If a plank is leaning on a wall it is making a right angles ladder with the ladder as its hypotenuse.So
Base^2 + Height^2 = 256 (where height is the distance o the top of the ladder to the ground)
at base = 8 feet
height = sqrt(256-64)= 8 sqrt(3)
Now
b^2+h^2=256
Differentiating both sides
2b db/dt + 2h dh/dt = 0
8*db/dt= -8sqrt(3) * .5
db/dt= -[Sqrt(3)]/2 feet /sec
Base moving towards the wall at the rate of [Sqrt(3)]/2 ft/sec
3. If you will visualise the situation it will turned out to be a right angled triangle but in this problem the height is constant.
So b. db/dt = l . dl/dt (where l is the distance between plane and antenna)
6 * db/dt = 10 *240
db/dt=400
Hence the speed of the plane is 400 miles per hour.
4. In this problem again height is constant the kite is moving in horizontal direction so
base = horizontal distance = sqrt(2500-1600)=30
b * db/dt = l * dl/dt
30*3= 50 *dl/dt
dl/dt= 1.8
Rate at which the string being let out = 1.8 ft/sec
5. Now in this question both the right angled sides and the hypotenuse is changing.let distances of east car base and south car height.So
b*db/dt +h*dh/dt = l *dl/dt (Got it from differentiating the pythagoras theorem with respect to time t)
Hypotenuse = distance between the cars=sqrt(22500+40000)=250
200*25+150*50/3 = 250 * dl/dt
dl/dt = 7500/250=30 ft/sec
The two cars approaching each other at the instant time = 30 ft/sec
6. Volume of the balloon = V = 4/3 *pi *r^3
Differentiating both sides with respect to t we get
dV/dt = 4pi r^2 * dr/dt
50= 4pi *25 dr/dt
dr/dt= 1/2pi
radius of balloon is increasing at the rate of 1/2pi inches/minute
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