State sales tax $S$ is directly proportional to retail price $p$. An item that sells for 186 dollars has a sales tax of 14.42 dollars.
Find a mathematical model that gives the amount of sales tax $S$ in terms of the retail price $p$.
Function: $S(p) = \square$
What is the sales tax on a 360 dollars purchase? Round to the nearest cent.
The sales tax on a 360 dollar purchase is $\$\square$.
Answer (2)
Establish the direct proportionality relationship: S = k ′ . ′ p .
Calculate the constant of proportionality k = 186 14.42 = 0.07752688172 .
Formulate the mathematical model: S ( p ) = 0.07752688172 p .
Calculate the sales tax for a $360 purchase and round to the nearest cent: 27.91 .
Explanation
Understanding Direct Proportionality We are given that the state sales tax S is directly proportional to the retail price p . This means we can write the relationship as S = k "." p , where k is the constant of proportionality.
Finding the Constant of Proportionality We are given that an item that sells for $186 has a sales tax of $14.42 . We can use this information to find the constant of proportionality k . Substituting p = 186 and S = 14.42 into the equation S = k "." p , we get 14.42 = k "."186 .
Calculating k To solve for k , we divide both sides of the equation by 186: k = 186 14.42 = 0.07752688172
Mathematical Model Now that we have found k , we can write the mathematical model that gives the amount of sales tax S in terms of the retail price p : S ( p ) = 0.07752688172 p
Calculating Sales Tax for 360 P u rc ha se T o f in d t h es a l es t a x o na $360 p u rc ha se , w es u b s t i t u t e p = 360 in t o t h ee q u a t i o n S(p) = 0.07752688172 p : S ( 360 ) = 0.07752688172 × 360 = 27.90967741935484 $
Rounding to Nearest Cent Rounding the result to the nearest cent, we get $27.91 .
Final Answer Therefore, the mathematical model is S ( p ) = 0.07752688172 p , and the sales tax on a $360 purchase is $27.91 .
Examples
Understanding direct proportionality is useful in many real-life scenarios. For example, the amount you earn might be directly proportional to the number of hours you work. If you earn $15 per hour, your total earnings can be modeled as E = 15 h , where E is your total earnings and h is the number of hours you work. Similarly, the distance a car travels at a constant speed is directly proportional to the time it travels. If a car travels at 60 miles per hour, the distance it covers can be modeled as d = 60 t , where d is the distance and t is the time.
The mathematical model for sales tax is S ( p ) = 0.07752688172 p . The sales tax on a $360 purchase is approximately $27.91.
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