To determine whether the box will float or sink in water and to find the mass of water displaced by the box, we need to follow a step-by-step solution:
1. Determine the density of the box:
The density ([tex]\( \rho \)[/tex]) of an object is defined as its mass ([tex]\( m \)[/tex]) divided by its volume ([tex]\( V \)[/tex]):
[tex]\[
\text{Density of the box} = \frac{\text{Mass of the box}}{\text{Volume of the box}}
\][/tex]
Given:
[tex]\[
\text{Mass of the box} = 500 \text{ g}
\][/tex]
[tex]\[
\text{Volume of the box} = 350 \text{ cm}^3
\][/tex]
Plug the values into the formula:
[tex]\[
\text{Density of the box} = \frac{500 \text{ g}}{350 \text{ cm}^3}
= 1.4285714285714286 \text{ g/cm}^3
\][/tex]
2. Compare the density of the box with the density of water:
The density of water is [tex]\( 1.0 \text{ g/cm}^3 \)[/tex]. If the density of the box is greater than the density of water, the box will sink. If the density of the box is less than or equal to the density of water, the box will float.
In this case:
[tex]\[
1.4285714285714286 \text{ g/cm}^3 > 1.0 \text{ g/cm}^3
\][/tex]
Since the density of the box is greater than the density of water, the box will sink.
3. Calculate the mass of water displaced by the box:
According to the principle of buoyancy, an object submerged in a fluid displaces a volume of fluid equal to the volume of the object. Given the volume of the box is 350 cm³, the mass of the water displaced by the box can be found by multiplying the volume of the box by the density of water:
[tex]\[
\text{Volume of the box} = 350 \text{ cm}^3
\][/tex]
[tex]\[
\text{Density of water} = 1.0 \text{ g/cm}^3
\][/tex]
[tex]\[
\text{Mass of water displaced} = \text{Volume of the box} \times \text{Density of water}
\][/tex]
[tex]\[
\text{Mass of water displaced} = 350 \text{ cm}^3 \times 1.0 \text{ g/cm}^3
= 350 \text{ g}
\][/tex]
Hence, the density of the box is [tex]\(1.4285714285714286 \text{ g/cm}^3\)[/tex], the mass of water displaced by the box is [tex]\(350 \text{ g}\)[/tex], and the box will sink in water.
