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The cost of inflation can be modeled by the following formula, where [tex]C[/tex] represents the cost [tex]x[/tex] years after 2010 of what cost \[tex]$100 in 1999. Use the given model to determine in which year the cost will be \$[/tex]160 for what cost \$100 in 1999.

[tex]C = 1.9x + 127.5[/tex]

In [tex]\square[/tex], the cost will be \[tex]$160 for what cost \$[/tex]100 in 1999. (Round to the nearest year as needed.)



Answer :

GinnyAnswer
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.

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