SOLUTION: Q5 A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/ sec.
a. How
Algebra.Com
Question 1033516: Q5 A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/ sec.
a. How fast is the top of the ladder sliding down the wall then?
b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?
c. At what rate is the angle between the ladder and the ground changing then?
Answer by addingup(3677) (Show Source): You can put this solution on YOUR website!
Let x be the distance between the base of the wall and the base of the ladder. Let y be the height the top of the ladder is off the ground. The given is
that dx/dt = +5 at the moment x = 12. The length of the ladder, 13 feet, is a constant, since its length doesn't change.
`````````````````````````````````````````````````````````````````````````
(a) How fast is the top of the ladder sliding down the wall then?
x^2+y^2 = 13^2
Solve for dy/dt in terms of x,y, and dx/dt:
use your calculator and you get y = 5. Therefore:
dy/dx = -x*dx/y*dt = (-12/5)*5 = -12 Note that the minus sign indicates the ladder is sliding down. So, the top of the ladder is sliding down at 12 ft/second
``````````````````````````````````````````````````````````````````````
b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing.
Here the unknown is the rate of change of the area, dA/dt
dA/dt = 1/2 (x*dy/dt+y*dx/dt)
Substitute with x = 12 and dx/dt = 5 and y=5 dy/dt = -12:
dA/dt = 1//2(12*(-12)5*5)= -144+25/2 = -119/2 = 59.5 ft^2/sec
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
c. At what rate is the angle between the ladder and the ground changing then?
ormula that relates the angle θ with the sides x and y of a right triangle:
Answer
1 rad/sec
RELATED QUESTIONS
Two ladders are leaning against a wall. The longer ladder is 7.2 ft long. The shorter... (answered by ankor@dixie-net.com)
A ladder leans against a house with its foot 15 ft from the house. When the foot of the... (answered by scott8148)
a 10 inch book is leaning against a bookshelf when its bottom starts to slide away. By... (answered by Alan3354)
if you have a ladder measuring 20 ft high and the rule of thumb is 4 to 1- meaning for... (answered by Cromlix)
The top of a 32 ft. ladder is leaning against a house 18 ft. above the ground. How far... (answered by CubeyThePenguin)
Twyla has a 26 ft. ladder leaning against her house. If the bottom of the ladder is 10... (answered by edjones)
A 40-ft ladder is leaning against a wall. If the ladder reaches 20 ft up the wall,how far (answered by rfer)
A 30-ft ladder is leaning against a building. If the foot of the ladder is 10 ft away... (answered by ikleyn)
Jon has a 26-ft ladder leaning against his house.If the bottom of the ladder is 10 ft.... (answered by checkley77,oscargut)