rate of change for base of triangle

Question

The altitude of a triangle is increasing at a rate of 3.5 centimeters/minute while the area of the triangle is increasing at a rate of 2.5 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 9.5 centimeters and the area is 86 square centimeters?

Answer 1

this is a "related rates" problem. In calculus textbooks there are usually a few pages on it, as a subtopic of the more general topic "derivatives" some problems get somewhat complicated. others relatively easy.

altitude increasing at 3.5 cm per minute, a=9.5 cm a'=3.5 = da/dt

area increasing at 2.5 cm^2 per minute, A=86 cm^2 A'=2.5 = dA/dt

at what rate is the base changing? find b' = db/dt = rate of change of the base length

A=ab/2

86=9.5b/2

9.5b = 172

b =172/9.5 = about 18.11 cm

A=ab/2 = Area of a triangle = one half the base times altitude

take the derivative with respect to time

A'=(1/2)(ba'+ab')

plug in the known numbers and solve for the remaining unknown b' = rate of change of the base length

2.5 = (1/2)[(172/9.5)(3.5) + 9.5b']

multiply by 2

5= (172/9.5)(3.5) + 9.5b'

5 = 172(7)/19 + 9.5b'

multiply by 19

95 = 1204 +180.5b'

180.5b' = -1109

b' = -1109/180.5 = about -6.14

b' = about -6.1 cm/minute

the base length is decreasing at the rate of 6.1 centimeters per minute

a very rough check on the answer is look at the new Area and new altitude in one minute and find the new base

new A = old A +A' = about 86+2.5 = 88.5

new a = old a + a' = 9.5+3.5 = 13

new b = 2(newA/newa)= 2(88.5)/13 = about 6.8(2) = 13.6

from old b to new b in the same one minute = about 18.11 to 13.6 = -4.51

which is in the right negative direction but not that close to the earlier calculated magnitude of -6.1 cm/min

the altitude is increasing faster than the Area, so the base must be decreasing. The negative sign looks accurate. just a question on the magnitude of the decrease. although this was a "rough" check and not that accurate,

the difference is the 4.51 rate of change is over an interval. The question asked for an instantaneous rate of change, so -6.1 cm/min may be the best answer.

Answer 2

Please see video above for a demonstration how to solve this. I felt for related rates seeing it worked out in real time is better than having a bunch of text variables explaining how to walkthrough a problem like this.