During a four-hour storm

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

During a four-hour storm, the rainfall rate

r(t)

(in inches per hour) is modeled by a trapezoidal profile: •

r(0)=0

and increases linearly to

r(1)=R

. •

r(t)=R

for

1\le t\le 3

. •

r(t)

decreases linearly to

r(4)=0

. Exactly

3.00

inches of rain accumulate on the ground over the four hours. (a) Derive an expression for the total accumulated rainfall from

t=0

t=4

in terms of the unknown peak rate

R

. (b) Solve for the numerical value of

R

so that the total rainfall equals

3.00

inches. (c) Let

F(t)

denote the accumulated rainfall (in inches) up to time

t

. Determine the time

T

(with

0<T<4

) at which

F(T)=1.50

inches. Show all steps leading to your answer. (d) Suppose a drainage system removes water from the site at a constant rate of

0.4

inches per hour throughout the storm. Write and simplify an expression for the net depth of standing water on the ground at

t=4

hours.