Question

A piece of high-density Styrofoam measuring 24.0 cm by 36.0 cm by 5.0 cm floats when placed in a tub of water. When a 1.5 kg book is placed on top of the Styrofoam, the Styrofoam partially sinks at 3.0 cm. Assuming that the density of water is 1.00 g/ml, what is the density of Styrofoam?

Answer 1

Explanation:

The given data is as follows.

         Width of Styrofoam = 24.0 cm

          Length of Styrofoam = 36.0 cm

          Height of Styrofoam = 5.0 cm

Therefore, volume of the Styrofoam will be calculated as follows.

                  Volume = length × width × height

                                =  (36.0 × 24.0 × 5.0) $cm^{3}$

                                 = 4320 $cm^{3}$

or,                             = $4.32 \times 10^{3} cm^{3}$

As Styrofoam partially sinks at 3.0 cm and total height of Styrofoam is 5.0 cm. Hence, height of Styrofoam above the water is (5.0 - 3 cm) = 2 cm.

So, volume of water displaced is as follows.

          24.0 cm × 36.0 cm × 2.0 cm

         = $1.73 \times 10^{3} cm^{3}$

Hence, mass of displaced water is as follows.

                 mass = density × volume

                           = $1.00 g/cm^{3} \times 1.73 \times 10^{3} cm^{3}$

                           = $1.73 \times 10^{3} g$

Since, book is placed on the Styrofoam. Therefore, mass of water displaced is also equal to the following.

             Mass of water displaced = mass of book + mass of Styrofoam

                  $1.73 \times 10^{3} g$ = 1500 g + mass of Styrofoam

                   (1730 - 1500) g = mass of Styrofoam

                   mass of Styrofoam = 230 g

Therefore, calculate the density of Styrofoam as follows.

                   Density = $\frac{mass}{volume}$  

                                 = $\frac{230}{4.32 \times 10^{3} cm^{3}}$

                                 = $53.24 \times 10^{-3} g cm^{-3}$

Thus, we can conclude that the density of Styrofoam is $53.24 \times 10^{-3} g cm^{-3}$.