Question

Every year, a town surveys the size of its wetland ecosystems. This equation estimates the total wetland area, in acres, over time, t, in years.
F(t)= 740(0.95)^t

Which statements are supported by the function?

When t=10, the town is predicted to have around 443 acres of wetlands.

At one time, the town had 740 acres of wetlands.

The wetland area in the town decreases by 5% per year.

When t=10, the town is predicted to have around 492 acres of wetlands.

The wetland area in the town increases by 95% per year.

Answer 1

Answer:

When t=10, the town is predicted to have around 443 acres of wetlands.

At one time, the town had 740 acres of wetlands.

The wetland area in the town decreases by 5% per year.

Step-by-step explanation:

Given

$f(t) = 740 * (0.95)^t$

Required

The true statements

$(a)\ f(10) = 443$

We have:

$f(t) = 740 * (0.95)^t$

Substitute 10 for t

$f(10) = 740 * (0.95)^{10$

$f(10) = 443$

Hence, (a) is true

$(b)\ f(t) = 740$

We have:

$f(t) = 740 * (0.95)^t$

Substitute 740 for f(t)

$740 = 740 * (0.95)^t$

Divide through by 740

$1 = 0.95^t$

Express 1 as 0.95^0

$0.95^0 = 0.95^t$

Cancel out the bases

$0 = t$

$t = 0$

Hence, the area had 740 acres initially;

(b) is true

$(c)\ r = 5\%$

We have:

$f(t) = 740 * (0.95)^t$

Using the general formula

$f(t) = a * b^t$

By comparison:

$b = 0.95$

0.95 < 1 means that:

$b = 1 - r$ --- where r = rates

$0.95 = 1 - r$

Collect like terms

$r = 1 - 0.95$

$r = 0.05$

Express as percentage

$r = 5\%$

Hence, (c) is true

$(d)\ f(10) = 492$

In (a)

$f(10) = 443$

Hence, (d) is false

$(e)\ r = 95\%$

In (c)

$r = 5\%$ ---- decrement

Hence, (e) is incorrect