E-RC03 · A shrinking similar box
A closed rectangular box keeps dimensions \(x\), \(2x\) and \(3x\) cm while shrinking. When \(\displaystyle x=2 \mathrm{cm}\), its volume decreases at \(72\ \mathrm{cm^3\,s^{-1}}\).
Find the signed rate \(\displaystyle \frac{dx}{dt}\) at this instant.
Find the total surface area and its signed time rate at that instant.
Use the instantaneous rates to estimate x and the surface area 0.02 s later. Explain why these are estimates rather than exact values.
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