This is a related rates (of change) type problem.
The variables of interest are
##a## = altitude
##A## = area and, since the area of a triangle is ##A=1/2ba##, we need
##b## = base.
The given rates of change are in units per minute, so the (invisible) independent variable is ##t## = time in minutes.
We are given:
##(da)/dt = 3/2## cm/min
##(dA)/dt = 5## cm##””^2##/min
And we are asked to find ##(db)/dt## when ##a = 9## cm and ##A = 81##cm##””^2##
##A=1/2ba##, differentiating with respect to ##t##, we get:
##d/dt(A)=d/dt(1/2ba)##.
We’ll need the on the right.
##(dA)/dt = 1/2 (db)/dt a + 1/2b (da)/dt##
We were given every value except ##(db)/dt## (which we are trying to find) and ##b##. Using the formula for area and the given values of ##a## and ##A##, we can see that ##b=18##cm.
Substituting:
##5= 1/2 (db)/dt (9)+1/2(18)3/2##
Solve for ##(db)/dt = -17/9##cm/min.
The base is decreasing at ##17/9## cm/min.
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