Answer:
The base of the triangle is shrinking at a rate of [tex]\frac{131}{32}[/tex] centimeters per minute.
Step-by-step explanation:
The formula of the area of a triangle is given by the following expression:
[tex]A = \frac{1}{2}\cdot b \cdot h[/tex]
Where:
[tex]A[/tex] - Area of the triangle, measured in square centimeters.
[tex]b[/tex] - Base of the triangle, measured in centimeters.
[tex]h[/tex] - Height of the triangle, measured in centimeters.
The base of the triangle is:
[tex]b = \frac{2\cdot A}{h}[/tex]
If [tex]A = 98000\,cm^{2}[/tex] and [tex]h = 8000\,cm[/tex], the base of the triangle is:
[tex]b = \frac{2\cdot (98000\,cm^{2})}{8000\,cm}[/tex]
[tex]b = 24.5\,cm[/tex]
The rate of change of the area of the triangle in time, measured in minutes, is obtained after differentiating by rule of chain and using deriving rules:
[tex]\frac{dA}{dt} = \frac{1}{2}\cdot h\cdot \frac{db}{dt} + \frac{1}{2}\cdot b \cdot \frac{dh}{dt}[/tex]
[tex]\frac{dA}{dt} = \frac{1}{2} \cdot \left(h\cdot \frac{db}{dt}+b \cdot \frac{dh}{dt} \right)[/tex]
The rate of change of the base of the triangle is now cleared:
[tex]2\cdot \frac{dA}{dt} = h\cdot \frac{db}{dt} + b\cdot \frac{dh}{dt}[/tex]
[tex]h\cdot \frac{db}{dt} = 2\cdot \frac{dA}{dt}-b\cdot \frac{dh}{dt}[/tex]
[tex]\frac{db}{dt} = \frac{2\cdot \frac{dA}{dt} - b \cdot \frac{dh}{dt} }{h}[/tex]
Given that [tex]\frac{dA}{dt} = 2000\,\frac{cm^{2}}{min}[/tex], [tex]b = 24.5\,cm[/tex], [tex]\frac{dh}{dt} = 1500\,\frac{cm}{min}[/tex] and [tex]h = 8000\,cm[/tex], the rate of change of the base of the triangle is:
[tex]\frac{db}{dt} = \frac{2\cdot \left(2000\,\frac{cm^{2}}{min} \right)-(24.5\,cm)\cdot \left(1500\,\frac{cm}{min} \right)}{8000\,cm}[/tex]
[tex]\frac{db}{dt} = -\frac{131}{32}\,\frac{cm}{min}[/tex]
The base of the triangle is shrinking at a rate of [tex]\frac{131}{32}[/tex] centimeters per minute.
