Question

A nugget of gold and quartz was found to contain \[x{\text{ }}g\] of gold and \[y{\text{ }}g\] of quartz and has density \[d\]. If the densities of gold and quartz are \[{d_1}\]and \[{d_2}\] respectively then the correct relation is:
(A) \[\dfrac{x}{{{d_1}}} + {\text{ }}\dfrac{y}{{{d_2}{\text{ }}}} = \dfrac{{x + y}}{d}\]
(B) \[x{d_1}{\text{ }} + {\text{ }}y{d_2}{\text{ }} = {\text{ }}\left( {x{\text{ }} + {\text{ }}y} \right)d\]
(C) \[\dfrac{x}{{{d_2}{\text{ }}}}{\text{ }} + {\text{ }}\dfrac{y}{{{d_1}{\text{ }}}} = {\text{ }}\dfrac{{x + y}}{d}\]
(D) \[\dfrac{{x + y}}{d} + {\text{ }}\dfrac{x}{{{d_1}{\text{ }}}} + {\text{ }}\dfrac{x}{{{d_2}}}\]

Answer 1

Hint: As per question first we identify the total density of a nugget of gold and quartz by using the formula
\[density\left( d \right){\text{ }} = {\text{ }}\dfrac{{Mass\left( m \right)}}{{Volume\left( v \right)}}\], after putting these values, then we can able to establish the relationship.

Complete step by step answer: As we know the Mass is the measure of the amount of matter. It is approximately the measure of the number of atoms in a given object. Mass is also the measure of an object’s resistance to gravity. The Kilogram is the basic SI unit of mass and Volume is a measure of the amount of three-dimensional space that is being occupied by a liquid, solid, or a gas. The basic SI unit for volume is cubic meter.
Density - Density refers to the measurements of how compact an object is.
The formula of \[Density{\text{ }}\left( d \right){\text{ }} = {\text{ }}Mass{\text{ }} \div {\text{ }}Volume\]
Here given,
Mass of Gold \[ = \;x\]
Mass of quartz \[ = {\text{ }}y\]
Density of gold \[ = {\text{ }}{d_1}\]
Density quartz \[ = {\text{ }}{d_2}\]
by using the formula- \[density\left( d \right){\text{ }} = {\text{ }}\dfrac{{Mass\left( m \right)}}{{Volume\left( v \right)}}\]
\[Density\left( d \right){\text{ }} = \dfrac{{\left( {mass{\text{ }}of{\text{ }}gold{\text{ }} + {\text{ }}mass{\text{ }}of{\text{ }}quartz} \right){\text{ }}}}{{\left( {volume{\text{ }}occupied{\text{ }}by{\text{ }}gold{\text{ }} + {\text{ }}volume{\text{ }}occupied{\text{ }}by{\text{ }}quartz} \right)}}{\text{ }}\]
Now,
volume of gold = \[\dfrac{x}{{{d_1}}}\]
volume of quartz \[ = \dfrac{{{\text{ }}y}}{{{d_2}}}\]
So we have,
\[d{\text{ }} = \dfrac{{\left( {x{\text{ }} + {\text{ }}y} \right)}}{{\left( {{\text{ }}x/{d_1}{\text{ }} + {\text{ }}y/{d_2}} \right)}}{\text{ }}\]
Multiply and divide the denominator above by \[({d_1} \times {d_2})\]we get,
\[d{\text{ }} = {\text{ }}\dfrac{{\left( {x{\text{ }} + {\text{ }}y} \right)}}{{\left( {{d_2}x{\text{ }} + {\text{ }}{d_1}y} \right)}}\]
Or we can write,
\[\dfrac{{\left( {{d_2}x{\text{ }} + {\text{ }}{d_1}y} \right)}}{{\left( {x{\text{ }} + {\text{ }}y} \right){\text{ }}}}{\text{ }} = \dfrac{1}{d}\]
or arranging the equation we get,
\[\dfrac{{x{\text{ }}}}{{{d_2}}} + {\text{ }}\dfrac{y}{{{d_1}}} = \dfrac{{\left( {x{\text{ }} + {\text{ }}y} \right)}}{d}{\text{ }}\]

so the option (C) is correct.

Note: According to the formula \[Density\; = {\text{ }}Mass/Volume\;\] it also means that the larger the volume of an object compared to its mass, the less dense it is and volume increases the mass of the object increases in direct proportion. The gradient of the graph equals the density of the material. Density and volume are scientific concepts pertaining to physical properties and characteristics of matter. Volume refers to the measurement of the amount of three-dimensional space occupied by an object.