30. The retail price of a certain item at a store is x dollars, where x > 5 If a coupon is used that reduces the retail price by $5, the reduced price r, in dollars, of the item is given by the function r(x) = x - 5 for x > 5 After a 5 percent sales tax is charged by the store on the reduced price of the item, the cost c, in dollars, of the item is given by the function c(r) = 1.05r Which of the following gives the cost c of the item as a function of its retail price x for x > 5 ?
Answer: B
c(x) = 1.05(x - 5)
To find the cost of the item as a function of its retail price x, we first apply the coupon discount of $5, leading to a reduced price of r(x) = x - 5. Following this, we apply a 5 percent sales tax, resulting in the final cost c(x) = 1.05(x - 5).
A) c(x) = 1.05x - 5
This option incorrectly suggests that the cost is simply 1.05 times the retail price minus $5. This does not account for the fact that the coupon is applied before the sales tax, which should be calculated on the reduced price, not the original price.
B) c(x) = 1.05(x - 5)
This option accurately represents the cost of the item after applying the coupon and then calculating the sales tax. The function correctly applies the $5 discount first, followed by the 5 percent tax on the reduced price.
C) c(x) = (1.05x)(x - 5)
This option incorrectly combines the retail price and the discount in a multiplication format, which does not reflect the proper sequential application of the discount and tax. The calculation should involve applying the coupon before the tax, not multiplying both together.
D) c(x) = 1.05x + (x - 5)
This option incorrectly adds the full retail price (1.05x) to the reduced price (x - 5), leading to an inflated cost that does not reflect the correct process of applying a discount followed by a tax.
E) c(x) = (1.05x)/(x - 5)
This option presents a division which is not relevant to the calculation of the cost after applying a discount and tax. It misrepresents the relationship between the retail price and the final cost of the item.
Conclusion
The correct function c(x) = 1.05(x - 5) accounts for the coupon discount applied to the retail price before calculating the sales tax. All other options misrepresent the order of operations or the mathematical relationships involved in determining the final cost, making them incorrect.