A formula in mathematics is an equation that involves more than one variable. I've written some fairly common formulas over here on the board.
A =(l)(w); that's the formula for the area of a rectangle. P = 2(l) + 2(w); that's the formula for the perimeter of a rectangle.
F = 9/5C + 32; that's the formula that gives the relationship between the two different temperature scales, Fahrenheit and Celsius.
y = mx + b; this turns out to be an important formula in mathematics;
it's the formula for the equation of a line with a slope of m and a y-intersect of b.
E = 8(l)(w)(h) is the formula for the energy in an ocean wave.
And z = x - μ, this is the Greek letter mu, over s, is the formula for z-score in statistics. There's many, many formulas in mathematics.
What links them all together is that they're equations that involve more than one variable.
Let's see some of the problems that we can work involving formulas. Suppose that G = (H)(R).
And we want to find G if H = 36 hours and R = 8 dollars per hour. So, this formula right here is the formula for gross pay,
if you work H hours per week at a rate of R dollars per hour. So, I know that H = 36; R = 8; and G = (H)(R).
So, G will be equal to H, which is 36; times R, which is 8. And that comes out to be,
let's see, if I multiply 8(6) = 48, carry the 4; 8(3) = 24 and 4 is 28: 288. So, this will be $288.
That will be the gross pay that you will make if you work 36 hours a week at a rate of 8 dollars per hour.
Now, that's something that you already know, but all we did was put it in the form of a formula.
Let's do the same problem again, but with H= 20 hours and R = $6 3/4 per hour.
So, this time, what we're going to have is that G = (H)(R). Since H = 20 and R = 6 3/4, I'll have 20(6 3/4).
20(6 3/4): 20(6) = 120. And, then 20 times 3/4;
4 divides into 20, 5 times; 5(3) = 15. So, this comes out to be 135. Now, there's other ways to do that arithmetic right there.
I just did kind of a shortcut, thinking that 6 3/4 is 6 + 3/4, and I used the distributive property. 20(6) = 120; 20(3/4) = 15.
And so, 120 + 15 = 135. So again, we simply used this formula right here, G = (H)(R), to find G when we were given H and R.
Let's look at that formula that compares the two temperatures systems.
We have C = 5/9(F - 32) a little different version of the formula
that we showed right in the beginning of this lesson. Now, we want to find C if F = 158.
So, I'm simply going to take 158, replace F with that, and then calculate what C is.
So, C = 5/9(158 - 32). So, let's see, 158 - 32, this would be 5/9 times, 2 from 8 is 6; 3 from 5 is 2; so, I have 126.
Let's see, does 9 divide in there evenly? I hope so. This is going to be 5 times, 9 divides into 12, let's see, once; and then I have 3 left over, 4
So, 5(4) = 20, carry the 2, times this; so, I end up with 70.
So, a temperature of 158 degrees Fahrenheit is equivalent to a temperature of 70 degrees Celsius.
So, I didn't put the little units on here: degrees C and degrees F. But, they're there if we need them.
So, if F is 158, I substitute it back in here, do this arithmetic, and what comes out is 70.
So, 70 is the Celsius temperature that corresponds to a Fahrenheit temperature of 158 degrees. Let's try one more.
What if F is 32? 32 degrees Fahrenheit is freezing, that's the temperature at which water freezes. Let's see what C is.
C = 5/9(F - 32). So, I take 32 and substitute it in for F.
That gives me C = 5/9(32 - 32) that will be 5/9(0), which will be 0.
So, a Fahrenheit temperature of 32 degrees corresponds to a Celsius temperature of 0 degrees. And, that's true.
Water freezes at 32 degrees Fahrenheit and 0 degrees Celsius.
Let's look at a problem that involves the area of a circle: A = (π)r². Find A if π = 22/7 and r = 7 feet.
So, A = (π)r². π = 22/7, so I'll substitute that in; r = 7; I'll substitute that in.
And I end up with A = (22/7)(7)². So, I'll have 22/7, (7)(7) because that's 7 feet squared.
Now, let's see, those 7's divide out. So, I'll multiply 22(7), right there: 7(2) = 14, carry the 1; 14 + 1 = 15; 154 square feet.
So, this gives us the area of a circle that has a radius of 7 feet, if we use this approximation for π of 22/7.
So, notice that when my units involve feet right here and then they're squared in the formula, that the units are squared, also.
So, I start with feet, and I end up with square feet. Let's try another one: r = 3/4 foot; π is equal to 22/7;
and the A = (π)r². So, I'll substitute both these values, both this value and this value, into this formula,
and I end up with A = (22/7)(3/4)². And then, I'll just put my feet squared on the outside.
So, 22/7 times, let's see 3² = 9, over 16. And what have I got here? A common factor of 2, so that's 11; and 8 divides in there.
Anything else? No. So, 11(9) = 99. And then, 8(7) = 56; so, 99/56 square feet.
And I could change that, if I wanted to, into an improper fraction also. But I think I'll just leave it right there.
So, if π = 22/7 and my radius is 3/4 of a foot, then I end up with an area of 99/56 square feet.