Respuesta :
Answer:
Therefore the volume of cube is change at the 29.4 cube in./s at that instant time.
Explanation:
Formula
- [tex]\frac{dx^n}{dx} =nx^{n-1}[/tex]
Cube :
The volume of a cube is = [tex]side^3[/tex]
The side of length is x in.
Then volume of the cube is (V) = [tex]x^3[/tex]
∴ V = [tex]x^3[/tex]
Differentiate with respect to t
[tex]\frac{d}{dt}(V)=\frac{d}{dt} (x^3)[/tex]
[tex]\Rightarrow \frac{dV}{dt} =3x^2\frac{dx}{dt}[/tex]....(1)
Given that the side of the cube is increasing at the rate of 0.2 in/s.
i.e [tex]\frac{dx}{dt} = 0.2[/tex] in/s.
And the sides of the cube are 7 in i.e x= 7 in
Putting [tex]\frac{dx}{dt} = 0.2[/tex] and x= 7 in equation (1)
[tex]\therefore \frac{dV}{dt} =3 \times 7^2 \times 0.2[/tex] cube in./s
=29.4 cube in./s
Therefore the volume of cube is change at the 29.4 cube in./s at that instant time.
The volume of the cube is changing at the rate of 29.4 in³/s.
Time derivative:
The rate of change of the sides of the cube is 0.2 in/s, which can be mathematically represented as :
[tex]\frac{dx}{dt}=0.2 \;in/s[/tex]
Now the volume of the cube is given by:
V = x³
If we take the time derivative of the above equation then it gives the rate of change of volume with time, so:
[tex]\frac{dV}{dt}= \frac{d}{dt}x^3\\\\ \frac{dV}{dt}= 3x^2\frac{dx}{dt}[/tex]
At the instant x = 7 in, the rate of change of volume will be:
[tex]\frac{dV}{dt}= 3\times7^2\times0.2\;in^3/s\\\\\frac{dV}{dt}=29.4\;in^3/s[/tex]
So the volume is changing at 29.4 cubic inches per second.
Learn more about differential equation:
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