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Consider the following case of exponential growth:

The average price of a home in a town was [tex]$\$ 172,000$[/tex] in 2013, and home prices are rising by [tex]$4 \%$[/tex] per year.

a. Find an exponential function of the form [tex]$Q = Q_0 \times (1 + r)^t[tex]$[/tex] (where [tex]$[/tex]r > 0$[/tex]) for growth to model the situation described.

[tex]$Q = \[tex]$[/tex]\ \square\ (1 + \square) ^ t$[/tex]

(Type an integer or a decimal.)

Sagot :

GinnyAnsweridn
To find the exponential function describing the growth in the average price of a home in the town, we start by identifying the given information:

1. The initial price of a home ([tex]$Q_0$[/tex]) in 2013: \[tex]$172,000 2. The annual growth rate (r): 4% We need to express the 4% annual growth rate as a decimal. A 4% increase per year corresponds to an increase factor of 0.04. An exponential growth function can be modeled using the formula: \[ Q = Q_0 \times (1 + r)^t \] Now, substituting the given values into the formula: 1. $[/tex]Q_0 = 172,000[tex]$ (the initial price of the house) 2. $[/tex]r = 0.04[tex]$ (the growth rate) Thus, the base of our exponential function (1 + r) will be: \[ 1 + 0.04 = 1.04 \] Putting it all together, the exponential function modeling the situation is: \[ Q = 172,000 \times 1.04^t \] Therefore, the values for the blanks are: \[ Q = \$[/tex] 172,000 (1.04)^t \]

This is the exponential function that models the growth in the average price of a home in the town.
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