Let's solve each step: ### 1. **Assign \( x \) values to each year** Let \( x \) be the number of years after 2003: - 2003: \( x = 0 \) → 2.52 million - 2004: \( x = 1 \) → 5.436 million - 2005: \( x = 2 \) → 27.433 million - 2006: \( x = 3 \) → 68.642 million - 2007: \( x = 4 \) → 118.772 million - 2008: \( x = 5 \) → 173.6 million --- ## **(a) Find the quadratic function \( s(x) = ax^2 + bx + c \) for sales** Let's use three points to find the quadratic: Use points: (0, 2.52), (2, 27.433), (4, 118.772) Write the system: \[ \begin{cases} s(0) = c = 2.52 \\ s(2) = 4a + 2b + c = 27.433 \\ s(4) = 16a + 4b + c = 118.772 \end{cases} \] Substitute \( c = 2.52 \): \[ 4a + 2b + 2.52 = 27.433 \implies 4a + 2b = 24.913 \] \[ 16a + 4b + 2.52 = 118.772 \implies 16a + 4b = 116.252 \] Now solve for \( a \) and \( b \): Divide the second by 4: \[ 4a + b = 29.063 \] Subtract the first: \[ (4a + 2b = 24.913) \\ (4a + b = 29.063) \\ \implies b = 24.913 - 29.063 = -4.15 \] Now substitute \( b \) back: \[ 4a + (-4.15) = 29.063 \implies 4a = 33.213 \implies a = 8.303 \] So, - \( a = 8.303 \) - \( b = -4.15 \) - \( c = 2.52 \) ### **Final quadratic model:** \[ \boxed{s(x) = 8.303x^2 - 4.150x + 2.520} \] (Rounded to three decimals.) --- ## **(b) Derivative of the sales model** The derivative is: \[ s'(x) = 2a x + b \] So, \[ s'(x) = 2 \times 8.303 x - 4.150 = 16.606x - 4.150 \] (Rounded to three decimals.) --- ## **(c) Rate of change in 2007 (\( x = 4 \))** Plug \( x = 4 \) into \( s'(x) \): \[ s'(4) = 16.606 \times 4 - 4.150 = 66.424 - 4.150 = 62.274 \] **Rounded to three decimals:** \[ \boxed{62.274} \text{ million iPods per year} \] --- ### **Interpretation** In 2007, cumulative iPod sales were **increasing** by **62.274** million per year. --- ## **Summary Table:** | Part | Answer | |------|--------| | (a) | \( s(x) = 8.303x^2 - 4.150x + 2.520 \) | | (b) | \( s'(x) = 16.606x - 4.150 \) | | (c) | \( 62.274 \) million iPods per year | | Interpretation | In 2007, cumulative iPod sales were **increasing** by **62.274** million per year. | Let me know if you want a detailed breakdown of the algebraic steps!