Question

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3. Over the week of Thanksgiving, friends are decluttering their living space. They plan to store some of their belongings in a storage unit. They will move it all into the storage root, and move things home every day. The unit costs \(\$ 0.10\) per day for every \(f t^{3}\) of storage. The amount of storage needed (in \(f t^{3}\) ) is given by \(S(t)=90-3 t\) where \(t\) is measured in days. How much do your friends have to pay to store their stuff for 7 days? Explain your reasoning.

This question is part of an integral/antiderivative assignment. But, I don't understand what I'm supposed to integrate here. At first glance it seems it's just S(7) x 0.1 x 7 = $48.3 but the professor said this:

But on the first day, they would have needed S(0)=90 cu ft of storage space.  The amount of space needed is decreasing as time progresses because they are clearing out the storage unit.

I still can't figure out how to do this one.

Answer 1

Given that \(S(t)=90-3 t\) The amount of storage for 7 days:

$$ \begin{aligned} S(7) &=90-3(7) \\ &=90-21 \\ &=69 \end{aligned} $$

The unit cost per day is \(\$ .10\) The unit cost for 7 days:

\(=69 \times 0.10\)

=$ 6.9

The unit cost per day is $ .10 . so, multiply the total amount of storage to unit cost per day.