Mathematical Problem

Question

A toy tractor sold for $236 in 1979 and was sold again in 1990 for $437. Assume that the growth in the value V of the collectors items was exponential.

A). Find the value k of the exponential growth rate. Assume V=236

B). Find the exponential growth function in which terms of f, where T is the number of years since 1979.

C). Estimate the value of the toy tractor in 2010.

D). Find the amount of time after which the value of the toy tractor will be $1,447.

Answer 1

Happy to help.

(A) Exponential growth follows the formula:

F = Ve^kt

V is the initial amount ($236)

k is the growth rate (what we're solving for)

t is the time in years that has elapsed since 1990. Ex. t = 1990-1979 = 11 years

F is the final amount ($437)

e is just the mathematical symbol which corresponds to 2.718...

So plug everything in the equation...

437 = 236e^k(11)

Now it looks a little tricky, but remember we can use natural logarithms and addition properties to cancel out exponents:

Ln (437) = Ln (236e^k(11) ) = Ln(236) + Ln (e^k(11)) = Ln(236) + k(11)

Using a calculator we find that Ln(437) = 6.08 and Ln(236) = 5.46

6.08 = 5.46 + 11k -> solve for k:

k = 0.056015

(B) The exponential growth function is what we used earlier, but now we can include the value of k in the equation:

F = ($236)e^(0.056)t This equation represents the value of the collector's goods

(C) The value of the toy tractor in 2010 can be calculated from our formula in (B).

First, calculate the elapsed number of years after 1979:

t = 2010-1979 = 31

Then plug t into the equation and solve with your calculator:

F = ($236)e^(0.056)t -> F = ($236)e^(0.056)(31) = $1339.21 This is the value of the toy tractor in 2010

(D) To find the amount of time until the tractor will be valued at $1447, we must solve for t in the equation:

$1447 = ($236)e^(0.056)t

Again, using logarithms we find:

Ln (1447) = Ln( 236e^(0.056)t) = Ln(236)+ Ln(e^(0.056)t) = Ln(236)+ (0.056)t

solve with your calculator:

7.28 = 5.46+0.056t

Then solve for t and you're done!

t = 32.43 years