Tickets for a concert have been in incredibly high demand, and as the date for the concert draws closer, the price of tickets increases exponentially. The cost of a pair of concert... Tickets for a concert have been in incredibly high demand, and as the date for the concert draws closer, the price of tickets increases exponentially. The cost of a pair of concert tickets was $150 yesterday, and today it is $162. As you complete the parts below, assume that each day's percent increase from the day before is the same. a. What is the daily percent rate of increase? What is the multiplier? b. What will be the cost of a pair of concert tickets one week from now? c. What was the cost of a pair of tickets two weeks ago? d. What will be the cost of a pair of tickets two weeks from now? Does your answer make sense? Why or why not? Read more

Steps to Solve

1. Calculate the Daily Percent Increase

To find the daily percent increase, use the formula:

$$ \text{Percent Increase} = \frac{\text{New Price} - \text{Old Price}}{\text{Old Price}} \times 100 $$

Substituting the given values:

$$ \text{Percent Increase} = \frac{162 - 150}{150} \times 100 = \frac{12}{150} \times 100 = 8% $$

2. Determine the Multiplier

The multiplier can be found with:

$$ \text{Multiplier} = 1 + \frac{\text{Percent Increase}}{100} $$

So,

$$ \text{Multiplier} = 1 + \frac{8}{100} = 1.08 $$

3. Calculate Future Price (One Week from Now)

To find the price one week from now (7 days), use the formula:

$$ \text{Future Price} = \text{Current Price} \times \text{Multiplier}^{\text{Number of Days}} $$

Insert the values:

$$ \text{Future Price} = 162 \times 1.08^{7} $$

Calculating this gives:

$$ \text{Future Price} \approx 162 \times 1.7138 \approx 277.57 $$

4. Calculate Past Price (Two Weeks Ago)

To find the price two weeks ago (-14 days), modify the formula:

$$ \text{Past Price} = \frac{\text{Current Price}}{\text{Multiplier}^{\text{Number of Days}}} $$

Insert the values:

$$ \text{Past Price} = \frac{162}{1.08^{-14}} $$

Calculating this gives:

$$ \text{Past Price} \approx \frac{162}{0.3935} \approx 412.23 $$

5. Calculate Future Price (Two Weeks from Now)

Using the same future price formula for two weeks from now (14 days):

$$ \text{Future Price} = 162 \times 1.08^{14} $$

Calculating this gives:

$$ \text{Future Price} \approx 162 \times 2.8396 \approx 459.86 $$

6. Evaluate the Reasonableness of the Prices

Check if the calculated price for two weeks from now makes sense considering the growth trend.