the total amount spent by some number of people on clothing and footwear in the years 2000 - 2009 can be…

the total amount spent by some number of people on clothing and footwear in the years 2000 - 2009 can be modeled by the quadratic function ( f(x)=-4.799x^{2}+71.91x + 97.35 ), where ( x = 0 ) represents january 1, 2000, ( x = 1 ) represents january 1, 2001, and so on, and ( f(x) ) is in billions of dollars. according to the model, in what year during this period was the amount spent on clothing and footwear a maximum?\nin the year ( square ), ( $ square ) billion was spent on clothing and footwear.\n(round the final answer for the amount spent to the nearest whole number as needed. round all intermediate values to the nearest tenth as needed.)

the total amount spent by some number of people on clothing and footwear in the years 2000 - 2009 can be modeled by the quadratic function ( f(x)=-4.799x^{2}+71.91x + 97.35 ), where ( x = 0 ) represents january 1, 2000, ( x = 1 ) represents january 1, 2001, and so on, and ( f(x) ) is in billions of dollars. according to the model, in what year during this period was the amount spent on clothing and footwear a maximum?\nin the year ( square ), ( $ square ) billion was spent on clothing and footwear.\n(round the final answer for the amount spent to the nearest whole number as needed. round all intermediate values to the nearest tenth as needed.)

Answer

Explanation:

Step1: Identify the coefficients of the quadratic function

For the quadratic function (f(x)=ax^{2}+bx + c), here (a=-4.799), (b = 71.91), (c=97.35).

Step2: Find the (x) - value of the vertex

The formula for the (x) - coordinate of the vertex of a quadratic function (y = ax^{2}+bx + c) is (x=-\frac{b}{2a}). Substitute (a=-4.799) and (b = 71.91) into the formula: (x=-\frac{71.91}{2\times(- 4.799)}=\frac{71.91}{9.598}\approx7.5)

Step3: Determine the year

Since (x = 0) represents 2000, when (x\approx7.5), the year is (2000 + 7.5\approx2007) (because we consider the whole - number year closest to (x) value in the context of the problem).

Step4: Find the value of (f(x)) at (x = 7.5)

Substitute (x = 7.5) into (f(x)=-4.799x^{2}+71.91x + 97.35) (f(7.5)=-4.799\times(7.5)^{2}+71.91\times7.5+97.35) First, calculate ((7.5)^{2}=56.25) (-4.799\times56.25=-4.799\times\frac{225}{4}=-269.94375) (71.91\times7.5=\frac{71.91\times15}{2}=539.325) Then (f(7.5)=-269.94375 + 539.325+97.35) (f(7.5)=366.73125\approx367)

Answer:

In the year (2007), ($367) billion was spent on clothing and footwear.