Answer:
The base of the triangle is decreasing at a rate 8 centimeter per minute.
Step-by-step explanation:
We are given the following in the question:
[tex]\dfrac{dh}{dt} = 2.5\text{ cm per minute}\\\\\dfrac{dA}{dt} = 2\text{ square cm per minute}[/tex]
Instant height = 7.5 cm
Instant area = 96 square centimeters
Area of triangle =
[tex]A=\dfrac{1}{2}\times b\times h[/tex]
where b is the base and h is the height of the triangle.
[tex]96 = \dfrac{1}{2}\times b \times 7.5\\\\b = \dfrac{96\times 2}{7.5} = 25.6[/tex]
Rate of change of area of triangle =
[tex]\dfrac{dA}{dt} = \dfrac{d}{dt}(\dfrac{bh}{2})\\\\\dfrac{dA}{dt} =\dfrac{1}{2}(b\dfrac{dh}{dt} + h\dfrac{db}{dt})[/tex]
Putting values, we get,
[tex]2 = \dfrac{1}{2}(7.5\dfrac{db}{dt}+25.6(2.5))\\\\4 = 7.5\dfrac{db}{dt} + 64\\\\-60 = 7.5\dfrac{db}{dt} \\\\\Rightarrow \dfrac{db}{dt} = \dfrac{-60}{7.5} = -8[/tex]
Thus, the base of the triangle is decreasing at a rate 8 centimeter per minute.