Example 4:
You are interested in two second-hand gaming consoles. The first console’s price is
given by the function , and the second console’s price is given
by , where represents the time in years. Your budget
constraints require you to choose the more affordable option within the next 2
years.
Which statement below best represents the outcome?

You buy the first console.

You buy the second console.

Both consoles cost the same amount.

Ask by Bright Whittaker. in the United States

Nov 14,2024

Question

Example 4:
You are interested in two second-hand gaming consoles. The first console’s price is
given by the function , and the second console’s price is given
by , where represents the time in years. Your budget
constraints require you to choose the more affordable option within the next 2
years.
Which statement below best represents the outcome?

You buy the first console.

You buy the second console.

Both consoles cost the same amount.

Ask by Bright Whittaker. in the United States

Nov 14,2024

UpStudy AI · Accepted Answer

To determine which gaming console is more affordable within the next 2 years, we need to compare the prices of the two consoles at $$ t = 2 $$ years.

Given:
- The price function for the first console is $$ f(t) = 600(0.75)^t $$.
- The price function for the second console is $$ g(t) = 800(0.6)^t $$.

We need to find the prices of both consoles at $$ t = 2 $$ years and compare them to determine which one is more affordable.

Let's calculate the prices of both consoles at $$ t = 2 $$ years:

1. For the first console:
$$ f(2) = 600(0.75)^2 $$

2. For the second console:
$$ g(2) = 800(0.6)^2 $$

Now, we can calculate the prices of both consoles at $$ t = 2 $$ years and compare them to determine which one is more affordable.
Calculate the value by following steps:
- step0: Calculate:
  $$600\times 0.75^{2}$$
- step1: Convert the expressions:
  $$600\left(\frac{3}{4}\right)^{2}$$
- step2: Evaluate the power:
  $$600\times \frac{9}{16}$$
- step3: Multiply:
  $$\frac{675}{2}$$
Calculate or simplify the expression $$ 800*(0.6)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$800\times 0.6^{2}$$
- step1: Convert the expressions:
  $$800\left(\frac{3}{5}\right)^{2}$$
- step2: Evaluate the power:
  $$800\times \frac{9}{25}$$
- step3: Multiply:
  $$288$$
The price of the first console at $$ t = 2 $$ years is $$ 337.5 $$ dollars, and the price of the second console at $$ t = 2 $$ years is $$ 288 $$ dollars.

Since $$ 288 < 337.5 $$, the second console is more affordable within the next 2 years.

Therefore, the statement that best represents the outcome is:
- You buy the second console.
You buy the second console.

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