Gabrielle and Shauna are scuba diving. Gabrielle is at an elevation of -40 feet, and she is descending at a rate of 12 feet per minute. Shauna is at an elevation of -14 feet and she is descending at a rate of 16 feet each minute. The variable t represents the time in minutes. After how many minutes will Gabrielle and Shauna be at the same elevation

1. The problem is a system of equations. To do this we first need to make the formulas of each person's descent. We plug the information into the linear equation, [tex]y=mt+b[/tex] , where y represents the elevation and t representing time for Gabrielle's and Shauna's respective information. While m represents the slope or speed of descent, and b representing the starting elevation.

Gabrielle: Starting at -40 ft elevation, the is descending at 12 ft per minute.

[tex]y=-12t-40[/tex]

Shauna: Starting at -14 ft elevation, the is descending at 16 ft per minute.

[tex]y=-16t-14[/tex]

Do note, the slopes are negative because they are descending.

2. Now that we have the equations for Gabrielle and Shauna, we can set up a system of equations. In this case we can use either substitution or elimination. For this instance we will use elimination.

[tex]y=-12t-40\\ y=-16t-14[/tex] Change the signs of equation 2 to cancel out the y's

[tex]y=-12t-40\\ -y=16t+14[/tex] Combine them to form one equation

[tex]0 = 4t-26[/tex] Add 26 to both sides to move the 26

[tex]26=4t[/tex] Divide by 4 to isolate t

[tex]6.5 = t[/tex]

3. We found out that t = 6.5, which means that it takes 6.5 minutes for Gabrielle and Shauna to be at the same elevation.

semsee45 semsee45 · Answer 2 · 2024-01-26T13:51:42+01:00

Answer:

6.5 minutes

Step-by-step explanation:

Given that Gabrielle is at an initial elevation of -40 feet and she is descending at a rate of 12 feet per minute, the function representing her elevation at time t is:

[tex]G(t) = -40 - 12t[/tex]

Given that Shauna is at an initial elevation of -14 feet and she is descending at a rate of 16 feet each minute, the function representing her elevation at time t is:

[tex]S(t) = -14 - 16t[/tex]

To find out when Gabrielle and Shauna will be at the same elevation, we need to set their elevation functions equal to each other and solve for the time t:

[tex]\begin{aligned}G(t)&=S(t)\\\\-40 - 12t &= -14 - 16t\\\\-40 - 12t + 16t &= -14 - 16t + 16t\\\\-40 + 4t &= -14\\\\-40 + 4t + 40 &= -14 + 40\\\\4t &= 26\\\\t &= 6.5\; \sf minutes \end{aligned}[/tex]

Therefore, Gabrielle and Shauna will be at the same elevation after 6.5 minutes.