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How does $g(t) = 3^t$ change over the interval from $t = 3$ to $t = 4$ ? $\newline$ Choices: $\newline$ (A) $g(t)$ decreases by $3$ $\newline$ (B) $g(t)$ increases by $3$ $\newline$ (C) $g(t)$ decreases by $3\%$ $\newline$ (D) $g(t)$ increases by $200\%$

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Exponential functions over unit intervals

Full solution

Q. How does

g(t) = 3^t

change over the interval from

t = 3

to

t = 4

?

\newline

Choices:

\newline

(A)

g(t)

decreases by

3

\newline

(B)

g(t)

increases by

3

\newline

(C)

g(t)

decreases by

3\%

\newline

(D)

g(t)

increases by

200\%

Evaluate $g(t)$ at $t = 3$ : We need to evaluate the function $g(t) = 3^t$ at $t = 3$ and $t = 4$ to determine how it changes over this interval. $\newline$ First, let's find the value of $g(3)$ . $\newline$ Substitute $t = 3$ into $g(t) = 3^t$ . $\newline$ $g(3) = 3^3$ $\newline$ $g(3) = 27$
Evaluate $g(t)$ at $t = 4$ : Next, we need to find the value of $g(4)$ . $\newline$ Substitute $t = 4$ into $g(t) = 3^t$ . $\newline$ $g(4) = 3^4$ $\newline$ $g(4) = 81$
Determine the changes in the function: Now, we compare the values of $g(3)$ and $g(4)$ to determine the change. $\newline$ We have $g(3) = 27$ and $g(4) = 81$ . $\newline$ Since $81$ is greater than $27$ , $g(t)$ increases over the interval from $t = 3$ to $t = 4$ .
Calculate percentage increase: $\newline$ $\text{Percentage Change} = \frac{(\text{final value} - \text{initial value})} {(\text{initial value})} \times 100$ $\newline$ $\ = \frac{g(4) - g(3)} {g(3)} \times 100$ $\newline$ $= \frac{81 - 27} {27} \times 100$ $\newline$ $= \frac{54} {27} \times 100$ $\newline$ $= 2 \times 100$ $\newline$ $= 200$ $\newline$ Therefore, the percentage change is $200\%$ .
Choose correct option: We found that $g(t)$ increases over the interval from $t = 3$ to $t = 4$ and the percentage change is $200\%$ . $\newline$ Correct option: $g(t)$ increases by $200\%$

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