2006 Pre-Calculus Free Response
Question 2 (non-calculator)
2. In order to combat erosion caused by the tide, a pumping station is adding sand to a beach. The rate at which the pumping
station adds sand to the beach is modeled by the function S defined by
16t
.
S (t ) =
1 + 4t
The rate at which the tide is removing sand from the beach is modeled by the function R defined by
⎛π ⎞
R(t ) = 3 + 5sin ⎜ t ⎟ .
⎝6 ⎠
Both S(t) and R(t) are measured in cubic yards per hour, and time t is measured in hours.
(a) What is the rate at which sand is being removed from the beach when t = 2? Evaluate all trigonometric functions with
exact values and indicate the units.
(b) What is the average rate of change of R (t ) over the interval [ 0, 1] ? Evaluate all trigonometric functions with exact
values and indicate the units.
(c) For the time interval 0 ≤ t ≤ 6 hours, at what time is the maximum amount of sand being removed?
(d) Using a trapezoid, estimate the total amount of sand being removed by the tide from t = 0 to t = 3 . Show the work that
leads to your answer and indicate the units.
(e) What is the rate at which the total amount of sand on the beach is changing at time t = 3 ? Show the work that leads to
your answer and indicate the units.
yds3
5
⎛π
⎞
(a) R (2) = 3 + 5sin ⎜ (2) ⎟ = 3 +
3
2
hr
⎝6
⎠
(cubic yards per hour)
yds3
π⎞
5 yds3 / hr
⎛
(b) ⎜ 3 + 5sin ⎟ − 3 =
or
6⎠
2
hr
hr 2
⎝
(cubic yards per hour per hour)
(c) Maximum value occurs when t = 3 hours.
1
33
(3)(3 + 8) =
2
2
33
1
or 16 cubic yards of sand.
Approximately
2
2
(d) R (0) = 3 , R (3) = 8
16(3)
48
=
1 + 4(3) 13
56
4
cubic yards per hour
S (3) − R (3) = − or − 4
13
13
or, the amount of sand is decreasing at the rate of
56
4
or 4
cubic yards per hour.
13
13
(a) 1 point:
1 point: correct answer
(b) 1 point:
1 point: correct answer
(c) 1 point:
1 point: t = 3
(d) 3 points:
1 point: correct value for R(3)
1 point: correct setup
1 point: correct answer
(e) R (3) = 8 , S (3) =
(e) 2 points: 1 point: calculates S (t ) − R (t )
1 point: correct answer
(Units)
(Units)
1 point: correct units in (b) and
in one of (a), (d), or (e)
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