Question

directions: answer each question. be sure to label your answers.
name:
date:

1. a new car that sells for $18,000 depreciates 25% each year. write a function that models the value of the car. find the value of the car after 4 years.

1. a new truck that sells for $29,000 depreciates 12% annually. write a function that models the value of the truck. find the value of the truck after 7 years.

the bear population increases at a rate of 2% per year. there are 1,573 bears right now. write a function that models the bear population. how many bears will there be in 10 years?

an investment of $75,000 increases at a rate of 12.5% each year. write a function that models the value of the investment over the years. find the value of the investment after 30 years.

a bird population is decreasing at a rate of 0.75% per year. there are currently... write a function that models the bird population. how many birds...

Explanation:

Problem 1:

Step1: Identify the depreciation formula

The formula for exponential depreciation is $V(t) = P(1 - r)^t$, where $P$ is the initial value, $r$ is the annual depreciation rate (as a decimal), and $t$ is the time in years. Here, $P = 18000$, $r = 0.25$, and we want to find the value after $t = 4$ years. First, write the function: $V(t)=18000(1 - 0.25)^t=18000(0.75)^t$.

Step2: Calculate the value after 4 years

Substitute $t = 4$ into the function: $V(4)=18000(0.75)^4$. Calculate $(0.75)^4 = 0.75\times0.75\times0.75\times0.75=\frac{81}{256}\approx0.3164$. Then $V(4)=18000\times0.3164 = 5695.2$.

Step1: Identify the depreciation formula

Using the exponential depreciation formula $V(t)=P(1 - r)^t$, where $P = 29000$, $r = 0.12$. The function is $V(t)=29000(1 - 0.12)^t=29000(0.88)^t$.

Step2: Calculate the value after 7 years

Substitute $t = 7$ into the function: $V(7)=29000(0.88)^7$. Calculate $(0.88)^7\approx0.4140$. Then $V(7)=29000\times0.4140\approx12006$.

Step1: Identify the growth formula

The formula for exponential growth is $P(t)=P_0(1 + r)^t$, where $P_0$ is the initial population, $r$ is the annual growth rate (as a decimal), and $t$ is the time in years. Here, $P_0 = 1573$, $r = 0.02$. The function is $P(t)=1573(1 + 0.02)^t=1573(1.02)^t$.

Step2: Calculate the population after 10 years

Substitute $t = 10$ into the function: $P(10)=1573(1.02)^{10}$. Calculate $(1.02)^{10}\approx1.2190$. Then $P(10)=1573\times1.2190\approx1917$.

Answer:

The function is $V(t) = 18000(0.75)^t$, and the value after 4 years is $\$5695.2$.

Problem 2: