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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Get The Answer The altitude of a triangle is increasing at a rate of 1.5 cm/min while the area of the triangle is increasing at a rate of 5 square cm/min. At what rate is the base of the triangle changing when the altitude is 9 cm and the area is 81 square cm?
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.
Previous answers to this question
This is a preview of an assignment submitted on our website by a student. If you need help with this question or any assignment help, click on the order button below and get started. We guarantee authentic, quality, 100% plagiarism free work or your money back.
Get The Answer