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The altitude (i.e., height) of a triangle is increasing at a rate of 2 cm/minute while the area of the triangle is increasing at a rate of 2.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 8 centimeters and the area is 96 square centimeters?

Respuesta :

The base base of the triangle is decreasing at 5.375cm/min.

Step-by-step explanation:

Area of triangle =  0.5 x Base x Altitude

A=0.5bh

Differentiating with respect to time

                 [tex]\frac{dA}{dt}=\frac{d}{dt}\left ( 0.5bh\right )\\\\\frac{dA}{dt}=0.5\left ( b\frac{dh}{dt}+h\frac{db}{dt}\right )[/tex]

We have

            A= 96 cm²

             h = 8 cm

             A = 0.5bh

              96 = 0.5 x b x 8

                b = 24 cm

               [tex]\frac{dA}{dt}=2.5cm^2/min\\\\\frac{dh}{dt}=2cm/min[/tex]

Substituting

              [tex]2.5=0.5\left ( 24\times 2+8\frac{db}{dt}\right )\\\\48+8\frac{db}{dt}=5\\\\\frac{db}{dt}=-5.375cm/min[/tex]

The base base of the triangle is decreasing at 5.375cm/min.

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