Area and Volume AP
Calculator Problem
Diego R., Ben C., Dustin K., Jack Z.,
Samantha G.
The Problem
Colorado has experienced a flood season and the city of Westminster was
worried about the overflow of Standley Lake in 2013. The average flow of water per day
from rainfall going into the lake, represented by R, R=-(x-8)^2/32+10 meters, and
variables depleting the lake would be W=-1.5sin(20t)+2, from 0<=t<=25. Picture
Standley lake as being represented by the space created by the function
r(h)=1200/sqrt(29)*sqrt(h+29). When the lake is at maximum capacity, the max depth
is approximately 29 meters, while the average height of the lake is approximately 13
meters. At time t=0, the depth of the lake is at the average.
Part A
With the information given, find the surface area of the lake on an average day
before the flood season.
Part A grading
Point system:
+1 Plug into R(H)
+1 Plug radius into area of a
circle.
+1 Correct answer to three
decimal places with correct
units.
Part B
What is the average volume of Standley Lake before the flooding season? Use
appropriate units in your answer.
Part B grading
Point System:
+1 For correct answer
+1 Correct units
Part C
At t=3 days, what would the volume of the lake be at, is this over the maximum
capacity? Justify your answer.
Part C grading
Point System:
+1 Adding in the additional
rainfall to the lake volume
+1 Taking the integral
+1 Finding a solution with
units
Part D
At what time t would Standley lake be at maximum capacity?
Part D grading
Point system:
+1 Finding the change in
water level
+1 Knowing to use max
height
+1 Taking the integral
+2 Correct Answers
Overall Scoring
Total Possible Points: 13 Points
AP Grades:
A: 3 Points
5: 11-13
B: 2 Points
4: 9-10
C: 3 Points
3: 7-8
D: 5 Points
2: 6
1: 0-5
Questions???