Question

Write a set of linear equations for each of the following problems and solve them using either determinants or row reduction. An object composed of \(x \mathrm{gm}\) of potassium (specific gravity \(0.8\) ) and \(y \mathrm{gm}\) of cesium (specific gravity \(2.0\) ) cannot be weighed in air or water because it would react with either. However, it weighs \(86 \mathrm{gm}\) in oil of specific gravity \(0.6\) and \(124 \mathrm{gm}\) in oil of specific gravity \(0.4\). Find \(x\) and \(y\).

Short answer

x = 120 gm, y = 80 gm

Step-by-step solution

Set up the equations based on the problem

We are given specific gravities and weights in different oils. Let the weight in air be the sum of the masses of potassium and cesium: \[ x + y = W \]For specific gravities, this fact holds: Weight in fluid = Actual weight - Buoyant force. Hence, for buoyant force: \[ \text{Buoyant force} = \text{Displaced fluid volume} \times \text{Specific gravity of fluid} \times \text{mass of fluid} \] Given in oil of specific gravity 0.6, \[ x + y - 0.6(x/0.8 + y/2) = 86 \] Given in oil of specific gravity 0.4, \[ x + y - 0.4(x/0.8 + y/2) = 124 \]

Simplify the system of equations

Start by simplifying the buoyancy terms: For the first equation (in oil of 0.6 specific gravity): \[ x + y - 0.6 \left( \frac{x}{0.8} + \frac{y}{2} \right) = 86 \] Simplifying further: \[ x + y - 0.6 \left( 1.25x + 0.5y \right) = 86 \] \[ x + y - 0.75x - 0.3y = 86 \] \[ 0.25x + 0.7y = 86 \] Similarly, simplify the second equation (in oil of 0.4 specific gravity): \[ x + y - 0.4 \left( 1.25x + 0.5y \right) = 124 \] \[ x + y - 0.5x - 0.2y = 124 \] \[ 0.5x + 0.8y = 124 \]

Solve the simplified system of equations

Use either determinants or row reduction to solve: From the simplified system, \[ 0.25x + 0.7y = 86 \] \[ 0.5x + 0.8y = 124 \] We use row reduction: Multiply the first equation by 2 to match coefficients of x: \[ 0.5x + 1.4y = 172 \] Subtract the second equation from this result: \[ 0.5x + 1.4y - (0.5x + 0.8y) = 172 - 124 \] \[ 0.6y = 48 \] \[ y = 80 \] Now substitute back to find x: Using 0.25x + 0.7y = 86: \[ 0.25x + 0.7(80) = 86 \] \[ 0.25x + 56 = 86 \] \[ 0.25x = 30 \] \[ x = 120 \]

Key concepts

specific gravity

Specific gravity is a measure of the density of a substance compared to the density of water. It is a dimensionless quantity, meaning it has no units. For example, the specific gravity of potassium is 0.8, which means potassium is less dense than water. On the contrary, the specific gravity of cesium is 2.0, indicating it is denser than water. This concept is crucial in calculating the buoyant force, as we see in the problem. By understanding an object's specific gravity, you can determine how it will interact with different fluids.
Specific gravity is calculated by:
\[\text{Specific Gravity} = \frac{\text{Density of Substance}}{\text{Density of Water}}\]
In the exercise, the different specific gravities of the objects and the oils help us set up the system of equations. We use specific gravity to account for the displacement of fluid volume and to find the buoyant force acting on the object in different oils.

determinants

Determinants are a special number that can be calculated from a square matrix. They play an important role in linear algebra, particularly in solving systems of linear equations. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant \(\det(A)\) is calculated as:
\[\det(A) = ad - bc\]
In our problem, while we opted for row reduction as the method of solution, determinants are another useful method to solve the system of linear equations. By organizing the coefficients of the equations into a matrix, the determinant helps determine whether the system has a unique solution. If the determinant is non-zero, the system has a unique solution. While not used directly in this solution, understanding determinants can be very helpful for learning more advanced linear algebra concepts.

row reduction

Row reduction, also known as Gaussian elimination, is a method for solving systems of linear equations. It involves performing operations on the rows of an augmented matrix to reduce it to row echelon form. For the problem given, we start with the system:
\[0.25x + 0.7y = 86\]
\[0.5x + 0.8y = 124\]
Here’s a simplified process to solve this through row reduction:

• Multiply the first equation by 2 to align the coefficients of \(x\):
  \[0.5x + 1.4y = 172\]
• Subtract the second equation from this result:
  \[0.5x + 1.4y - (0.5x + 0.8y) = 172 - 124\]
  \[0.6y = 48\]
• Solve for \(y\):
  \[y = 80\]

Substitute back to find \(x\):
\[0.25x + 0.7(80) = 86\]
\[0.25x + 56 = 86\]
\[0.25x = 30\]
\[x = 120\]
Row reduction simplifies step-by-step equations into an easier form to solve. It is particularly useful when dealing with larger systems.

buoyant force

Buoyant force is the upward force exerted by a fluid that opposes the weight of an immersed object. This is the principle behind why objects float or sink. The magnitude of the buoyant force is equal to the weight of the fluid displaced by the object. For the problem, the formula for buoyant force is:

• \[\text{Buoyant force} = \text{Displaced fluid volume} \times \text{Specific gravity of fluid} \times \text{mass of fluid}\]

Given the specific gravities and weights of the different objects in various oils, you can calculate the buoyant force and set up the respective equations. For example, in oil with specific gravity 0.6, the weight of the object is countered by the buoyant force, which includes the specific gravity and the volumes displaced. Learning and applying the concept of buoyant force helps in solving static fluid problems and understanding principles of flotation.
Key formula to remember:

• Weight in fluid = Actual weight - Buoyant force
  

Seeing how the buoyant force affects weight measurement in different fluids helps you grasp why the object's weight appears different in oils of different specific gravities.