He altitude of a triangle is increasing at a rate of 3000 centimeters/minute while the area of the triangle is increasing at a rate of 3500 square centimeters/minute. at what rate is the base of the triangle changing when the altitude is 10000 centimeters and the area is 89000 square centimeters?

By definition, the area of a triangle is given by:
 A = (1/2) * (b) * (h)
 Where,
 b: base of the triangle
 h: height of the triangle
 Deriving the area we have:
 A '= (1/2) * ((b') * (h) + (b) * (h '))
 Clearing b' we have:
 b '= (2A' - (b) * (h ')) / (h)
 We must look for the value of the base:
 A = (1/2) * (b) * (h)
 Substituting values:
 89000 = (1/2) * (b) * (10000)
 b = (89000) / (5000)
 b = 17.8 cm
 Then, the speed at which the base changes is:
 b '= (2 * (3500) - (17.8) * (3000)) / (10000)
 b '= -4.64 cm / min
 
 The base of the triangle is decreasing at:
 4.64 cm / min

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