Mathematical Methods practice

Your number is correct. Your answer may still be incomplete.

In an applied question, a bare number can hide whether you differentiated the right quantity, substituted the right value or understood what the result measures. Units, notation and working carry that information.

By StudyDesign AI editorial team. Drafted with AI assistance, then checked by differentiating and substituting independently. The question and solutions are original. No teacher review is claimed.

An original question

Water in a test tank has volume V(t) = 1200 + 80t − 5t² litres, where t is the number of minutes after a pump starts. Find the instantaneous rate of change of the volume at t = 6.

The correct number, written badly

The number is right, but it does not show what was differentiated and it does not answer the applied part of the question. Is 20 a volume, a time, or a rate? The reader has to reconstruct the method.

A complete worked answer

V′(t) = 80 − 10t

Differentiate the volume function. The derivative is a rate of change, not a volume.

V′(6) = 80 − 10(6) = 20

Substitution is visible. The notation also ties the result to the requested time.

The volume is increasing at 20 litres per minute.

The sign and units turn the calculation into an answer about the tank.

What each detail proves

Notation: V′(6)

This identifies the derivative of volume at six minutes. Writing only “20” loses the connection between the value and the quantity being measured.

Working: 80 − 10(6)

A visible substitution lets a marker follow the method. It can also preserve evidence of a correct approach if a later arithmetic slip changes the final number.

Units and direction: litres per minute, increasing

The derivative's units come from volume divided by time. Since the result is positive, the tank's volume is increasing at that instant.

A ten-second check before you move on

Have you named the quantity you calculated?
Do the units match the operation you performed?
Can someone see the equation or substitution that produced the result?
If the value has a sign, have you explained what that sign means in context?

Exact mark allocation depends on the question and its marking scheme. The safe habit is to write enough mathematics to make your reasoning unambiguous, not to add working for its own sake.

Check a full response, not just the final number

StudyDesign AI can mark work you have already completed and explain where the written reasoning becomes unclear. It will not complete assessed work for you.

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