Question

a snack stand at the basketball game charges a setup fee plus a cost for each snack you buy. the total cost, c, can be represented by the function c(x) = 2x + 10, where x is the number of snacks you buy.
part a:
explain what the numbers 2 and 10 mean in this situation.
part b:
another stand uses the function c(x) = 3x + 6 to show their prices. at what number of snacks will both stands charge the same total cost? show your work and explain how you know your answer makes sense.

Explanation:

Part A:

Step1: Analyze the function form

The function \(C(x) = 2x+10\) is in the form of a linear function \(y = mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the \(y\) - intercept.
In the context of cost, the coefficient of \(x\) (which is \(2\)) represents the cost per snack. The constant term \(10\) represents the setup fee (a fixed cost that does not depend on the number of snacks \(x\)).

Part B:

Step1: Set the two cost functions equal

We want to find when \(C_1(x)=C_2(x)\), where \(C_1(x)=2x + 10\) and \(C_2(x)=3x+6\).
Set up the equation \(2x + 10=3x + 6\).

Step2: Solve for \(x\)

Subtract \(2x\) from both sides:
\(2x-2x + 10=3x-2x + 6\)
\(10=x + 6\)
Subtract \(6\) from both sides:
\(x=10 - 6\)
\(x = 4\)

Step3: Check the meaning of \(x = 4\)

For \(x = 4\):

• For \(C(x)=2x + 10\), \(C(4)=2\times4+10=8 + 10=18\)
• For \(C(x)=3x + 6\), \(C(4)=3\times4+6=12 + 6=18\)

The number \(2\) in \(C(x)=2x + 10\) is the cost per snack for the first stand, and \(10\) is its setup fee. The number \(3\) in \(C(x)=3x + 6\) is the cost per snack for the second stand, and \(6\) is its setup fee. When \(x = 4\) snacks are bought, the total cost from both stands is the same (\(C = 18\)). This makes sense because we are solving for the equilibrium point where the two linear cost - models intersect.

Answer:

Part A:

The number \(2\) represents the cost per snack, and \(10\) represents the setup fee.

Part B:

When \(x = 4\) snacks are bought, both stands charge the same total cost. For \(x = 4\), \(C(4)=2\times4 + 10=18\) and \(C(4)=3\times4+6=18\). The values \(2\) (cost per snack) and \(10\) (setup fee) for the first function, and \(3\) (cost per snack) and \(6\) (setup fee) for the second function are consistent with the linear cost - structure (total cost=cost per unit\times number of units+fixed cost), and solving \(2x + 10=3x + 6\) gives the intersection point of the two cost - lines.