Question

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learning goal
i can graph exponential functions that are expressed symbolically. this means i can show intercepts and end - behavior
lesson reflection (circle one)
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complete the previous problems, check your solutions, then complete the lesson checkpoint below.
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1. jessica recently purchased her dream car, a porsche 911, for $55,000. the value of this car will depreciate by 8% each year. find the value of the car after 5 years.

write an exponential growth or decay function to model this situation.

1. simone just purchased a home for $300,000. the value of her home will likely appreciate 3% each year in the long term. if t represents the number of years shes owned her home, and v(t) represents the value of her home, which of the following functions models this situation? select all that apply.

v(t)=300,000(1 +.03)^t yes no
v(t)=300,000(1 -.03)^t yes no
v(t)=1.03(300,000)^t yes no

Explanation:

Step1: Identify the decay - formula

The general formula for exponential decay is $V(t)=V_0(1 - r)^t$, where $V_0$ is the initial value, $r$ is the rate of decay, and $t$ is the time. For the car problem, $V_0 = 55000$, $r=0.08$, and $t = 5$.

Step2: Substitute values into the formula

$V(5)=55000(1 - 0.08)^5=55000\times0.92^5$.

Step3: Calculate $0.92^5$

$0.92^5=0.92\times0.92\times0.92\times0.92\times0.92 = 0.6590815232$.

Step4: Calculate the value of the car

$V(5)=55000\times0.6590815232\approx36249.48$.

For the house - value problem:
The general formula for exponential growth is $V(t)=V_0(1 + r)^t$, where $V_0$ is the initial value, $r$ is the rate of growth, and $t$ is the time. Here, $V_0 = 300000$ and $r = 0.03$.

• For $V(t)=300000(1 + 0.03)^t$, this is the correct exponential - growth formula, so the answer is Yes.
• For $V(t)=300000(1 - 0.03)^t$, this is an exponential - decay formula, so the answer is No.
• For $V(t)=1.03(300000)^t$, this is not the correct form of the exponential - growth formula. The correct form has the exponent on the $(1 + r)$ term, not on the initial value, so the answer is No.

Answer:

1. The value of the car after 5 years is approximately $\$36249.48$.
2. $V(t)=300000(1 + 0.03)^t$: Yes

$V(t)=300000(1 - 0.03)^t$: No
$V(t)=1.03(300000)^t$: No