A Walk in the (Dog) Park
The town council has purchased a rectangular plot of land to use as a dog park. Part of the land will be used for benches and parking, but the design will center around four circular dog runs of four different diameters. Your company has been asked to develop estimates for the cost based on the following investigation. The model you will use is a scale model where 1 cm on the drawing represents a number of feet.
Designing a park
Using the rectangular paper provided, construct four circles of varying sizes to represent the dog runs in the park. Be careful to have four different diameters. Label the runs A, B, C, and D.
Measure the four diameters to the nearest centimeter and record the measurements in the table at the bottom of this page.
Fencing
Each dog run must be enclosed by fencing. Estimate the amount of fencing using the Cheerio cereal. Each “O” represents $25 in fencing. Record the number of “Os” needed to enclose each of the four dog runs. Compute the estimated price for each fence and record it below.
Sod
Each dog run must also be sodded with a dog-friendly turf. The pieces of this special turf are laid in non-overlapping, close proximity to each other. Once planted, the grass will spread into the non-sodded slivers between the pieces.
Each piece of turf costs $10. The waffle-type cereal pieces represent the size and shape of these turf pieces. Estimate the number of pieces of turf needed to cover each of the dog runs and record this number in the table below. Compute the estimated price for the turf in each dog run.
Diameter in CM / Fencing units in Os / Estimated cost of fencing / Number of sod units needed / Estimated cost of sod / Total estimated cost of runA
B
C
D
The town council is reluctant to commit to your design because the costs of parking lot and sidewalks will also be affected by the sizes of your dog runs. While the council members engage in the practice of bureaucracy, you wisely devise a plan so you can predict the cost of a dog run from the diameter of the run.
Regression for the cost of fencing
Enter the diameters of the dog runs you designed in L1 of your calculator. Enter the corresponding costs of the fencing and the sod in L2 and L3 respectively.
Create a scatterplot using the diameters and the fencing costs. Sketch the pattern and describe the pattern you observe.
Use the regression functions on your calculator tofind a model for this relationship. Sketch the pattern below. If you want to graph the model against your scatterplot, include Y1 at the end of your instructions to the calculator, for instance LinReg L1,L2,Y1.
Which model matches your scatterplot best? Write the formula for cost of fencing in terms of diameter, F(x).How did you decide what “best” means?
Regression for the cost of sod
Repeat the process using the diameter and the sod cost and L1 and L3.
Which model works best for this scatterplot? Write the formula for cost of sod in terms of diameter, S(x). Why is this the best model?
Combine your formulas to create one formula for the cost of the dog run, D(x).
Using the model to predict the cost
The council has decided. Estimate the total costs of the dog runs if the diameters of the runs in the drawingare 5, 10, 20, and 25 cm.