Question

a motorcycle can be purchased for $9100 or leased for a down payment of $500 and $260 per month. find a function that describes how the cost of the lease depends on time. assuming that the monthly payments are made, how long can the motorcycle be leased before more than the purchase price has been paid?

the function that models the situation is \\(p = 260t + 500\\), where \\(p\\) is the amount paid on the lease in dollars and \\(t\\) is the time in months.
(simplify your answer.)

the motorcycle can be leased for month(s) before more than the purchase price is paid.
(round down to the nearest integer as needed.)

Explanation:

Model the lease cost function

Using the Linear Function Modeling knowledge point

$$ p = 260t + 500 $$

Set up the inequality for lease cost vs purchase price

To find how long the motorcycle can be leased before the total lease cost exceeds the purchase price of \$9100, we set up an inequality where the lease cost is less than or equal to the purchase price:

$$ p \le 9100 $$

Substitute the function for \(p\):

$$ 260t + 500 \le 9100 $$

Solve the inequality for time t

Subtract 500 from both sides of the inequality:

$$ 260t \le 8600 $$

Divide both sides by 260:

$$ t \le \frac{8600}{260} \approx 33.0769 $$

Determine the maximum number of months

Since the question asks how long the motorcycle can be leased before more than the purchase price is paid, and specifies to round down to the nearest integer:

$$ t = 33 \text{ months} $$

At 33 months, the amount paid is:

$$ 260(33) + 500 = 8580 + 500 = 9080 \le 9100 $$

At 34 months, the amount paid is:

$$ 260(34) + 500 = 8840 + 500 = 9340 > 9100 $$

Thus, the maximum duration is 33 months.

Answer:

Question 1

The function that models the situation is \(p =\) <blank>\(260t + 500\)</blank>, where \(p\) is the amount paid on the lease in dollars and \(t\) is the time in months.

Question 2

The motorcycle can be leased for <blank>33</blank> month(s) before more than the purchase price is paid.