In 2004, an art collector paid $106,124,000 for a particular painting. The same painting sold for $34,000 in 1950. Complete parts (a) through (d). a) Find the 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. V(t) = 34,000e.149t (Type an expression. Type integers or decimals for any numbers in the expression. Round to three decimal places as needed.) b) Predict the value of the painting in 2026. $ 2,815,000,000 (Round to the nearest million as needed.) c) Estimate the rate of change of the painting's value in 2026. 419,000,000 dollar(s) per year. (Round to the nearest million as needed.) d) How long after 1950 will the value of the painting be $3 billion? year(s) (Do not round until the final answer. Then round to the nearest year as needed.)View an example I Allparts showing x ' In 2004. an art collector paid 541554.000 for a particular painting. The same painting sold for $21,000 In 1950. Complete parts (a) through (d). n) a) Find the exponential growth rate It. 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. W An exponential growth function is of the forth I = lt - V[t) tor some constant V0. The price of the painting in 1950. when t = 0. is given. sothe initial value Va. is 321.000. Thus. v(t) = 21,0009\". The value of k Is not given but it can be computed using the other information. We are given that the price of the painting in 1950. or the initial value Va. is $21,000. We're also given that the price of the painting in the year 2004 is 942554.000. The value oft is as follows. t22004-1950254 So when t = 54. the value ofthe painting is $42,554,000. that is. V154) = 42,554,000. Substitute tor t and W54). v(1) = 21,0009"1 v(54) = 2100:1005" 42,554,000 = 21 0004041 Solve for It. First. isolate the exponential expression. Divide both sides of the equation by 21,000 and simplify. 42.554 _ 54k 21 " Now use logarithms to isolate in rounding to three decimal places. In 4225154 = In (-5\" Take the natural logarithm of both sides. 42.554 In 21 = 54k Applyln [9'] :3. 0.141 = It Divide both sides by 54 and simplify. Determine the exponential growth function V, for which Wt} is the painting's value. in dollars. I years after 1950. vii) = 21.00091\"m b) Predict the value of the painting in 2000. The value of the painting is determined by Wt] = 21.0004'0'1'". where t is the number of years after 1950. The value oft is as follows. t=2008-1950= 53 Find the value of WSB}, given the equation V(t) = 21000:" lan' rounding to the nearest million, visa} = 21.000it-1\"'5' = 75.000000 Therefore. the value of the painting was approximately 575000.000 in 2008. 0) Estimate the rate of change of the painting's value in 2008. To calculate the rate of change of the value. the equation forV'm must be used. Use the fact that the derivative of the exponential model is the rate constant times the exponential function and k = 0 141 to calculate \\i"[t). V'itl = k'vn) v'm = [1141 . 21 _m.-141= Calculate v'(t}. Substitute Tim and It. 01153} = [1141 . 21 [mp-\""591 Evaluate V'(t) at t = 58. = 11,000,000 Simplify. Therefore. the rate of change of the painting was approximately $11,000,000 per year in 2000. d) How long after 1950 will the value of the painting be $8 billion? 0.1411 Find the value oft for which V[t) : 21,000:- = 3 billion. Then use that value to determine the year. Substitute the given value tor Wt). V(t) = 21,0009'1\"\" d) How long after 1950 will the value of the painting be $8 billion? Question 17 Find the value of t for which V(t) = 21,000e0-141 = 8 billion. Then use that value to determine the year. Substitute the given value for V(t). Question 18 V(t) = 21,000e0.141t 8,000,000,000 = 21,000e0-141t Question 19 Solve for t. First, isolate the exponential expression. Divide both sides of the equation by 21,000 and simplify. 8,000,000 0.141t =e 21 Question 20 Now use logarithms to isolate t, rounding to the nearest year. 8,000,000 In 0.141t Take the natural logarithm of both sides. 21 8,000,000 In = 0.141t 21 Apply In (e* ) = x. 91 = t Divide both sides by 0.141 and simplify. Therefore, the value of the painting will have reached $8 billion 91 years after 1950, or 2041. Print Close Help me solve this View an example Get more help - Clear all Check