Question

restful resort charges a $980 lodging fee plus $270 per person for a weekly meal plan.
sunset spa charges a $440 lodging fee plus $360 per person for a weekly meal plan.
which statement is true?

1. for 4 people, the restful resort is less expensive than the sunset spa.
2. for 5 people, the cost is the same at both the restful resort and sunset spa.
3. it costs $2,600 for the same number of people to stay at either the restful resort or sunset spa.
4. it costs $2,870 for 7 people to stay at either the restful resort or sunset spa.

Explanation:

Step1: Define cost functions

Let $x$ = number of people.
Restful Resort cost: $C_R = 980 + 270x$
Sunset Spa cost: $C_S = 440 + 360x$

Step2: Test Option 1 (x=4)

Calculate $C_R = 980 + 270(4) = 980 + 1080 = 2060$
Calculate $C_S = 440 + 360(4) = 440 + 1440 = 1880$
$2060 > 1880$, so Option 1 is false.

Step3: Test Option 2 (x=5)

Calculate $C_R = 980 + 270(5) = 980 + 1350 = 2330$
Calculate $C_S = 440 + 360(5) = 440 + 1800 = 2330$
$2330 = 2330$, so Option 2 is true.

Step4: Verify remaining options (optional)

For Option3: Set $980+270x=2600$ → $270x=1620$ → $x=6$; Set $440+360x=2600$ → $360x=2160$ → $x=6$. But $C_R(6)=980+1620=2600$, $C_S(6)=440+2160=2600$, but this is a single case, not the general claim. However, Option2 is already confirmed true.
For Option4: $C_R(7)=980+270(7)=980+1890=2870$; $C_S(7)=440+360(7)=440+2520=2960≠2870$, so false.

Answer:

1. For 5 people, the cost is the same at both the Restful Resort and Sunset Spa.