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Question

The 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 1500 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 10500 centimeters and the area is 100000 square centimeters?

 

Answer must be in centimeters/minute.

 

Note: The "altitude" is the "height" of the triangle in the formula "Area=(1/2)*base*height". Draw yourself a general "representative" triangle and label the base one variable and the altitude (height) another variable. Note that to solve this problem you don't need to know how big nor what shape the triangle really is.
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Answer 1

The problem is asking us to find db/dt when h = 10,500 and A = 100,000

 

We know that A = 1/2 bh, so when A = 100,000 cm² we have

 

100,000 = 1/2 b(10,500)

100,000 = 5250b

b = 19.05 cm

 

Now, we are also given that dA/dt = 1500 cm/min and dh/dt = 3000 cm/min

 

If A = 1/2 bh, and we differentiate both sides with respect to t, we get:

 

dA/dt = 1/2[b * dh/dt + h * db/dt] by the Product Rule

 

We have values for each of the above items except for db/dt, which is what we are trying to determine

 

1500 = 1/2[19.05 * 3000  + 10,500 db/dt]

 

1500 = 1/2(57150 + 10,500 db/dt)

 

1500 = 28575 + 5250 db/dt

 

-27075 = 5250 db/dt

 

db/dt = -5.157 cm/min

Answer 2

First, we use A=½•bh to solve for b when A=100,000, h=10,500 and we get b = 400/21.

 

Now we differentiate that formula, assuming that all variables are functions of time; the product rule applies:

dA/dt = ½[h•db/dt + b•dh/dt] and we plug in lots of values for the instant described by my first sentence. 

 

1,500 =  ½[h•10,500(db/dt) + (400/21)(3,000]

 

From this I obtain db/dt = -5.156  cm/min but of course you must check that I've put all numbers in the right places and done the arithmetic correctly.