High School

The altitude of a triangle is increasing at a rate of 3000 centimeters per minute, while the area of the triangle is increasing at a rate of 3500 square centimeters per minute. At what rate is the base of the triangle changing when the altitude is 10,000 centimeters and the area is 89,000 square centimeters?

Answer :

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
Answer:
The base of the triangle is decreasing at:
4.64 cm / min

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