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ka.mukiteo.wednet.edu WinterBreakQuin2015 Kamiak High School: Classroom Home Page: Home Page YouTube 44. long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Consider the triangle formed by the side of the house,the ladder, and the ground. Find the rate at which the area triangle is changed when the base of the ladder is 7 feet fromthe wall. Round your answer to two decimal places. 25 ft sec O A. 1967 ft /sec O B. 45.91 ft /sec O C. 107.54 ft /sec O D. 60.34 ftelsec 4 E. 22.95 ft /secExplanation / Answer
We have given the base of ladder is 7ft from the wall and rate of change in base 2ft/sec
Let the triangle with side c, base b and height a
side c will be ladder length 25 ft,b will be base of the triangle with 7ft and a will be the side of the house
from right triangle we get
a^2+b^2=(25)^2
we have given b=7ft
a^2=625-49=576 implies a=24 ft
a^2+b^2=(25)^2
differentiating with respect to t
2a*(da/dt)+2b*(db/dt)=0
2*24*(da/dt)+2*7*2=0 since we have given db/dt=2ft/sec,a=24ft b=7ft
48*(da/dt)=-28 implies da/dt =-28/48 =-7/12
we know the area of triangle is
A=(1/2)*base*height =(1/2)*b*c
A=(1/2)*b*c
differentiating with respect to t
dA/dt =(1/2)*[a*(db/dt)+b*(da/dt)]
dA/dt=(1/2)*[(24*2)+7*(-7/12)]=(1/2)*[48-49/12)]=(1/2)*(576-49)/12 =527/24 ft^2/sec
dA/dt=527/24 ft^2/sec =21.95 ft^2/sec
The rate is changing in triangle area is 21.95 ft^2/sec when the base of ladder is 7ft from the wall.
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