Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 18 gallons of fuel, the airplane weighs 2199 pounds. When carrying 44 gallons of fuel, it weighs 2342 pounds. How much does the airplane weigh if it is carrying 50 gallons of fuel?
To find the weight of the airplane when carrying 50 gallons of fuel, we need to determine the equation that represents the relationship between the weight and the amount of fuel. Since the weight is given as a linear function of the fuel, we can use the slope-intercept form of a linear equation, which is y = mx + b.
First, let's determine the slope (m) of the equation. The slope represents the rate of change, or how much the weight increases for each additional gallon of fuel. We can calculate the slope using the formula:
m = (change in y) / (change in x)
In this case, the change in weight is (2342 - 2199) = 143 pounds, and the change in fuel is (44 - 18) = 26 gallons.
m = (143 pounds) / (26 gallons) = 5.5 pounds/gallon
Now that we have the slope, we can determine the y-intercept (b) of the equation. The y-intercept represents the weight when the fuel is zero, which in this case is not mentioned. However, we can still solve for b by substituting the weight and fuel values from one of the given points.
Let's use the point (18 gallons, 2199 pounds):
y = mx + b
2199 = (5.5 pounds/gallon) * 18 gallons + b
2199 = 99 pounds + b
To isolate b, we subtract 99 pounds from both sides:
b = 2199 - 99 = 2100 pounds
Now we have the equation for the weight (y) in terms of the fuel (x):
y = 5.5x + 2100
To find the weight when carrying 50 gallons of fuel, we substitute x = 50 into the equation:
y = 5.5 * 50 + 2100
y = 275 + 2100
y = 2375 pounds
Therefore, the airplane weighs 2375 pounds when carrying 50 gallons of fuel.
w = mg+b
so,
18m+b = 2199
44m+b = 2342
Solve for m and b, and figure w(50)