Question

In 2004, an art collector paid $84,275,000 for a particular painting. The same painting sold for $27,000 in 1950. Complete part (a) through (d)? (a) Find exponential growth rate k, to three decimal places, and determine the exponential growth function V, for which V(t) is the painting’s value, in dollars, t years after 1950.

Answer 1

In 2004, an art collector paid $84,275,000 for a particular painting, the exponential growth rate k is 0.103 and determine the exponential growth function V(t) for the painting's value is 27,000 * e^(0.103 * t).

Step-by-step explanation:

To find the exponential growth rate k and determine the exponential growth function V(t) for the painting's value, we can use the formula for exponential growth.

Let's define t as the number of years after 1950, and V(t) as the value of the painting in dollars at time t.

We have two data points: in 1950, the painting sold for $27,000, and in 2004, it was sold for $84,275,000.

First, we need to find the growth rate k.

We can use the formula V(t) = V(0) * e^(k*t)

Where V(0) is the initial value of $27,000.

Plugging in the values for 2004 and 1950, we get 84,275,000 = 27,000 * e^(k*54).

Solving for k, we can take the natural logarithm of both sides and divide by 54: k = ln(84,275,000/27,000) / 54.

Calculating this expression to three decimal places, we find k ≈ 0.103.

Now, we can determine the exponential growth function V(t).

Substituting the known values for V(0) and k into the formula, we get V(t) = 27,000 * e^(0.103 * t).

This function represents the value of the painting, in dollars, t years after 1950.

So therefore the exponential growth rate k is 0.103 and determine the exponential growth function V(t) for the painting's value is 27,000 * e^(0.103 * t).

Answer 2

To find the exponential growth rate k and the exponential growth function V(t), we can set up equations using the initial value and the final value of the painting's price. Solving these equations, we find that the growth rate is approximately 0.074, and the exponential growth function is V(t) ≈ 27000 * e^(0.074t).

Step-by-step explanation:

To find the exponential growth rate k and the exponential growth function V(t), we need to use the formula V(t) = Ce^(kt), where C is the initial value and e is the base of natural logarithm.

Given that the painting was sold for $27,000 in 1950 and $84,275,000 in 2004, we can set up the following equations:

27000 = Ce^(k*0)

84275000 = Ce^(k*54)

Simplifying these equations, we get:

C = 27000

84275000 = 27000 * e^(k*54)

Solving the second equation for k, we find:

k ≈ 0.074

So the exponential growth function V(t) is:

V(t) ≈ 27000 * e^(0.074t)