notes: direct and inverse variations #2 (ppt)
TRANSCRIPT
The general equation for DIRECT VARIATION is y =kx with k≠0.
k is called the constant of variation.
We will do an example together.
If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation
(b) Find y when x=2
(a) Find the constant of variationy=kx Write the general equation
24=k⋅3 Substitute
k=8
(b) Find y when x=2
First we find the constant of variation, which was k=8
Now we substitute into y=kx.
y=kxy=8⋅2
y=16
Another method of solving direct variation problems is to use proportions.
If y1 =kx1, then k=y1x1
and
If y2 =kx2, then k=y2x2
Therefore...
y1x1
=y2x2
So lets look at a problem that can by solved by either of these two methods.
If y varies directly as x and y=6 when x=5, then find y when x=15.
Proportion Method:65 = y
15 Let x1 =5, y1 =6, x2 =15, y2 =y
5y=90y=18
Now lets solve using the equation.
y=kx6=k⋅5k=6
5
y=kx
y=65⋅15
y=18
Either method gives the correct answer, choose the easiest for you.
Now you do one on your own.
y varies directly as x, and x=8 when y=9. Find y when x=12.
Answer: 13.5
What does the graph y=kx look like?A straight line with a y-intercept of 0.
5
-5
-10 10
f x( ) = 3⋅x
Looking at the graph, what is the slope of the line?
Answer: 3
Looking at the equation, what is the constant of variation?
Answer: 3The constant of variation and the slope are the same!!!!
We will apply what we know and try this problem.
According to Hook’s Law, the force F required to stretch a spring x units beyond its natural length varies directly as x. A force of 30 pounds stretches a certain spring 5 inches. Find how far the spring is stretched by a 50 pound weight.
F1x1
=F2x2
Set up a proportion
305 =50
xSubstitute
30x=250x=81
3 inches
Now try this problem.
Use Hook’s Law to find how many pounds of force are needed to stretch a spring 15 inches if it takes 18 pounds to stretch it 13.5 inches.
Answer: 20 pounds
Inverse Variation
y varies inversely as x if k≠0such that xy=k or y=k
x
Just as with direct variation, a proportion can be set up solve problems of indirect variation.
x1y2
=x2y1
A general form of the proportion
Lets do an example that can be solved by using the equation and the proportion.
Find y when x=15, if y varies inversely as x and x=10 when y=12
Solve by equation:xy=k
10⋅12=k120=k
xy=k15⋅y=120
y=8
Solve by proportion:x1y2
=x2y1
1512=10
y
15y=120
y=8
Solve this problem using either method.
Find x when y=27, if y varies inversely as x and x=9 when y=45.
Answer: 15
Lets apply what we have learned.
The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.
Step #1: Set up a proportion.
x1y2
=x2y1
40200= x
600
200x=24000
x=120 pounds/ in2
Now try this one on your own.
A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle’s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.
Answer: 80 pounds/ in2
What does the graph of xy=k look like? Let k=5 and graph.
6
4
2
-2
-4
-6
-10 -5 5 10
f x( ) = 5
x
This is a graph of a hyperbola.
Notice: That in the graph, as the x values increase the y values decrease.
alsoAs the x values decrease the y values increase.