Questions: During the 1980s, sales of compact discs surpassed record sales. From 1985 to 1990, sales of compact discs in millions can be modeled by the formula f(x)=48.6(x-1985)+8.5, whereas sales of LP records in millions can be modeled by g(x)=-29.2(x-1985)+45.3. Approximate the year x when sales of LP records and compact discs were equal by using the intersection-of-graphs method. Each had equal sales in the year (Round to the nearest whole number as needed.)

During the 1980s, sales of compact discs surpassed record sales. From 1985 to 1990, sales of compact discs in millions can be modeled by the formula f(x)=48.6(x-1985)+8.5, whereas sales of LP records in millions can be modeled by g(x)=-29.2(x-1985)+45.3. Approximate the year x when sales of LP records and compact discs were equal by using the intersection-of-graphs method.

Each had equal sales in the year
(Round to the nearest whole number as needed.)

Transcript text: During the 1980s, sales of compact discs surpassed record sales. From 1985 to 1990, sales of compact discs in millions can be modeled by the formula $\mathrm{f}(\mathrm{x})=48.6(\mathrm{x}-1985)+8.5$, whereas sales of LP records in millions can be modeled by $g(x)=-29.2(x-1985)+45.3$. Approximate the year $x$ when sales of $L P$ records and compact discs were equal by using the intersection-of-graphs method. Each had equal sales in the year $\square$ (Round to the nearest whole number as needed.)

Solution

Solution Steps

To find the year when sales of LP records and compact discs were equal, we need to find the intersection point of the two given functions. This involves setting the two equations equal to each other and solving for $ x $. The solution will give us the year when the sales were equal.

Step 1: Define the Functions

The sales of compact discs and LP records are modeled by the following equations: \[ f(x) = 48.6(x - 1985) + 8.5 \] \[ g(x) = -29.2(x - 1985) + 45.3 \]

Step 2: Set the Functions Equal

To find the year when sales were equal, we set $ f(x) $ equal to $ g(x) $: \[ 48.6(x - 1985) + 8.5 = -29.2(x - 1985) + 45.3 \]

Step 3: Solve for $ x $

Rearranging the equation gives: \[ 48.6x - 96462.5 = 58007.3 - 29.2x \] Combining like terms results in: \[ 77.8x = 154469.8 \] Solving for $ x $ yields: \[ x \approx 1985.473 \]

Step 4: Round the Result

Rounding $ x $ to the nearest whole number gives: \[ x \approx 1985 \]