Inverse Variation If a quantity y varies inversely

Inverse Variation If a quantity, y "varies inversely as" x, then as x increases, y decreases and as x decreases, y increases. The phrase, "y varies inversely as x", has equation form: Here, k is a positive constant called the constant of proportionality. The phrase, "y varies inversely as x", is commonly written as "y is directly proportional to x". The phrase, "y varies inversely as the square of x", has equation form: Table of Contents

Inverse Variation The phrase, "y varies inversely as the cube of x", has equation form: The phrase, "y varies inversely as square root of x", has equation form: Example 1: The diameter of a pulley varies inversely as the rate at which it spins (see figure). If the larger pulley has a diameter of 2. 5 inches and spins at a rate of 20 revolutions per minute, (rpm), find the rate at which the smaller pulley is spinning if it has a diameter of 1. 5 inches. Table of Contents Slide 2

Inverse Variation Let, y represent the rate at which a pulley spins and x represent its diameter. (It does not matter which is x and which is y. ) Then write the algebraic equation for this inverse relationship. Next, find k by substituting the information given about the larger pulley. 50 = k Next, rewrite the inverse relationship equation with this value of k. Next, use this equation to answer the question by substituting the 33. 3 rpm diameter of the smaller pulley. The smaller pulley revolves at a rate of 33. 3 rpm. Table of Contents Slide 3

Inverse Variation Example 2: Determine if the data in the table can be modeled by an equation of the form, natural number. where n is a x y One way to approach this is to by trying values for n. For example, let n = 1, so 2 400 4 50 Then find the value of k by 8 6. 75 10 3. 2 substituting any pair of x and y-values from the table: k = 200, so Table of Contents Slide 4

Inverse Variation However, the equation, will not model all the other x, y-pairs from the table. For example, x = 10 and y = 3. 2, will not fit the model. x y 2 4 8 10 400 50 6. 75 3. 2 The next try would be to let n = 2. If this does not work then, successive trials would be n = 3, n = 4, etc. Try this approach and see which, if any, work. The answer will be revealed on the next left mouse click. The data can be modeled with n = 3. NOTE: This can be solved later in a direct way using exponential functions. Table of Contents Slide 5

Inverse Variation Table of Contents