Direct Inverse and VARIATION
k is called the constant of proportionality. The general equation for DIRECT VARIATION is k is called the constant of proportionality. We will do an example together.
(a) Find the constant of variation If y varies directly as x, and y=24 and x=3 find: (a) the constant of proportionality (b) Find y when x=2 (a) Find the constant of variation Write the general equation Substitute
(b) Find y when x=2 First we find the constant of proportionality, which was k=8 Now we substitute into y=kx.
Another method of solving direct variation problems is to use proportions. Therefore...
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:
Now lets solve using the equation. 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.
Looking at the graph, what is the slope of the line? Answer: 3 Looking at the equation, what is the constant of proportionality? Answer: 3 The constant of proportionality (k) 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.
Set up a proportion Substitute
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 such that xy=k or Just as with direct variation, a proportion can be set up solve problems of indirect variation.
Alternate method for solving: x1 y1 = x2 y2 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:
Solve by the alternate method x1 y1 = x2 y2 15 y = 10 12
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. x1 y1 = x2 y2 x 200 = 40 600
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.
What does the graph of xy=k look like? Let k=5 and graph.
This is a graph of a hyperbola. Notice: That in the graph, as the x values increase the y values decrease. also As the x values decrease the y values increase.
When graphed, it must pass through the origin (0,0) Direct Variation Inverse Variation x1 y1 = x2 y2 When graphed, it must pass through the origin (0,0) Graph is a hyperbola Doesn’t match either one? 1) Solve for y 2) Check for a match 3) Still no match – neither direct nor inverse