Question

select the correct answer from each drop-down menu.

tonya and leo each bought a cell phone at the same time.

the trade-in values, in dollars, of the cell phones are modeled by the given functions, where x is the number of months that each person has owned the phone

tonya’s phone: ( f(x) = 490(0.90)^x )

leo’s phone:

x	g(x)
2	360
4	270

____ phone has the greater initial trade-in value

during the first four months, the trade-in value of tonya’s phone decreases at an average rate ____ the trade-in value of leo’s phone

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Explanation:

To solve the problem, we analyze the initial trade - in values and the average rate of decrease for both phones.

Step 1: Find the initial trade - in values

• For Tonya's phone, the function is \(f(x)=490(0.90)^{x}\). The initial trade - in value occurs when \(x = 0\). Using the property of exponents \(a^{0}=1\) (where \(a = 0.90\) in this case), we have \(f(0)=490\times(0.90)^{0}=490\times1 = 490\) dollars.
• For Leo's phone, we look at the table. When \(x = 0\), \(g(0)=400\) dollars.

Since \(490>400\), Tonya's phone has the greater initial trade - in value.

Step 2: Calculate the average rate of decrease for Tonya's phone

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by the formula \(\frac{f(b)-f(a)}{b - a}\).
For Tonya's phone, we want to find the average rate of decrease over the first 4 months, so \(a = 0\) and \(b = 4\).
First, find \(f(4)\):
\(f(4)=490\times(0.90)^{4}\)
\((0.90)^{4}=0.90\times0.90\times0.90\times0.90 = 0.6561\)
\(f(4)=490\times0.6561=321.489\)
Now, use the average rate of change formula:
\(\frac{f(4)-f(0)}{4 - 0}=\frac{321.489 - 490}{4}=\frac{- 168.511}{4}\approx - 42.13\)

Step 3: Calculate the average rate of decrease for Leo's phone

For Leo's phone, using the table, when \(x = 0\), \(g(0)=400\) and when \(x = 4\), \(g(4)=270\).
The average rate of change over the interval \([0,4]\) is \(\frac{g(4)-g(0)}{4 - 0}=\frac{270 - 400}{4}=\frac{-130}{4}=- 32.5\)

Since \(\vert - 42.13\vert>\vert - 32.5\vert\), the trade - in value of Tonya’s phone decreases at an average rate greater than the trade - in value of Leo’s phone.

Answer:

s:

• The phone with the greater initial trade - in value: Tonya's
• The relationship between the average rate of decrease of Tonya's phone and Leo's phone: greater than