For the period 1995-2002, the total personal expenditures (in billions of dollars) in the United States for food, f(x), and for housing, h(x), can be described by the equations shown below, where x is the number of years past 1995. f(x) = 40.74x + 742.65 and h(x) = 47.93x + 725 Find the year in which these expenditures were equal and the amount spent on each. (Round the year up. Give the amount correct to the nearest whole billion.)
1998
well, set them equal:
40.74x + 742.65 = 47.93x + 725
Now just find x and add 1995.
842.6586
Oh, I see you're trying to crunch some numbers there! Let's solve this math problem with a sprinkle of humor.
To find the year when the expenditures on food, f(x), and housing, h(x), were equal, we need to set up an equation and solve for x.
So, let's set f(x) equal to h(x):
40.74x + 742.65 = 47.93x + 725
Alright, let's solve this equation and find x. But before that, let me tell you a food-related joke to lighten the mood.
Why did the tomato turn red?
Because it saw the salad dressing!
Now, let's solve the equation and find x:
40.74x - 47.93x = 725 - 742.65
-7.19x = -17.65
To make things simple, let's divide both sides by -7.19:
x = -17.65 / -7.19
x ≈ 2.45
Now, let's convert x into the nearest whole number. We'll round up because we want to find the year when the expenditures were equal.
Round up x to 3.
So, the expenditures were equal in the year 1995 + 3 = 1998.
Now, let's find out how much was spent on each:
f(3) = 40.74(3) + 742.65 ≈ 864.87 billion dollars
h(3) = 47.93(3) + 725 ≈ 869.79 billion dollars
In 1998, approximately 865 billion dollars were spent on food and 870 billion dollars were spent on housing. Oh, expenses, they really know how to add up!
Hope that helps! If you have more questions or need some more laughs, feel free to ask!
To find the year in which the expenditures were equal and the amount spent on each, we need to set the two equations equal to each other and solve for x.
Setting f(x) equal to h(x), we have:
40.74x + 742.65 = 47.93x + 725
Now, let's solve for x.
First, let's collect like terms:
40.74x - 47.93x = 725 - 742.65
Simplifying further:
-7.19x = -17.65
Now, let's divide both sides of the equation by -7.19 to isolate x:
x = -17.65 / -7.19
Calculating this, we find:
x ≈ 2.45
Since x represents the number of years past 1995, we need to round up to the nearest whole number, which gives us 3. Thus, in the year 1998 (1995 + 3), the expenditures for food and housing were equal.
To find the amount spent on each, we substitute this value of x back into either equation. Let's use f(x):
f(x) = 40.74x + 742.65
f(3) = 40.74(3) + 742.65
Calculating this, we find:
f(3) ≈ 124.92
Therefore, in the year 1998, the amount spent on food was approximately 125 billion dollars.
To find the amount spent on housing, we substitute x = 3 into the h(x) equation:
h(x) = 47.93x + 725
h(3) = 47.93(3) + 725
Calculating this, we find:
h(3) ≈ 868.79
Therefore, in the year 1998, the amount spent on housing was approximately 869 billion dollars.