Ask a New Question
Menu

A stack of small cups has a height, h, in centimeters of h = 1.5n + 6. A stack of large cups has a height, h, in centimeters of h = 0.5n + 9

11 answers

To analyze the height of stacks of small and large cups based on the number of cups (n), we can look at each equation and compare them.

  1. Small Cups: The height \( h \) of a stack of small cups is given by the equation: \[ h = 1.5n + 6 \] Here, \( n \) represents the number of small cups. The term \( 1.5n \) indicates that each small cup adds 1.5 cm to the height of the stack, and the constant 6 cm is the initial height or offset when no cups are present (i.e., when \( n = 0 \)).

  2. Large Cups: The height \( h \) of a stack of large cups is given by the equation: \[ h = 0.5n + 9 \] Similar to the small cups, \( n \) indicates the number of large cups. In this case, each large cup adds 0.5 cm to the height of the stack, and the constant 9 cm is the initial height or offset when no cups are present.

Comparison of the Two Equations

To determine how the heights of the stacks relate as \( n \) increases, we can set the two equations equal to each other to find out when the heights of the stacks are the same:

\[ 1.5n + 6 = 0.5n + 9 \]

Solve for \( n \):

  1. Subtract \( 0.5n \) from both sides: \[ 1.5n - 0.5n + 6 = 9 \] This simplifies to: \[ n + 6 = 9 \]

  2. Subtract 6 from both sides: \[ n = 3 \]

Height at \( n = 3 \):

We can now find the height for both types of cups when \( n = 3 \):

  • Small Cups: \[ h = 1.5(3) + 6 = 4.5 + 6 = 10.5 \text{ cm} \]

  • Large Cups: \[ h = 0.5(3) + 9 = 1.5 + 9 = 10.5 \text{ cm} \]

Conclusion:

The two stacks of cups will have the same height of 10.5 cm when there are 3 cups of each type.

For values of \( n < 3 \), the stack of small cups is shorter than the stack of large cups, and for values of \( n > 3 \), the stack of small cups is taller than the stack of large cups.

write ten things you know about this

Sure! Here are ten things about the stack of small and large cups based on the equations provided:

  1. Linear Relationship: Both height equations represent linear relationships between the number of cups (n) and the height of the stack (h).

  2. Formulas:

    • Small cups: \( h = 1.5n + 6 \)
    • Large cups: \( h = 0.5n + 9 \)

  3. Initial Height: The initial height (or y-intercept) of the small cup stack is 6 cm when no cups are present, while for the large cup stack, it is 9 cm.

  4. Height Increase per Cup: Each small cup adds 1.5 cm to the height of the stack, while each large cup adds only 0.5 cm.

  5. Intersection Point: The heights of both stacks equal 10.5 cm when there are 3 cups of each type. This is the point where the two linear equations intersect.

  6. Growth Rates: The small cup stack grows at a faster rate (1.5 cm per cup) compared to the large cup stack (0.5 cm per cup).

  7. Height Comparison: For \( n < 3 \), the large cup stack is taller than the small cup stack. For \( n = 3 \), they are the same height, and for \( n > 3 \), the small cup stack is taller.

  8. Application: Such equations can be applied in real-world scenarios such as determining how many cups can fit in a given height restriction or calculating the total height of cup stacks for display purposes.

  9. Graphical Representation: If plotted on a graph, the two equations would be straight lines with different slopes, showing how the height changes as more cups are added.

  10. Practical Significance: Understanding these equations can help in designing spaces for displaying cups, ensuring that stacks meet specific height requirements in storage or sale displays.

name one more
  1. Dependency on Variables: The height of each stack is directly dependent on the number of cups (n), meaning if you increase or decrease the number of cups, the height will change accordingly based on the constants and coefficients in the equations. This emphasizes the importance of understanding how variables interact within mathematical models.
Identify the number of solutions to the system of equation:
y = x + 3
3x + y = -1

How can you tell from the equations?

How can you tell from the graph?

To determine the number of solutions to the given system of equations, we can analyze the equations and their graphical representation.

