Respuesta :
Answer:
The base of the triangle is decreasing at a rate of 1.4167 inch/hour.
Step-by-step explanation:
Area of a triangle:
The area of a triangle of base b and height h is given by:
[tex]A = bh[/tex]
In this question:
We have to derivate the equation of the area implicitly in function of time. So
[tex]\frac{dA}{dt} = b\frac{dh}{dt} + h\frac{db}{dt}[/tex]
The altitude of a triangle is increasing at a rate 2 inch/hour while the area of the triangle is decreasing at a rate of 0.5 square inch per hour.
This means that:
[tex]\frac{dh}{dt} = 2, \frac{dA}{dt} = -0.5[/tex]
At what rate is the base of the triangle is changing when the altitude is 6 inch and the area is 24 square inch?
This is [tex]\frac{db}{dt}[/tex] when [tex]h = 6[/tex]
Area is 24, so the base is:
[tex]A = bh[/tex]
[tex]24 = 6b[/tex]
[tex]b = \frac{24}{6} = 4[/tex]
Then
[tex]\frac{dA}{dt} = b\frac{dh}{dt} + h\frac{db}{dt}[/tex]
[tex]-0.5 = 4(2) + 6\frac{db}{dt}[/tex]
[tex]6\frac{db}{dt} = -8.5[/tex]
[tex]\frac{db}{dt} = -\frac{8.5}{6}[/tex]
[tex]\frac{db}{dt} = -1.4167[/tex]
The base of the triangle is decreasing at a rate of 1.4167 inch/hour.
