Last year, Mitch inherited a first-edition book from his grandfather. When he first received it,
he looked online and found that the value of the book was listed at . He checked again
one year later and saw that the value has increased to . The value of the book is
expected to continue to increase over the next few years.
Write an exponential equation in the form that can model the value of Mitch’s book,
, in years.
Use whole numbers, decimals, or simplified fractions for the values of a and b.

Ask by Moran Daniel. in the United States

Jan 24,2025

Question

Last year, Mitch inherited a first-edition book from his grandfather. When he first received it,
he looked online and found that the value of the book was listed at . He checked again
one year later and saw that the value has increased to . The value of the book is
expected to continue to increase over the next few years.
Write an exponential equation in the form that can model the value of Mitch’s book,
, in years.
Use whole numbers, decimals, or simplified fractions for the values of a and b.

Ask by Moran Daniel. in the United States

Jan 24,2025

UpStudy AI · Accepted Answer

To model the value of Mitch's book using an exponential equation of the form $$ y = a(b)^x $$, we'll follow these steps:

1. **Identify the Initial Value ($$ a $$)**:
   - When Mitch first received the book (consider this as $$ x = 0 $$ years), its value was $$ \$250 $$.
   - So, $$ a = 250 $$.

2. **Determine the Growth Factor ($$ b $$)**:
   - After 1 year ($$ x = 1 $$), the book's value increased to $$ \$265 $$.
   - The growth factor $$ b $$ can be calculated using the formula:
     $$
     b = \frac{\text{Value after 1 year}}{\text{Initial value}} = \frac{265}{250} = 1.06
     $$
   - This means the book's value increases by 6% each year.

3. **Construct the Exponential Equation**:
   - Plugging $$ a $$ and $$ b $$ into the exponential form:
     $$
     y = 250 \times (1.06)^x
     $$

**Final Model:**
$$
y = 250 \times (1.06)^x
$$

This equation models the value of Mitch's book $$ y $$ in $$ x $$ years, assuming the value continues to increase by 6% annually.

**Answer:**  
$$ y = 250 \times (1.06)^{\,x} $$
$$ y = 250 \times (1.06)^x $$

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