Question

The altitude (i.e., height) of a triangle is increasing at a rate of 1.5 cm/minute while the area of the triangle is increasing at a rate of 4 square cm/minute. At what rate is the base of the triangle changing when the altitude is 10 centimeters and the area is 85 square centimeters?

Answer 1

The answer is -1.75cm/min

Explanation:

We begin by calculating the value of the base when the height of the triangle is 10cm and area is 85cm2. We get,

`A=85=hb/2=10b/2\\b=17cm`

Now that we have all three values of A,b and h at a particular moment in time, lets call it time t=0 , where t will be used to symbolized the time in minutes passed from this instant in time, the instant of time at which we also have to calculate the rate of change of the triangle base. So to accomplish this very task we have to write down the equation to calculate the area of the triangle at any moment in time with respect to time t and the B the rate of change of the triangles base, which should be just A=85+4t , where the height is given by h=10+1.5t and base is b=17+Bt, plugging this into the area of the triangle given by,

`A=bh/2\\85+4t=(10+1.5t)*(17+Bt)/2`

if we make B the subject of the equation in terms of t we get,

`B=-17.5/(1.5t+10)`

but from our definition of time t we know that we need to evaluate the rate of change of the base at time t=0. We can do that by plugging t=0 in the above equation, we get,

`B = -1.75cm/min`

which is our final answer.