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A car dealer will sell you the car you want for \(\$ 18,194,\) which is just $\$ 200$ over the dealer's invoice price (the price the dealer pays the manufacturer for the car). You tell him that you will think about it. The dealer is anxious to meet his monthly quota of sales, so he calls the next day to offer you the car for \(\$ 17,994\) if you agree to buy it tomorrow, October \(31 .\) You decide to accept the deal. a. What is the decay factor associated with the decrease in the price to you? Write your answer in decimal form to the nearest thousandth. b. What is the percent decrease of the price, to the nearest tenth of a percent? c. The sales tax is \(6.5 \% .\) How much did you save on sales tax by taking the dealer's second offer?

Step by step solution

01

Find the decrease in price

Initially, the car price was \(\$ 18,194\). The dealer offers to sell the car for \(\$ 17,994\). So, we need to calculate the decrease in the price. Decrease in price = Initial Price - Offer Price Decrease in price = \(18194 - 17994\) Decrease in price = \(200\) The decrease in price is \(\$ 200\).

02

Calculate the decay factor

The decay factor is the ratio of the decrease to the initial price. We can compute it as follows: Decay factor = \(\frac{\text{Offer Price}}{\text{Initial Price}}\) Decay factor = \(\frac{17994}{18194}\) Decay factor ≈ 0.989 The decay factor associated with the decrease in price is approximately 0.989.

03

Calculate the percent decrease

The percent decrease is the ratio of the decrease in price to the initial price, multiplied by 100%. We can compute it as follows: Percent decrease = \(\frac{\text{Decrease in price}}{\text{Initial Price}} \times 100 \%\) Percent decrease = \(\frac{200}{18194} \times 100 \%\) Percent decrease ≈ 1.1% The percent decrease of the price is approximately 1.1%.

04

Calculate the sales tax and savings

The sales tax rate is given as 6.5%. Let's calculate the sales tax for both prices and find the savings. Initial sales tax = Initial Price × Sales tax rate Initial sales tax = \(18194 \times 0.065\) Initial sales tax ≈ \(1182.61\) Offer sales tax = Offer Price × Sales tax rate Offer sales tax = \(17994 \times 0.065\) Offer sales tax ≈ \(1169.61\) Sales tax savings = Initial sales tax - Offer sales tax Sales tax savings = \(1182.61 - 1169.61\) Sales tax savings = \(13\) You save \(\$13\) on sales tax by taking the dealer's second offer.

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