Linear Applications Wednesday, November 13, 2013

Linear ApplicationsWednesday, November 13, 2013

Bathtub Problem

• You pull out the plug from the bathtub. After 40 seconds, there are 13 gallons of water left in the tub. One minute after you pull the plug, there are 10 gallons left. Assume a constant rate for the amount of gallons that drain each second.

Step 1: Define Variables

• Let Output be gallons ()• Let Input be seconds () •We need to find the slope.•We know the values for and at two

different times. •What can we do to find the slope?

Step 2: Find the Slope

•After 40 seconds there are 13 gallons left, so one point is (40, 13). •After 60 seconds, there are 10 gallons

left, so another point is (60, 10). Now we just use the slope formula. •Rate =

Step 3: Use Point-Slope to Write the Equation

• Pick a point: (40, 13) and use the slope:

How many gallons after 20 seconds? After 50 seconds?

16 gallons remain after 20 seconds.

11.5 gallons remain after 50 seconds.

At what time will there be 7 gallons left in the tub?• g = 19 .15(s) •Which variable, gallons or seconds

should we replace 7 with? • 7 = 19 .15(s) • .15(s) = 12 • s = 80 gallons

Find the y-intercept. What does it mean?• To find the y-intercept, we look for the b-

value when our equation is in slope-intercept formWhich equation do we use? Equation A: Equation B:

• y-intercept is 19•Number of gallons in tub after 0 seconds.

Find the x-intercept (time-intercept). What does it mean?

• To find the x-intercept, we set our output (gallons) equal to 0 and solve for our input.

What variable are we trying to solve for?

• After seconds, there is no water in the tub.

• You drive home from a football game. Assume that the distance varies linearly with time. Suppose you are 11km from home when you have been driving 10 minutes, and 8 km away from home when you drove for 15 minutes.

Find an equation to represent this scenario.

Define the Variables

• Let D be distance.• Let t be time in minutes. •Rate: =

Find the slope

• (10, 11) (15, 8) • •What does the slope mean in terms of

the problem?

•Write the equation expressing the number of kilometers (D) you are from home in terms of the number of minutes (t) since you left the game.

Predict your distance from home after driving 20 min, 25 min, and 30 min.

•D(20) = 5km•D(25) = 2km•D(30) = -1

When were you 7 km from home? How many minutes have you been traveling

• minutes.

Find the distance-intercept. What does this number represent in the real world?

•17km This is the number of miles you are away from home when you start your journey.

Find the time-intercept. What does this number represent in the real world?

• This is the amount of time it takes to get home.