To determine the year when the cost will be [tex]$160 for an item that cost $[/tex]100 in 1999, we need to solve the given cost model formula:
[tex]\[ C = 1.9x + 127.5 \][/tex]
Here, \( C \) represents the cost, and \( x \) represents the number of years after 2010. We are given that \( C = 160 \). Therefore, we substitute 160 for \(C\) and solve for \( x \).
The equation is:
[tex]\[ 160 = 1.9x + 127.5 \][/tex]
To find \(x\), we perform the following steps:
1. Subtract 127.5 from both sides of the equation to isolate the term with \(x\):
[tex]\[ 160 - 127.5 = 1.9x \][/tex]
2. Calculate the left-hand side:
[tex]\[ 32.5 = 1.9x \][/tex]
3. Divide both sides of the equation by 1.9 to solve for \( x \):
[tex]\[ x = \frac{32.5}{1.9} \][/tex]
Upon performing the division, we get:
[tex]\[ x \approx 17.105263157894736 \][/tex]
Since \( x \) represents the number of years after 2010, to find the actual year, we add 2010 to \( x \):
[tex]\[ \text{Year} = 2010 + x \][/tex]
Rounding \( x \) to the nearest integer (which is 17), we calculate:
[tex]\[ \text{Year} = 2010 + 17 = 2027 \][/tex]
Therefore, in the year 2027, the cost will be [tex]$160 for an item that cost $[/tex]100 in 1999.
