Answer :
The rate at which the volume of the box of height 10 inches is changing when the width is 8 inches and the length is 6 inches is 440 cubic inches per second
What is the volume of a rectangular box?
The volume of a rectangular box is the product of the length, the width and the height of the box.
The height of the rectangular box, h = 10 inches
The rate of change of the length of the box is; [tex]\frac{dl}{dt}[/tex] = 2 in/sec
The rate at which the width is decreasing is; [tex]\frac{dw}{dt}[/tex] = 4 in/sec
When the length is 8 inches and the width is 6 inches, we get;
[tex]\dfrac{dV}{dt} = \dfrac{d}{dt} \left(l\cdot w \cdot h\right)[/tex]
[tex]\dfrac{dV}{dt} =h\times \left(l\cdot \dfrac{dw}{dt} + w \cdot \dfrac{dl}{dt} \right)[/tex]
When l = 8, and w = 6, we get;
[tex]\dfrac{dV}{dt} =10\times \left(8\times 4 + 6 \times 2 \right)=440[/tex]
The rate at which the volume of the rectangular box is changing in cubic inches per second is therefore;
[tex]\dfrac{dV}{dt} =440 \ in^3/sec[/tex]
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