A community library starts a digital collection with 5,000 e-books. The library's goal is … — SAT

A community library starts a digital collection with $5,000$ e-books. The library's goal is to increase the number of e-books in the collection by $8\%$ each year. Which of the following functions best models the number of e-books, $E(t)$ , in the collection $t$ years after the collection was started?

A
$E(t) = 400t + 5,000$
B
$E(t) = 5,000(0.08)^t$
$E(t) = 5,000(1.08)^t$ Cevap
D
$E(t) = 1.08t + 5,000$

Cevap

E(t) = 5,000(1.08)^t

The correct option is the function that represents exponential growth, where the quantity increases by a constant percentage each year. The general formula for exponential growth is

E(t) = a(1 + r)^t

, where

a

is the initial value,

r

is the growth rate as a decimal, and

t

is the number of time periods. Here, the initial number of e-books is

a = 5,000

and the annual growth rate is

r = 8\% = 0.08

. Therefore, the growth factor is

1 + r = 1.08

. Substituting these values into the formula yields the correct model:

E(t) = 5,000(1.08)^t

Adım Adım Çözüm

Determine if the growth is linear or exponential by analyzing the type of increase described.

The problem states that the collection increases by

8\%

each year. Since the increase is a constant percentage rather than a constant quantity, this represents exponential growth.

Identifying the growth type determines whether to use a linear equation or an exponential equation.

State the general form of an exponential growth function.

The general formula is

E(t) = a(1 + r)^t

, where

a

is the initial amount,

r

is the percent growth rate expressed as a decimal, and

1 + r

is the growth factor.

This formula provides the template for constructing the model.

Identify the values of

a

and

r

from the given context and substitute them into the template.

The initial value

a = 5,000

. The growth rate

r = 8\% = 0.08

, so the growth factor is

1 + 0.08 = 1.08

. Substituting these gives

E(t) = 5,000(1.08)^t

This completes the model construction to find the correct expression.

Anahtar Kavram

Linear and Exponential Growth

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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