Two environmental cleanup projects, Project A and Project B, begin treating separate, iden… — SAT

Two environmental cleanup projects, Project A and Project B, begin treating separate, identical bodies of water that each contain $V_0$ gallons of a certain chemical. The volume of the chemical remaining in the water treated by Project A is modeled by the exponential decay function $A(t) = V_0(k)^t$ , where $t$ is the number of days since treatment began and $k$ is a constant. The volume of the chemical remaining in the water treated by Project B is modeled by the linear decay function $B(t) = V_0 - c \cdot t$ , where $c$ is a positive constant. After $1$ day of treatment, the volume of the chemical remaining in both bodies of water is the same. After $2$ days of treatment, the volume of the chemical remaining in the water treated by Project A is exactly $\frac{16}{15}$ times the volume of the chemical remaining in the water treated by Project B. If the treatment for Project B continues at this constant rate, after how many days will the chemical in Project B's water be completely removed?

Cevap: 5 days

Cevap

Equating both models at

t = 1

gives

\frac{c}{V_0} = 1 - k

. At

t = 2

, Project A's remaining volume is

V_0 k^2

and Project B's is

V_0(2k - 1)

. Setting

k^2 = \frac{16}{15}(2k - 1)

results in the quadratic equation

15k^2 - 32k + 16 = 0

. Factoring this yields

k = 0.8

(discarding

1.33

because the scenario represents decay). The chemical in Project B's water is completely removed when

B(t) = 0 \implies t = \frac{V_0}{c} = \frac{1}{1-k} = \frac{1}{0.2} = 5

days.

Adım Adım Çözüm

Relate parameters

k

and

c

using the condition

A(1) = B(1)

\frac{c}{V_0} = 1 - k

Establishing a link between the rates of linear decay and exponential decay.

Express the remaining volumes at

t = 2

in terms of

k

A(2) = V_0 k^2

and

B(2) = V_0(2k - 1)

Preparing equations to apply the ratio condition at

t = 2

using a single variable.

Apply the condition

A(2) = \frac{16}{15}B(2)

to solve for

k

k = 0.8

Setting up the quadratic equation

15k^2 - 32k + 16 = 0

, factoring it to find roots

0.8

and

1.33

, and selecting

0.8

because

k < 1

for a decay process.

Calculate the time

t

when

B(t) = 0

t = 5

Solving

V_0 - c \cdot t = 0

yields

t = \frac{V_0}{c} = \frac{1}{1-k} = \frac{1}{0.2} = 5

days.

Anahtar Kavram

Linear and exponential decay modeling

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