Question

A baseball card sold for 213 in 1978 and was sold again in 1989 for 466. Assume that the growth in the value v of the collector's item was exponential. What is the growth rate of the value of the baseball card?

1) 5%
2) 10%
3) 15%
4) 20%

Answer 1

The exponential growth rate of the value of the baseball card is 7.7% per year. Since this is not an option, the closest provided rate is 5%.

Step-by-step explanation:

To calculate the exponential growth rate of the value of a baseball card sold for $213 in 1978 and for $466 in 1989, we use the formula for exponential growth:

V = P * e^(rt)

Where V is the final value, P is the initial value, r is the rate of growth, t is the time in years, and e is the base of the natural logarithm. We can rearrange this formula to solve for the growth rate r:

r = (ln(V/P)) / t

Substituting the given values (V = $466, P = $213, t = 1989 - 1978 = 11 years), we get:

r = (ln(466/213)) / 11

Calculating this gives us a growth rate of approximately 0.077 or 7.7% per year. However, since this value is not one of the provided options, we must choose the closest one, which is 5%.

Graphs are often used to display data or evidence, providing a visual representation of numerical patterns, including exponential growth. This can be helpful in understanding how values change over time, especially when the rate of change increases each period to maintain a constant growth rate.