Two new rides are being compared by a local amusement park in terms of their annual operating costs. The two rides are assumed to be able to generate the same level of revenue (and thus the focus on costs). The Tummy Tugger has fixed costs of $10,000 per year and variable costs of $2.50 per visitor. The Head Buzzer has fixed costs of $4000 per year, and variable costs of $4 per visitor. Provide answers to the following questions so the amusement park can make the needed comparison. (a) Mathematically determine the breakeven number of visitors per year for the two rides to have equal annual costs. (b) Develop a graph that illustrates the following: (Note: Put visitors per year on the horizontal axis and costs on the vertical axis.

Two new rides are being compared by a local amusement park in terms of their annual operating costs. The two rides would generate the same level of revenue (thus the focus on costs). The Tummy Tugger has fixed costs of $10,000 per year and variable costs of $2.50 per visitor. The Head Buzzer has fixed costs of $4000 per year and variable costs of $4 per visitor.

Provide answers to the following questions so the amusement park can make the needed comparison.

(a) Mathematically determine the breakeven number of visitors per year for the two rides to have equal annual costs.

(b) Develop a graph that illustrates the following

i. Accurate total cost lines for the two alternatives (show lines, slopes, and equations).

ii. The breakeven point for the two rides in terms of number of visitors.

iii. The ranges of visitors per year where each alternative is preferred.

Solution to the problem

(a) Mathematically determine the breakeven number of visitors per year for the two rides to have equal annual costs.

Determine the slope-intercept equation of each ride

For the Tummy Tugger:

y = mx + b

m = slope = variable costs (per visitor) = 2.50

b = fixed costs = $10,000

y = 10,000 + 2.50x

For the Head Buzzer

y = mx + b

m = variable costs = 4

b = fixed costs = 4,000

y = 4,000 + 4x

For the two rides to have equal annual costs, make the two functions equal:

10,000 + 2,50x = 4,000 + 4x

Solve:

4x - 2.50x = 10,000 - 4,000

1.50x = 6,000

x = 6,000 / 1.50

x = 4,000

Hence, for 4,000 visitors per year the two rides have equal annual costs.

(b) Develop a graph that illustrates the following

i. Accurate total cost lines for the two alternatives (show lines, slopes, and equations).

See the figure attached.

For the Tummy Tugger, the graph is the red line:

The equation is y = 10,000 + 2.50x

The slope is $ 2.50/visitor (the coefficient of the variable)

To draw the line use two points:

Use the y-intercept: (0, 10000)

Use the breake evenpoint: (4000, 20000)

For the Head Buzzer, the graph is the blue line:

The equation of the line is y = 4,000 + 4x

The slope is 4 $ per visitor

To draw the line use two points:

Use the y-intercept: (0, 4000)

Use the breakevenpoint (4000, 20000)

The draw must only show the first quadrant (negative values do not count).

ii. The breakeven point for the two rides in terms of number of visitors.

The breakeven is 4,000 visitors and $20,000 of cost: (4,20000)

iii. The ranges of visitors per year where each alternative is preferred.

For the Tummy Tugger it is below the intersection point, i.e [0, 4000)

Note that 0 visitors is the absolute minimum, because there cannot be negative visitors)

For the Head Buzzer it is above the intersection point, i.e. (4000, ∞)

Note that there not exist an upper bound; the more visitors the better, and the Head Buzzer ride is preferred.