Jason is a trainer for the Mamaroneck Jets football team. He keeps track of the amount of water that the players consume throughout practice. Jason observes that the amount of water consumed is linear to the temperature on each given day. Jason finds that when it is 90°F the players consume 220 gallons of water and when it is 76°F the players consume 178 gallons of water.
Predict how many gallons of water would be consumed if the temperature was 83°F. *
Predict what the temperature would be if the players consumed 150 gallons of water (to the nearest tenth).
You have two points: (90,220) and (76,178)
The slope of the line joining those two points is (220-178)/(90-76) = 3
so the equation of the line is
y-178 = 3(x-76)
So plug in x=83 to find y (the # of gallons)
plug in y=150 to find x (the temp)
To predict the amount of water consumed if the temperature is 83°F, we can use linear interpolation.
Step 1: Determine the slope of the linear relationship between temperature and water consumption.
- The change in water consumption = 220 gallons - 178 gallons = 42 gallons
- The change in temperature = 90°F - 76°F = 14°F
Slope = change in water consumption / change in temperature
= 42 gallons / 14°F
= 3 gallons/°F
Step 2: Use the slope to predict the amount of water consumed at 83°F.
- Assume that the starting point is the temperature of 76°F and the amount of water consumed is 178 gallons.
- Calculate the change in temperature from 76°F to 83°F: 83°F - 76°F = 7°F.
Predicted change in water consumption = Slope * Change in temperature
= 3 gallons/°F * 7°F
= 21 gallons
Predicted amount of water consumed at 83°F = Starting amount + Predicted change
= 178 gallons + 21 gallons
= 199 gallons
Therefore, if the temperature is 83°F, it is predicted that the players would consume approximately 199 gallons of water.
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To predict the temperature if the players consumed 150 gallons of water, we can use linear interpolation again.
Step 1: Determine the slope of the linear relationship between temperature and water consumption.
- The change in water consumption = 220 gallons - 178 gallons = 42 gallons
- The change in temperature = 90°F - 76°F = 14°F
Slope = change in water consumption / change in temperature
= 42 gallons / 14°F
= 3 gallons/°F
Step 2: Use the slope to predict the temperature corresponding to 150 gallons of water.
- Assume that the starting point is the temperature of 76°F and the amount of water consumed is 178 gallons.
- Calculate the change in water consumption from 178 gallons to 150 gallons: 178 gallons - 150 gallons = 28 gallons.
Predicted change in temperature = Change in water consumption / Slope
= 28 gallons / 3 gallons/°F
= 9.333°F
Predicted temperature = Starting temperature + Predicted change
= 76°F + 9.333°F
≈ 85.3°F (to the nearest tenth)
Therefore, if the players consumed 150 gallons of water, it is predicted that the temperature would be approximately 85.3°F.
To predict the amount of water that would be consumed if the temperature was 83°F, we can use the concept of linear regression.
Step 1: Find the equation of the linear relationship between temperature and water consumed.
We have two data points: (90, 220) and (76, 178). Let's use these points to find the equation of the line.
First, calculate the slope of the line:
slope = (change in y / change in x) = (220 - 178) / (90 - 76) = 42 / 14 = 3.
Now, we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Let's use the first data point (90, 220):
y - 220 = 3(x - 90)
Simplifying the equation, we get:
y = 3x - 270 + 220
y = 3x - 50
Step 2: Predict the amount of water consumed at 83°F.
To predict how many gallons of water would be consumed if the temperature was 83°F, substitute x = 83 into the equation and solve for y:
y = 3(83) - 50
y = 249 - 50
y = 199
Therefore, if the temperature was 83°F, the players are predicted to consume approximately 199 gallons of water.
Now let's move on to predict the temperature given the amount of water consumed.
Step 1: Rearrange the equation to solve for x (the temperature):
y = 3x - 50
3x = y + 50
x = (y + 50) / 3
Step 2: Predict the temperature if the players consumed 150 gallons of water.
Substitute y = 150 into the equation and solve for x:
x = (150 + 50) / 3
x = 200 / 3
x ≈ 66.7
Therefore, if the players consumed 150 gallons of water, the predicted temperature would be approximately 66.7°F (rounded to the nearest tenth).