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 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.

Explanation:

Answer 2

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.

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