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Consider the following problem:
The total number of subscribers Zhang Wei has for his video page is changing at a rate of
r(t)=21-2t subscribers per week (where
t is the time in weeks). At time
t=8 weeks, Zhang Wei has 120 subscribers. How many subscribers does Zhang Wei have by week 20 ?
Which expression can we use to solve the problem?
Choose 1 answer:
(A)
r(20)-r(8)+120
(B)
int_(20)^(20)r(t)dt
(C)
int_(8)^(20)r(t)dt+120
(D)
r(20)

Consider the following problem: $\newline$ The total number of subscribers Zhang Wei has for his video page is changing at a rate of $r(t)=21-2 t$ subscribers per week (where $t$ is the time in weeks). At time $t=8$ weeks, Zhang Wei has $120$ subscribers. How many subscribers does Zhang Wei have by week $20$ ? $\newline$ Which expression can we use to solve the problem? $\newline$ Choose $1$ answer: $\newline$ (A) $r(20)-r(8)+120$ $\newline$ (B) $\int_{20}^{20} r(t) d t$ $\newline$ (C) $\int_{8}^{20} r(t) d t+120$ $\newline$ (D) $r(20)$

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Write two-variable inequalities: word problems

Full solution

Q. Consider the following problem:

\newline

The total number of subscribers Zhang Wei has for his video page is changing at a rate of

r(t)=21-2 t

subscribers per week (where

t

is the time in weeks). At time

t=8

weeks, Zhang Wei has

120

subscribers. How many subscribers does Zhang Wei have by week

20

?

\newline

Which expression can we use to solve the problem?

\newline

Choose

1

answer:

\newline

(A)

r(20)-r(8)+120

\newline

(B)

\int_{20}^{20} r(t) d t

\newline

(C)

\int_{8}^{20} r(t) d t+120

\newline

(D)

r(20)

Rate of Change Calculation: We are given the rate of change of subscribers as $r(t) = 21 - 2t$ . To find the total change in subscribers from week $8$ to week $20$ , we need to integrate this rate of change over the interval from $t = 8$ to $t = 20$ . This will give us the total number of new subscribers added between these two times.
Integration of Rate of Change: The integral of $r(t)$ from $t = 8$ to $t = 20$ is represented mathematically as $\int_{t=8}^{t=20} (21 - 2t) \, dt$ . This integral will calculate the total number of subscribers gained or lost over the interval from week $8$ to week $20$ .
Performing Integration: To perform the integration, we integrate the function $21 - 2t$ with respect to $t$ . The antiderivative of $21$ with respect to $t$ is $21t$ , and the antiderivative of $-2t$ with respect to $t$ is $-t^2$ . So the integral becomes $21t - t^2$ evaluated from $t = 8$ to $t$ $0$ .
Calculating Upper Limit: Plugging in the upper limit of the integral, we get $21(20) - (20)^2 = 420 - 400 = 20$ . Plugging in the lower limit of the integral, we get $21(8) - (8)^2 = 168 - 64 = 104$ .
Calculating Lower Limit: Now we subtract the value of the integral at the lower limit from the value at the upper limit to find the total change in subscribers from week $8$ to week $20$ . This gives us $20 - 104 = -84$ . However, this is a mistake because when we subtract the lower limit from the upper limit, we should actually get the upper limit value minus the lower limit value, which should be $20 - (-104) = 20 + 104$ .

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