Question

One side of the base of a square base prism A is increased by \(50\)% and the other side is decreased by \(50\)% keeping the height of the prism \(10\) cm in both cases. If the side of the square is a, then what change in the total surface area of the prism is expected?

Answer 1

Let \(a\) cm be the side of the square base prism and the height of the prism is \(10\) cm.

Total surface area of the square base prism \(=\) Area of the two bases \(+\) Area of \(4\) lateral faces
\(= 2a^2+4 \times (a \times 10)\)
\(= 2a^2 + 40 a\)

As one of the sides of the square of the square base prism is increased by \(50\)% and the the other side is decreased by \(50\)%, we have:

The dimensions of the rectangular base are \(1.5a\) cm and \(0.5a\) cm.

Total surface area of the rectangular base prism \(= 2 \times (l \times w + w \times h + h \times l)\)

\(= 2 \times (1.5a \times 0.5a +0.5a \times 10 + 10 \times 1.5a)\)

\(= 2 \times (0.75a^2 +5a + 15a)\)

\(= 2 \times (0.75a^2 +20a)\)

\( = 1.5a^2 +40a\)

Change in the the surface area \(=\) Final surface area \(-\) Initial surface area

\(= 1.5a^2 +40a - (40a + 2a^2)\)

\(= 1.5a^2 +40a - 40a - 2a^2\)

\(= - 0.5a^2\)

Negative sign indicates that the total surface area decreases.

Hence, the decrease in the surface area \(= 0.5a^2\)