The Equations

  1. \( y = x + 3 \) (Equation 1)
  2. \( 3x + y = -1 \) (Equation 2)

Step 1: Check for Intersections Algebraically

We can substitute Equation 1 into Equation 2 to see if there is a solution:

From Equation 1, we have: \[ y = x + 3 \]

Substituting this into Equation 2: \[ 3x + (x + 3) = -1 \] Combine like terms: \[ 4x + 3 = -1 \] Subtract 3 from both sides: \[ 4x = -4 \] Divide by 4: \[ x = -1 \]

Now substitute \( x = -1 \) back into Equation 1 to find \( y \): \[ y = -1 + 3 = 2 \]

Thus, the solution is: \[ (x, y) = (-1, 2) \]

Step 2: Determine the Number of Solutions

The system of equations has one solution at the point \((-1, 2)\) because we found a specific point where both equations intersect.

Step 3: Analyze the Graph

To confirm the solution graphically, we can visualize both equations.

  1. Equation 1 (\( y = x + 3 \)): This represents a straight line with a slope of 1 and a y-intercept of 3. It rises diagonally from left to right, crossing the y-axis at the point (0, 3).

  2. Equation 2 (\( 3x + y = -1 \)): We can rewrite this in slope-intercept form: \[ y = -3x - 1 \] This line has a slope of -3 and a y-intercept of -1, meaning it falls from left to right, crossing the y-axis at (0, -1).

Graphical Analysis: When you plot both lines on the same grid, you will see that they intersect at a single point, which confirms that there is one solution to the system.

Conclusion

  • From the equations: We determined there is one solution (\(-1, 2\)) through substitution and algebraic manipulation.
  • From the graph: The two lines intersect at exactly one point, confirming that the system has one unique solution.
2x = y + 4
-4x + 2y = 2

How can you tell from the equations?

To determine the number of solutions to the given system of equations algebraically, we can analyze the equations provided:

  1. First equation: \[ 2x = y + 4 \] We can rewrite this equation in slope-intercept form (\(y = mx + b\)): \[ y = 2x - 4 \]

  2. Second equation: \[ -4x + 2y = 2 \] We'll also convert this into slope-intercept form by isolating \(y\): \[ 2y = 4x + 2 \] \[ y = 2x + 1 \]

Step 1: Analyze the Slopes

Now, we have both equations in slope-intercept form:

  1. \( y = 2x - 4 \) (Slope = 2)
  2. \( y = 2x + 1 \) (Slope = 2)

Step 2: Compare the Equations

Since both equations have the same slope (2), they are parallel. The y-intercepts are different (one is -4 and the other is +1), so they will never intersect.

Step 3: Conclusion from the Equations

From the analysis of the equations, we can conclude that:

  • The system of equations has no solutions because the lines are parallel. They will never meet at any point.

This means that the equations describe two distinct lines that will never intersect. Thus, the number of solutions to the system of equations is zero.

Identify the number of solutions to the system of equation:
y = ⅓ x + 2
-x + 3y = 6

How can you tell from the equations?

To determine the number of solutions to the given system of equations, let's analyze both equations algebraically.

The Equations

  1. First equation: \[ y = \frac{1}{3}x + 2 \] This is already in slope-intercept form (\(y = mx + b\)), where the slope (\(m\)) is \(\frac{1}{3}\) and the y-intercept (\(b\)) is \(2\).

  2. Second equation: \[ -x + 3y = 6 \] To analyze it easily, we can rewrite it in slope-intercept form as well: \[ 3y = x + 6 \] Now, divide by 3: \[ y = \frac{1}{3}x + 2 \]

Step 1: Compare the Equations

Now we have:

  1. \(y = \frac{1}{3}x + 2\) (from the first equation)
  2. \(y = \frac{1}{3}x + 2\) (from the second equation)

Step 2: Analyze the Equations

Both equations are identical, meaning they represent the same line.

Conclusion from the Equations

Since the two equations describe the same line, there are infinitely many solutions to the system. This is because any point on the line defined by these equations is a solution.

In summary:

  • How can you tell from the equations? The two equations are the same (or equivalent), confirming that every point on the line is a solution. Therefore, this system of equations has infinitely many solutions.