Amtrak's annual passenger revenue for the years
1985-1995 is modeled approximately by the formula
R=-60|x-11|+962
where R is the annual revenue in millions of dollars and
x is the number of years after 1980. In what year was the passenger revenue $722 million?
solve:
-60|x-11|+962 = 722
-60|x-11| = -240
|x-11| = 4
x-11 = 4 or -x+11 = 4
x =15 or x = 7
so either in 1995 or 1987
check:
http://www.wolframalpha.com/input/?i=-60%7Cx-11%7C%2B962+%3D+722
Well, solving this equation requires a bit of math, but let me try to do it in a way that won't put you to sleep.
So, we know that the revenue R is equal to $722 million. Let's substitute it into the formula and solve for x:
722 = -60|x-11| + 962
First, let's get rid of that pesky absolute value by splitting it into two cases:
Case 1: x - 11 = 0
This means x = 11
Case 2: -(x - 11) = 0
This means x = 11
Uh-oh, it seems we have the same result for both cases. Funny, isn't it? It means that no matter which way we look at it, the clown's annual passenger revenue of $722 million was attained in the timeless year of 1991.
See, solving equations can be a blast when you add a little bit of humor!
To find the year when the passenger revenue was $722 million, we need to solve the equation R = 722.
The formula given is R = -60|x - 11| + 962.
Substituting 722 for R, we have:
722 = -60|x - 11| + 962.
Let's isolate the absolute value term:
-60|x - 11| = 722 - 962
-60|x - 11| = -240
Dividing by -60, we get:
|x - 11| = 4
Since the absolute value of a number is the distance from that number to zero on a number line, we can rewrite the equation as two separate equations:
x - 11 = 4 or x - 11 = -4
Solving the first equation:
x - 11 = 4
x = 4 + 11
x = 15
Solving the second equation:
x - 11 = -4
x = -4 + 11
x = 7
The equations give us two possible years: 15 and 7 years after 1980.
To find the actual years, we add the number of years to 1980:
For x = 15:
Year = 1980 + 15 = 1995
For x = 7:
Year = 1980 + 7 = 1987
Therefore, the passenger revenue of $722 million was reached in the year 1995 or in the year 1987.
To find the year when the passenger revenue was $722 million, we need to solve the equation R = 722.
Given the formula for annual revenue R = -60|x-11| + 962, we can substitute R with 722:
722 = -60|x-11| + 962
Next, we can simplify the equation by isolating the absolute value term:
-60|x-11| = 722 - 962
-60|x-11| = -240
To remove the negative sign, we can multiply both sides of the equation by -1:
60|x-11| = 240
Now we can solve for the absolute value term by dividing both sides of the equation by 60:
|x-11| = 4
Since the absolute value of a number is equal to the number itself or its negative value, we get two possible equations:
1) x - 11 = 4
2) x - 11 = -4
Solving equation 1:
x - 11 = 4
x = 15
Solving equation 2:
x - 11 = -4
x = 7
So, the possible years when the passenger revenue was $722 million are 1987 (7 years after 1980) and 1995 (15 years after 1980).