Question

sam and sonya are scuba diving. sam is at an elevation of -42 feet, and she is descending at a rate of 10 feet per minute. sonya is at an elevation of -12 feet and she is descending at a rate of 14 feet each minute. the variable t represents the time in minutes. after how many minutes will sam and sonya be at the same elevation? at what elevation will they be at that time? write an expression to represent sam’s elevation after t minutes, where t represents the number of minutes.

Explanation:

Step1: Write Sam's elevation expression

Sam starts at -42 feet and descends 10 ft/min.
Expression: $-42 - 10t$

Step2: Write Sonya's elevation expression

Sonya starts at -12 feet and descends 14 ft/min.
Expression: $-12 - 14t$

Step3: Set expressions equal, solve for $t$

Set elevations equal to find when they meet:
$$-42 - 10t = -12 - 14t$$
Add $14t$ to both sides:
$$-42 + 4t = -12$$
Add 42 to both sides:
$$4t = 30$$
Divide by 4:
$$t = \frac{30}{4} = 7.5$$

Step4: Find elevation at $t=7.5$

Substitute $t=7.5$ into Sam's expression:
$$-42 - 10(7.5) = -42 - 75 = -117$$

Answer:

Sam's elevation: $\boldsymbol{-42 - 10t}$
Sonya's elevation: $\boldsymbol{-12 - 14t}$
Time at same elevation: $\boldsymbol{7.5}$ minutes
Elevation at that time: $\boldsymbol{-117}$ feet