A once-popular children's doll is slowly declining in popularity. The quartic function f(x)=0.002x^(4)+0.025x^(3)-0.364x^(2)-7.243 x+86.993 where x is the number of years since 1993, can be used to estimate the number of dolls of this type that were sold (in thousands) during a given year from 1993 to 2003. Estimate how many dolls were sold in 1997.

Question

A once-popular children's doll is slowly declining in popularity. The quartic function  f(x)=0.002x^(4)+0.025x^(3)-0.364x^(2)-7.243 x+86.993 where  x is the number of years since 1993, can be used to estimate the number of dolls of this type that were sold (in thousands) during a given year from 1993 to 2003. Estimate how many dolls were sold in 1997.

Bytelearn · Accepted Answer

Identify Year 1997: Identify the value of x for the year 1997. Since x represents the number of years since 1993, we need to calculate the difference between 1997 and 1993 to find the value of x. Calculation: 1997 - 1993 = 4Substitute x in Function: Substitute the value of x into the quartic function to estimate the number of dolls sold. We have the function f(x) = 0.002x^4 + 0.025x^3 - 0.364x^2 - 7.243x + 86.993, and we found that x = 4 for the year 1997. Calculation: f(4) = 0.002(4)^4 + 0.025(4)^3 - 0.364(4)^2 - 7.243(4) + 86.993Perform Term Calculations: Perform the calculations for each term of the function. Calculation for the first term: 0.002(4)^4 = 0.002 * 256 = 0.512 Calculation for the second term: 0.025(4)^3 = 0.025 * 64 = 1.6 Calculation for the third term: -0.364(4)^2 = -0.364 * 16 = -5.824 Calculation for the fourth term: -7.243(4) = -28.972 Calculation for the fifth term: 86.993 (this term does not depend on x)Add Up Terms: Add up all the terms to find the value of f(4). Calculation: f(4) = 0.512 + 1.6 - 5.824 - 28.972 + 86.993Calculate Dolls Sold: Complete the calculation to find the estimated number of dolls sold in 1997. Calculation: f(4) = 0.512 + 1.6 - 5.824 - 28.972 + 86.993 = 54.309Interpret Result: Interpret the result. Since the function gives the number of dolls sold in thousands, we need to multiply the result by 1,000 to get the actual number of dolls sold. Calculation: 54.309 * 1,000 = 54,309

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