You are working in a factory that has a machine calibrated to make boxes based o

Question

You are working in a factory that has a machine calibrated to make boxes based on the following specifications: For a given height, the machine will make boxes with a length 7in. longer than the height and a w idth 5 in. longer than the height. A label is printed on each box listing its volume accurate to two decimal places. After receiving an order for boxes that must be precisely 3.01 in. high, your coworker begins to type the label and lists the volume at 241.34 in^2 - a number he found quickly and without a calculator by using a powerful estimating trick. When questioned about the technique, the coworker replied, "Because I know the volume is 240 in^3 when the height is 3 in., I can easily estimate the volume to be 241.34 in^3 when the height is

Explanation / Answer

The Volume of box with Height h, Length l and Width w is given as

Volume (V) = Length(l) * Height(h) * Width(w) = (h+7)*(h+5)*h

For height= 3;

            Length = height+7 = 10;

            Width = height+5 = 8;

Then Volume = 10*8*3 = 240

Moving to second part,

Here h is changing by small factor of 0.01.

To find amount of volume changed due to change in height, we need to find derivative of volume with respective to h

i.e.,

dV/dh = d[(h+7)(h+5)h]/dh = (h+7)(h+5)+(h+7)h+(h+5)h

dV = [(h+7)(h+5)+(h+7)h+(h+5)h]*dh

where h = 3 and dh = 0.01

dV = [80+30+24]*0.01

dV = 134*0.01

dV = 1.34

So, volume when height is 3.01 is 241.34 cubic inches.

which concludes he can put this number in label

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