a baseball card sold for $205 in 1979 and was sold again in 1985 for $488. assume that the growth in the…

a baseball card sold for $205 in 1979 and was sold again in 1985 for $488. assume that the growth in the value v of the collectors item was exponential.\na) find the value k of the exponential growth rate. assume ( v_0 = 205 ).\nk = 0.0488\n(round to the nearest thousandth.)

a baseball card sold for $205 in 1979 and was sold again in 1985 for $488. assume that the growth in the value v of the collectors item was exponential.\na) find the value k of the exponential growth rate. assume ( v_0 = 205 ).\nk = 0.0488\n(round to the nearest thousandth.)

Answer

Explanation:

Step1: Write the exponential growth formula

The exponential growth formula is (V = V_0e^{kt}). Here, (V_0 = 205), (V = 488), and (t=1985 - 1979=6) years. Substituting the values into the formula gives (488 = 205e^{6k}).

Step2: Solve for (e^{6k})

Divide both sides of the equation (488 = 205e^{6k}) by (205): (\frac{488}{205}=e^{6k}). Since (\frac{488}{205} = 2.3804878), the equation becomes (2.3804878=e^{6k}).

Step3: Take the natural logarithm of both sides

Take the natural logarithm of both sides: (\ln(2.3804878)=\ln(e^{6k})). Using the property (\ln(e^{x})=x), we get (\ln(2.3804878) = 6k). Since (\ln(2.3804878)\approx0.867), then (0.867 = 6k).

Step4: Solve for (k)

Divide both sides of (0.867 = 6k) by (6): (k=\frac{0.867}{6}). (k = 0.1445\approx0.145)

Answer:

(k\approx0.145)