Respuesta :
Answer:
The base is decreasing at a rate 3.27 centimeters per minute.
Step-by-step explanation:
We are given the following in the question:
[tex]\dfrac{dh}{dt}= 1.5\text{ centimeters per minute}\\\\\dfrac{dA}{dt} = 2.5\text{ square centimeters per minute}[/tex]
Instant altitude = 8.5 cm
Area = 93 square centimeters
Area of triangle:
[tex]A = \dfrac{1}{2}\times b \times h[/tex]
Putting values, we get,
[tex]93 = \dfrac{1}{2}\times b\times 8.5\\\\b = \dfrac{93\times 2}{8.5}[/tex]
where b is the base and h is the altitude of the triangle.
Rate of change of area =
[tex]\dfrac{dA}{dt} = \dfrac{1}{2}(h\dfrac{db}{dt} + b\dfrac{dh}{dt})[/tex]
Putting values, we get,
[tex]2.5 = \dfrac{1}{2}(8.5\dfrac{db}{dt} + \dfrac{186}{8.5}\times 1.5)\\\\5 = 8.5\dfrac{db}{dt} + \dfrac{186}{8.5}\times 1.5\\\\8.5\dfrac{db}{dt} = 5 - \dfrac{186}{8.5}\times 1.5 \\\\8.5\dfrac{db}{dt} = -27.82\\\dfrac{db}{dt} = -3.27[/tex]
Thus, the base is decreasing at a rate 3.27 centimeters per minute.
