To determine the correct exponential equation that fits the given points (0, 30000) and (1, 22500), we can use the general form of an exponential decay equation:
\[ y = a \cdot b^x \]
where:
- \( a \) is the initial value (the value of the car at year 0),
- \( b \) is the decay factor,
- \( x \) is the number of years since purchase.
Given the first point (0, 30000), we have:
\[ a = 30000 \]
Now, we also know from the second point (1, 22500) that:
\[ y = 22500 \] \[ x = 1 \]
Substituting these values into the equation gives us:
\[ 22500 = 30000 \cdot b^1 \]
To solve for \( b \):
\[ b = \frac{22500}{30000} \] \[ b = \frac{3}{4} \] \[ b = 0.75 \]
So the exponential equation representing the car's value over time is:
\[ y = 30000 \left( \frac{3}{4} \right)^x \]
The correct response that matches this equation is:
y = 30,000(3/4)^x
So the correct option is:
y equals 30,000 left parenthesis Start Fraction 3 over 4 End Fraction right parenthesis superscript x baseline.